ELSEVIER Physica B 230-232 (1997) 959-961

Phonon-induced anomalous Raman spectra in undoped high- Tc cuprates

J.D. Lee, B.I. Min*

Department of Physics, Pohang University of Science and Technology, Pohang 790-784, South Korea

Abstract

In order to describe a shoulder peak structure near 4J in the magnon Raman spectra of undoped high-T~ cuprates, we have explored the phonon contribution to the Raman spectra. Incorporating the magnon-phonon Hamiltonian in the spin-wave theory, we have evaluated the two-magnon Raman spectral function originating from the lowest-order magnon-phonon-magnon scattering. It is found that phonons induce a shoulder peak near 4J besides the dominant two-magnon peak near 3J, in agreement with experiments.

Keywords: Cuprates; Magnon-phonon interaction

Magnon Raman spectra of undoped La2CuO4 and YBa2Cu306 exhibit anomalous features, such as a very broad line width and a long tail far beyond the maximum energy of the magnon pair [1-3]. A shoulder peak structure exists near 4J, besides the dominant two-magnon peaks near 3J arising from a spin-pair coupling accompanied by electric-dipole transitions in antiferromagnetic sys- tems [4]. There have been several attempts to understand these phenomena theoretically [5 9]. Among those, Knoll et al. [3] found that the addi- tional spin-lattice interaction leads to a reduction of the magnon lifetime and a broadening of the two-magnon spectra with Big symmetry. More re- cently, Nori et al. [10] reported that the two-mag- non Raman peak is strongly modified indeed due to coupling to phonons.

In this paper, we have explored phonon effects on the line shapes of Raman spectra, taking into

* Corresponding author.

account the magnon-phonon interaction explicitly in the spin-wave theory for the 2D Heisenberg Hamiltonian. We have found that the magnon- phonon-magnon scattering yields a phonon-in- duced magnon peak near 4J, which is small com- pared to the two-magnon main peak, but has a nontrivial peak structure.

We consider the Heisenberg Hamiltonian to describe magnetic interactions on a 2D square lattice,

H = 2 J(ri - rJ) Si "Sj, (1) <ij>

where the summation is over nearest-neighbor Cu pairs. In a conventional antiferromagnet with spin- excitation energies less than typical phonon ener- gies, the momentum and energy conservation pre- dicts a negligible phonon contribution to the spin- excitation process. However high T¢ materials are known to have large exchange energies, and so the spin-excitation involving many-phonon processes will be allowed.

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960 J.D. Lee, B.L Min / Physica B 230-232 (1997) 959 961

The exchange interaction J is modulated by lat- tice vibrations varying ionic distances E11]. For the super-exchange interaction between the Cu-O-Cu, the variation of Cu-Cu distance due to the Cu-O stretching mode is expected to induce the first- order spin-phonon interaction [3]. Then, the ex- change energy integral can be expanded as

J(ri -- rj) ~ Jo + (ul - u j ) 'VJ(r i - rj)lo, (2)

where u~ is the ionic displacement. Phonon oper- ators are introduced via displacements u~, and thus the spin-phonon interaction arises from Eqs. (1) and (2).

We have used the Dyson-Maleev transforma- tion and the Bogoliubov transformation to express the spin-phonon Hamiltonian Hs_ph in terms of magnon boson operators [7]. In this expression, the magnon-phonon coupling constant 9(q, 2) is

given by V J / x / 2 N M f 2 ~ , where M is the reduced ionic mass of Cu-O, and ~2 ~ is the frequency of a phonon with a wave vector q, and a branch 2 [12]. We have considered the longitudinal phonon modes (Cu-O stretching modes).

Effective Loudon-Fleury Hamiltonian A, which describes B~g spin-pair mode, is given by

A = B Y [1Ei° o.eso - (3"Ei°o)(3"Eso)]Si'Si+., (3) i,6

where E~,~¢ and Esc are the polarization vectors of the incoming and outgoing photons, respectively, and B is an undetermined coupling constant [4]. Expressing the Raman operator A in terms of mag- non boson operators [7], the scattering intensity I(co) = - 2 Im GA(Og) can be obtained from the zero-temperature Green's function, i G A ( t - t ' ) = ( f ro lJA( t )A( t ' ) l f ro ) . The most important contri- butions to GA(Og) come from ladder-type two-mag- non diagrams with a repeated magnon-magnon scattering. As for the finite lifetime effects, it is known that the magnon damping due to the mag- non-magnon scattering vanishes [13]. However, magnon-phonon interaction leads to a finite mag- non damping, which will be considered phenom- enologically in this study.

In the case of ladder diagram including the repeated magnon-magnon scattering, all the dia- grams are of the same order, which makes a per-

turbation expansion not possible. On the contrary, the contributions from higher-order magnon- phonon scattering diagrams are shown to be rap- idly decreasing [12]. Thus, it is sufficient to consider only the second-order magnon-phonon- magnon scattering for the phonon part of the Green's function GPh(og) in Fig. 1. We have as- sumed that the zone boundary magnons are most relevant in the two-magnon process. Dominant zone boundary magnons also impose a restriction on the available momenta of phonons. Therefore, we can take f2~ h -~ ~ h ~ Oph, i.e. the frequency near the zone boundary, for acoustic or optical modes.

In the practical calculation, we consider the damping parameter of magnon Green's functions Ak as being constant, and carry out the numerical integration of G Ph(~o) over the 2D antiferromag- netic Brillouin zone. The most subtle parameter we should estimate is the coupling constant 9, which determines the strength of the magnon-phonon- magnon scattering contribution. We have evaluated g using the estimated values of(V J) ~ 1 eV x and ~ p h ~ 0 .5 ~r2max(~ma x ~ 2J ~ 2000 cm- 1: max- imum frequency of the spin-wave dispersion). We adopt A ,-~ 0.1Qma x as estimated in Ref. [3].

Numerical results for the Raman spectral func- tions including phonon scattering effects are pro- vided in Fig. 2. Comparison with the experiment is also given in the inset. As seen in the figure,

%1 ~,l t,p t

Fig. 1. Two-magnon propagator with the lowest order of the magnon-phonon-magnon scattering. The wiggled line denotes the phonon propagator.

ZD. Lee, B.I. Min / Physica B 230-232 (1997) 959 961 961

i 4~

~3

0 1 2 3 4

Fig. 2. The total Raman spectral functions taking into account the phonon contribution with various magnon phonon coup- ling parameter 0 = ~(VJ)2/(2M(2Ph). A and ~ph denote the magnon damping parameter and the phonon frequency, respec- tively. Note the subpeak structures near 2(2max(= 4J) increasing with 9. In the inset, we provide the comparison of experimental result (YBa2Cu306.2 from Ref. [5]) and theoretical Raman spectra, where 0 = 0.25, A = 0.115, and ~ph = 0.5 are assumed. All the values are in unit of ~2m,x(~ 2J).

We have found that the contribution from higher- order magnon-phonon vertex correction is negli- gible, as compared to that from the magnon-mag- non vertex, and thus the leading-order diagram with a single-phonon line is sufficient to take into account the phonon contribution. In this way, we have obtained the overall line shape in good agree- ment with the experimental line shape. In particu- lar, it is found that the phonon-induced two-mag- non spectral function well describes the shoulder peak structure in B~g Raman spectra near 4J.

Acknowledgements

This work was supported by the POSTECH- BSRI program of the Korean Ministry of Educa- tion, and also in part by the Korea Science Engin- eering Foundation through the SRC program of SNU-CTP. We are very grateful to J.H. Kim for helpful discussions.

although the intensity of the phonon-induced mag- non peak is less than 10% ot' the two-magnon intensity, the total Raman scattering intensity clearly exhibits a "shoulder-like" structure near 4J. Most notable is the fact that the phonon-induced peak near 4J is very robust, regardless of the size of phonon frequency [12], which reflects that the phonon-induced peak is an intrinsic nature of un- doped high-To cuprates. It should also be noted that the high-energy tail of the spectra is not yet fully understood. To appreciate this anomalous tail behavior, more extended theoretical works incor- porating an explicit single-magnon self-energy and multiple-magnon contributions should be done.

To summarize, we have investigated the line shapes of Raman spectra in 2D S = ½ HAFM, tak- ing into account the phonon contribution in the framework of the conventional spin-wave theory.

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