Raman scattering in YBa 2 Cu 3 O 7 : A comprehensive theoretical study in comparison with experiments C. Ambrosch-Draxl, H. Auer, R. Kouba, and E. Ya. Sherman Institute of Theoretical Physics, Karl-Franzens-University of Graz, A-8010 Graz, Austria P. Knoll and M. Mayer Institute for Experimental Physics, Karl-Franzens-University of Graz, A-8010 Graz, Austria Received 10 July 2001; published 3 January 2002 All Raman-active phonon modes of YBa 2 Cu 3 O 7 are investigated by first-principles linearized augmented plane-wave calculations based on density-functional theory for a fully optimized crystal structure. The calcu- lated frequencies as well as the Raman scattering intensities are in excellent agreement with measured Raman spectra. The effect of site-selective isotope substitution on the Raman spectra is investigated. The substitution not only shifts the phonon frequencies, but also leads to dramatic changes in the scattering intensities. DOI: 10.1103/PhysRevB.65.064501 PACS numbers: 74.25.Kc, 74.25.Jb, 78.30.-j, 71.15.Mb I. INTRODUCTION The electronic properties of high-T c superconducting cu- prates are strongly governed by their crystal structure. Dis- placements of ions from their equilibrium positions as caused by vibrational modes lead to changes in the electronic structure, due to strong electron-phonon coupling in these compounds. Electron-phonon coupling and the role of the lattice were subject of investigation since the discovery of high-T c superconductivity. Results of these investigations in- dicate the importance of the coupling of electrons to the lat- tice in the normal state as well as a significant relation be- tween phonons and the superconducting transition. When ions are displaced from their equilibrium positions, the increase of the total energy leads to lattice forces which push them back and thereby result in lattice vibrations. At the same time the crystal’s electronic structure and the related properties are changed. When an electromagnetic wave propagates through a crystal, its frequency is changed due to the modulation of the dielectric tensor caused by the time- dependent ionic displacements. This light scattering by lat- tice vibrations results in the Raman process. Since Raman scattering is determined by the influence of the ionic motions on the electronic states in a wide energy range, Raman spectroscopy provides valuable information on both, the lattice properties as well as the electronic sub- system. The total energy and the dielectric tensor as a func- tion of the atomic coordinates required for the theoretical description of the Raman spectra can be obtained by first- principles band-structure methods within the frozen-phonon approach. The total energy and the atomic forces acting on the displaced ions yield the force constants necessary to de- termine phonon frequencies and eigenvectors, while also the dependence of the dielectric tensor is used to calculate the Raman intensities. First-principles calculations appeared to be very useful for describing the properties of high-T c cuprates close to optimal doping, where they demonstrate good metallic behavior. While the Raman spectra of YBa 2 Cu 3 O 7 -x have been mea- sured on some underdoped samples, 1,2 the corresponding the- oretical investigations are still lacking. This is due to the fact that first-principles calculations require an ordered structure which would lead to huge supercells for arbitrary doping levels. In this paper we focus on YBa 2 Cu 3 O 7 which is best studied, experimentally as well as theoretically. It turned out that best agreement with experimental data is achieved when the crystal structure is optimized theoreti- cally, i.e., when unit-cell volume, axis ratios, and atomic po- sitions are obtained by searching for the lowest total energy. 3 In order to study vibrational properties and phonon Raman scattering we have performed frozen-phonon calculations within this optimized crystal structure. We focus our atten- tion on the Raman-active modes zero-momentum transfer which are divided into three symmetry classes: A 1 g , B 2 g , and B 3 g . Each class contains five eigenmodes which are coupled vibrations of Ba, the plane copper atom Cu2, the plane oxygens O2 and O3, and the apical oxygen O4. The A 1 g modes represent c-axis vibrations of these five atoms, while within the B 2 g , and B 3 g modes these atoms move in the a and b direction, respectively. The spectral densities of inelastic light scattering are quantitatively compared to mea- sured spectra. Our approach also gives us the possibility to consider the site-selective isotope substitution and its influ- ence on the phonon frequencies and scattering intensities. II. METHOD A. Raman scattering in the frozen-phonon approximation In Raman spectroscopy a light beam with frequency  rmI irradiates a crystal, where the spectrum of scattered light is investigated. The spectrum of scattered light contains peaks around the frequencies  S = I - ph Stokes process and  S = I + ph anti-Stokes process, where  ph is a phonon frequency. To calculate the scattering probability, first the frequencies and eigenvectors of the zero-momentum phonons are determined by solving the eigenvalue equation D ˆ Q  =  2 Q  . 2.1 PHYSICAL REVIEW B, VOLUME 65, 064501 0163-1829/2002/656/0645019/$20.00 ©2002 The American Physical Society 65 064501-1