Lattice vibrations in high-Tc superconductors: Optical spectroscopy and lattice dynamics

Physica C 159 ( 1989) 1-32 North-Holland, Amsterdam

REVIEW LATTICE VIBRATIONS IN HIGH-To SUPERCONDUCTORS: OPTICAL SPECTROSCOPY AND LATI'ICE DYNAMICS ¢r Rudolf FEILE Institut ~ r Physik, Johannes Gutenberg Universitiit, D-6500Mainz, Fed. Rep. Germany

Received 4 April 1989

Raman and IR-measurementson the high-Tomaterial series La2CuO4and YBa2Cu307 are reviewed. This summaryfocuseson lattice vibrations in these superconductingsystems, on impurity phases, the influence of oxygenstoichiometryand element substitution. The determination of the superconductinggap, the phonon renormalization in the superconductingstate, and the isotope effect with its implications on the theory of superconductivityin these materials are discussed. In addition, the measured lattice vibrations are compared with the results of lattice dynamicscalculations.

1. Introduction The discovery of superconductivity above 30 K in CuO-perovskites by Bednorz and Miiller [ 1 ] initiated tremendous efforts in solid state physics and material sciences with the aim to isolate the phases which are responsible for the superconductivity and to search for other substances beside the La2CuO4 family which exhibit this phenomenon. These activities succeeded in the discovery of superconductivity in YBa2CuaO7 at 92 K by Wu et al. [ 2 ] and the observation of a transition temperature above 100 K by Sheng et al. [3]. Parallel to the preparation of new materials a huge number of investigations have been performed in order to illuminate the nature of the superconductivity. One direction of experiments has the intention to find out what is the contribution of lattice vibrations to the superconductivity. Neutron scattering, as the method to determine the vibrational spectrum throughout the whole Brillouin zone, could at the beginning of these activities only measure the phonon density of states, as shown by Ramirez et al. [4], Renker et al. [5,6], Briiesch et al. [7], Biirer et al. [8 ], and Belushkin et al. [9 ], because large single crystals were not available. Raman and infrared (IR)-spectroscopy, however, can yield at least some ~r This work is dedicated to Doris, Nina and Jan-Erik.

of the phonon frequencies, namely the long wavelength optical phonons at the center of the Brillouin zone, from polycrystalline material. Even tiny single crystals embedded in polycrystalline samples can be investigated in Raman experiments with the use of a microscope. The published results of experimental work on Raman and IR-spectroscopy have been published in many papers. They cover detailed information about the lattice vibrations of the superconducting materials and their dependence on oxygen contents, element substitution and impurity phases, which appeared in various samples and partly yielded controverse results. In addition, the temperature variation of the optical phonon spectrum and the superconducting gap have been investigated. It is the intention of this review to summarize the literature about the Raman and IR-experiments. The author hopes to have considered every article in these fields which have appeared in the first two years of investigations on the La2CuO4 and YBa2CuaO7 series. Regarding the enormous number of papers, however, one or the other may have been overlooked, and the author apologizes to all colleagues whose works have been omitted here unintentionally. The article is organized as follows: First, after the discussion of the structure and the symmetry of the vibrational modes in the La2CuO4 compounds the Raman and IR-results are shortly summarized. Sec-

0921-4534/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )

2

R. Feile / Lattice vibrations in high-T,, superconductors

ondly, the articles on the YBa2Cu307 family are reviewed starting again with structural and symmetry considerations. The two sections, Raman and IRspectroscopy on these compounds, have both been subdivided in sections dealing with the impurity phases, the influence of oxygen stoichiometry, substitution of elements, single crystals and films, the temperature dependence of the vibrations and the superconducting gap, and, shortly, with the results from magnetic excitations. The following third section summarizes the lattice dynamics calculations performed for both compounds and compares the results with the experimental findings. Finally the isotope effect is discussed from which information about the relevance of the lattice vibrations for the superconductivity had been expected.

La2CuO4 and the related Sr and Ba-doped compounds have soon after their discovery been subject of spectroscopic investigations [ 10-63 ]. Yet the number of published articles on these materials is rather small compared to the investigations reported for YBa2Cu307.This may have been due to the strong attraction which this new 92 K material has gained and/or experimental difficulties with the La-compound as mentioned below.

is to be determined via standard group theoretical techniques (see, e.g. ref. [ 66] ). Table I gives the resuits of an analysis of the phonon symmetries at the F-point of the reciprocal lattice. In the high temperature phase four Raman-active and seven infraredactive modes should be observable. Below the structural phase transition the number of Raman- and IRmodes increases because the number of atoms in the unit cell is doubled: 18 Raman- and 21 IR-active modes are expected. Some of them can be deduced from the zone centre modes of the tetragonal phase, the others from zone boundary modes now becoming zone centre modes because of the cell doubling. Doping La2CuO4 with Sr (or Ba) shifts the tetragonal-orthorhombic transition to lower temperatures and yields the superconducting material (Lal_xSrx)2CuO4 with the highest transition temperature in this series of 40 K for x ~ 0.075. Recent investigations by Jorgensen et al. [67] revealed that the superconductivity sometimes observed in the undoped compound is due to a new oxygen rich phase of orthorhombic (D~3-Fmmm) or (Dr-F222) structure. Table I gives also the symmetry of the possible lattice vibrations for both structures. All the optical modes should be Ramanactive for the D 7 structure, whereas the number of Raman-active modes is reduced in D 23 because the inversion symmetry of this structure distinguishes between gerade (Raman-active) and ungerade (IRactive) modes.

2.1. Structure and s y m m e t r y

2.2. R a m a n spectroscopy

2. Lattice vibrations in La2Cu04 and related compounds

The stoichiometric compound LaaCuO4 crystallizes at high temperatures in the KzNiF4 structure (D4~-I4/mmm) with one chemical formula unit per unit cell (Jorgensenet al. [64]). At lower temperatures the crystals undergo a structural phase transition into an orthorhombic phase (D~8-Cmca) by doubling the unit cell in a v / 2 a × x/-2a fashion. The exact transition temperature depends on the oxygen stoichiometry of the sample and occurs between 430 K and 530 K (Johnston et al. [65 ] ). Sometimes the structure is also denoted as (DlSh-Bmab). Both assignments may be transferred into each other changing the lables from the orthorhombic axes from abe to a~b, respectively. This has to be taken into account if the symmetry of possible lattice vibrations

First results on the vibrational spectrum of (La,

Sr)2CuO4 have been obtained by Brunet al. [ 12] by means of Raman spectroscopy on polycrystalline sintered samples. The spectrum given there consists of four peaks as predicted by the symmetry considerations. The vibrations were assigned by a comparison with the sequence of Eg, Alg, E,, and AIg modes obtained by lattice dynamics calculations (see also section 4). These and other published results by Saito et al. [ 10 ], Blumenr~der et al. [ 14 ], Copic et al. [ 15 ], Batlogg et al. [ 16 ], Kourouklis et al. [ 17 ], Sugai et al. [ 13 ], McGreevy et al. [ 24 ], Weber et al. [27,68], and Burns et al. [56,28] on the Raman spectra of La2CuO4 and (La, Sr)2CUO4 are collected in fig. 1 in chronological order of their publication.

R. Feile / Lattice vibrations in high- T~ superconductors

3

Table I Symmetry of normal modes of (La, Sr)2CuO4 in four structures: D~7 and D~h 8 the structure of the tetragonal high temperature phase and orthorhombic low temperature phase, respectively, of (La, Sr)2CuO4 and La2CuO4. DId and D7 are possible structures of oxygen enriched La2CuO4÷~. D,~7

D~h8

atom

site

sitesymmetry

normal modes

site

sitesymmetry

normal modes

La Cu O,-, O v O-

e a

C4~ D4.

C

D2h

e

C4~

AI,+A2~+Es+E~ A2~+ Eu Aeu+ B2u+ 2Eu Aig+A2u+Eg+Eo

f a e f

C~': C~h C-~ Cw

2Ag+Ao+ B~,+2BI~+ B2g+2B2o+2B3g+B3o Au+2Blo+2B2u+Bsu 2Ag+ A~+ B1,+ 2Bu+ 2B2g+ 2B2.+ 2Bsg+ 2B3u 2Ag+Au+ Big+ 2BI~+ B2g+2B2o+ 2B3g+B3~

normal modes grouped according to their symmetry B2y A2u+ Eu 3A2u+ 4Eu 2Atg+ 2Eg

silent acoustic IR Raman

4Au Bju+ B2~+ B3u 6Blu+ 7B2u+ 4B3u 5A~+ 3B~g+4B2g+ 6B3g

D~

silent acoustic IR Raman D~

atom

site

sitesymmetry

normal modes

La Cu O,.,Ov O-

g a h g

C2 D2 C2 C~

AIg+A+BI +2B2+2Bs BI+B2+B3 A+BI+B2+B3 A+B~ +2B2+2B3

site

sitesymmetry

normal modes Ag+Bl~+B2g+B2o+ 2B3g+B3~ Biu+B2u+B3u Ag+ Bi. + B2a+B2o+ 2B3g+B3u Ag+ Bio+B2g+ B2u+2B3g+Bsu

i

C2~

a

D2h

e i

C~;~ C~v

normal modes grouped according to their symmetry

B~+ B2+ Bs 3A+3B~ +6B2+6B3

acoustic IR+ Raman

Au Blu+ B2u+ B3o 3B~+4B2u+4B3u 2A~+2B2g+ 2B3g

It can be seen that the conformity of the different results is not very good. Three groups may be identified: ~ 4 2 0 cm -~, ~ 2 2 0 cm -~, a n d ~ 120 cm - l . F r o m the observed modes above 400 c m - l the one around 500 c m - 1 is the most well defined. These four lines together may represent the R a m a n active modes predicted for La2CuO4. The two groups at 220 c m and 420 c m - ~can be identified as the two Alg modes (Ag in the o r t h o r h o m b i c c o m p o u n d s ) from single crystal m e a s u r e m e n t s ( M a k s i m o v et al. [20], Kourouklis et al. [ 17], Burns et al. [28] a n d Weber et al. [27,68 ] ). They reflect the symmetric in-phase a n d out-of-phase m o t i o n s of the La a n d O-atoms along the crystalline c-axis. The two others could exhibit the Eg modes. The group a r o u n d 100 c m - l can be

silent acoustic IR Raman

identified as the soft mode of the orthorhombic phase which becomes an X-point p h o n o n of the tetragonal phase as found by Birgeneau et al. [69] and B6ni et al. [ 70 ] in inelastic n e u t r o n scattering experiments. Burns et al. [28] observed this p h o n o n in single crystals of the orthorhombic phase. Weber et al. [27] could follow this p h o n o n with increasing temperature, a n d found it to soften towards the phase transition. The different assignment given by Copic et al. [ 15 ] is based on results from polycrystalline material and may therefore be incorrect. There are two possible reasons why other authors reported R a m a n spectra different from single crystal spectra. Pure La2CuO4 is k n o w n to be quite sensitive to the oxygen stoichiometry (Jorgensen et al. [67] )

4

R. Feile /Lattice vibrations in high-To superconductors Ramonactive phononsin

(La,Sr) 2 CuO 4

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ol

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Kouroukhs et Ot (S}

i !'

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Moksimov etal (s) Su£oi et el

I

Mc Greevy ~t ol

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t? I,

J

li

i, 200

i,i ~00

I I

Burns et el (s)

I ~)

200

~*00

, 600

~(?0 0

phonon frequency (cm -I)

Fig. 1. Results of Raman spectroscopy on lattice vibrations in (La, Sr)2CuO4. The short bars indicate the frequencies obtained by different authors. Some of the frequencies are labeled with the symmetry of the phonons as given by the authors. The thin vertical lines give an estimate of the most probable frequencies in the pure and doped compound.

and exhibits phase separation if the oxygen contents differs from the ideal value. This might also be the case for the doped material. Also impurities as Cu:O and La203 may contribute to the spectra. Weber et al. [68 ] discussed that this could be the reason why the spectra of some authors [ 14,13 ] deviate strongly from others (see also section 3.4). Because of the difficulties to obtain and interpret Roman spectra of the compounds La2CuO4 and (La, Sr):CuO4 Burns et al. [56,71] performed experiments on Sr2TiO4 which is isostructural to La:CuO4. It is an insulator and exhibits well defined Raman and IR-spectra. The interpretation given above [ 28 ] is consistent with these results.

[42], Sugai et al. [43,50], Herr et al. [45 ], Degiorgi et al. [49], Noh et al. [52], and Tanaka et al. [31 ], and from LaaBaCu50]3 and La2SrCu206 by Kaplan et al. [62]. From the calculated real and imaginary parts of the dielectric function the excitation spectra are deduced. In addition to lattice vibrations (see fig. 2) a peak in the conductivity around 0.5 eV was obtained which was attributed to a high-energy electronic excitation. The correlation between oscillator strength and the transition temperature has been interpreted as an indication for an excitonic pairing interaction in these superconductors. Orenstein et al. [44] modeled the anisotropic dielectric function of these materials with factorized dielectric functions representing the vibrational contributions and a Drude-ansatz for the electronic part from the CuO2 planes. The dielectric function was then averaged over the randomly oriented crystallites and gave reflectivity spectra which fitted the experimental results. This second way to analyse the data was followed also by other authors: Thomas et al. [46], Schlesinger et al. [47,60], Sulewski et al. [48], Nob et al. [52], Doll et al. [54,63], Sherwin et al. [55], Suzuki et al. [57], Geserich et al. [58], IR-active phononsin (La,Sr)2CuO4

I

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2.3. IR-spectroscopy

Schicsmget ct at

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The infrared spectra of (La, S r ) 2 C u O 4 reflect two contributions: one from the electronic response of the charge carriers and the other from the lattice vibrations. Both interfere strongly making the interpretation of the spectra more difficult than in the case of Raman spectra. One way to analyse the spectra is to perform a Kramers-Kronig transformation of the reflectivity data as done for La:CuO4 and (La,Sr) 2CuO4 by Saito et al. [29], Schlesinger et al. [32;41 ], Stavola et al. [35], Ogita et al. [37], Ohbayasi et al. [38,11], Sawada et al. [39], Nagasaka et al. [40], Bonn et al.

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i

Bums et al

I1'

i ~ !

Mazm at at

Ii i I

lii !I [I

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,i

200

~,[ii,, 400

600

,

O00

,! !,

200

~,00

m,

600

8~0

phonon frequency (cm i)

Fig. 2. Results of IR-spectroscopy on lattice vibrations in (La, Sr)2CuO4. The short bars indicate the frequencies obtained by different authors. Some of the frequencies are labeled with the symmetry of the phonons as given by the authors. The thin vertical lines give an estimate of the most probable frequencies in the pure and doped compound.

R. Feile /Lattice vibrations in high-T~superconductors

Mazin et al. [59], and Tajima et al. [72]. Their results for the phonon frequencies are included in fig. 2. Two different averaging procedures were used in these calculations. For wavelengths much smaller than the crystallite dimensions (2 << d) the total reflectivity of the polycrystalline samples was obtained by an average over the individual reflectivities of the grains [ 60]. In the opposite case (2 >> d) the dielectric functions of the individual crystallites were averaged in an effective medium theory [46,54]. The two methods gave dissimilar values for the Drude parameters, different also from recent single crystal results by Tajima et al. [72]. According to Orenstein et al. [ 73 ] the anisotropy of the dielectric function of these materials has been made responsible for the occurrence of the peak in the frequency dependent conductivity at 0.5 eV obtained from the Kramers-Kronig analysis. These authors showed that a quasi two-dimensional metallic crystal exhibits a m a x i m u m in the conductivity at 09~ 0 if the light does not impinge on the crystal perpendicular to the conducting plane. Averaging over different orientations leads to a maximum in the conductivity at a finite frequency. The frequencies of the IR-active phonons obtained from different experiments are collected in fig. 2. The summary of the phonon modes also given in this figure exhibits clear bands at 160, 360, 5 I0, and 690 c m - ~for the pure La2CuO4 and at 160, 240, 350, 500, 570, 570, and 670 cm-~ for the Sr-doped material. The symmetry of the vibrations has been assigned by Doll et al. [63]. An additional aspect has been treated by several authors: Sulewski et al. [31,34], Walter et al. [33], Schlesinger et al. [ 32,41,47,60 ], Bonn et al. [ 74,42 ], Orenstein et al. [44], Nob et al. [52], and Sherwin et al. [55]. They have observed a reflectance edge near 50 cm-~ in the superconducting state in (La, Sr)2CuO4. This edge was first assigned to the onset of absorption due to quasi particle excitations. The ratio of the gap energy and the transition temperature 2A/kaTc (ka is the Boltzmann-constant) ranges from 1.6 to 2.7, considerably smaller than the BCSvalue [75]. Bonn et al. [42] ascribed this edge to a zero crossing of the dielectric function caused by higher lying vibrational resonances. Sherwin et al. [55 ] showed that both assumptions yield to a similar temperature dependence. Thus the question

5

which model gives the correct interpretation for the reflectivity edge is still unanswered.

3. Lattice vibrations in YBa2Cu307 and related compounds In the course of the recent two years since the discovery of the new high Tc superconducting material YBa2Cu307 by Wu et al. [2] the Raman [76-183] and IR-investigations [ 184-239 ] focused mainly on this material. The different aspects of these measurements are grouped into several sections as mentioned above. 3.1. Structure and symmetry

Depending on the oxygen contents the YBa2Cu3Ox system is a semiconductor ( x = 6) or a metal with a superconducting Tc of 92 K (x = 7). The structure of both phases has been determined by various groups (e.g. Beno et al. [240], Greedan et al. [241 ], Capponi et al. [242], Beech et al. [243], Katano et al. [244], Jorgensen et al. [245], and Kajitani et al. [246]). It is tetragonal (D~h-P4/mmm) for x = 6 and orthorhombic ( Dlh-Pmmmm ) for x = 7 with one chemical formula unit per crystallographic unit cell. Reducing the oxygen stoichiometry from x = 7 lowers the superconducting transition temperature continuously and superconductivity vanishes below x = 6 . 5 (Jorgensen et al. [247], Chaillout et al. [248] ). The dependence of Tc on the oxygen contents is influenced by the preparation conditions of the oxygen deficient samples. An adjustment of the stoichiometry by annealing the samples at temperatures between 400 and 800°C in an argon atmosphere, followed by rapid quenching to room temperatures, yielded a more or less continuous transition temperature To(x). As shown by Cava et al. [ 249 ] annealing at lower temperatures using an oxygen gettering technique gave rise to samples which exhibited a pronounced plateau of Tc's at 60 K for x = 6.6-6.8 due to correlated oxygen vacancies in the CuO chains. Table II gives the sites for the different ionic species, their local symmetry, and the symmetry of the normal translational modes to which they contribute for the tetragonal and the orthorhombic phase. The

6

R. Feile /Lattice vibrations in high-T,, superconductors

Table II Symmetry of normal modes of orthorhombic YBa2Cu3Cu307and tetragonal YBa2Cu306with structures D2~hand D~h, respectively. D~h

D41h

atom

site

sitesymmetry

normal modes

site

sitesymmetry

Y Ba Cu( 1) Cu(2) O( 1) 0(2)

h t a q q s

D2, D2v D2, D~v D2v D2v

Blu+B2u+B3u Ag+ BLuWB2B-bB2u+ B3g+B3u

d h a g g i

D4h

A2u+Eu

C4v D4h C4v C4v C~v

Alg+A2u+Eg+Eu A2u+ E~ A,g+A2u+Eg+Eu A~g+A2u+E~+E~

0(3)

r e

D~ D2h

O (4)

BLu+B2u+B3u Ag+ Blu+ B2g+B2u+B3g+B3u Ag+ B~u+B2g+B2u+ B3g+B3u As+ Btu+ B2s+ B2u+ B3g+B3u A~+ Blu + B2g+B2~+ B3g+ B3. Biu+ B2u+ B3u

d

normal modes

Alg+ A2u+ Big+ Bzu+ 2Eg+ 2Eu included in 0 ( 2 )

not occupied

Normal modes grouped according to their symmetry B tu+ B2~q-B3~ 7BI~+ 7B2~+7B3~ 5Ag+ 5B2g+5B3g

acoustic IR Raman

labeling of the ions given here, especially of the oxygen ions, is according to Beech et al. [243 ]. The oxygen ions of the chains along y-direction which are removed in the tetragonal phase are n a m e d O ( 4 ) , the oxygen ions bridging the chains and the CuO2 planes (O ( 2 ) / O ( 3 ) ) are labeled O ( 1 ). This assignment is used throughout this article. A different notation as given by Jorgensen et al. [245], who labeled the chain oxygen O (1) and the bridging one O (4), has also been used by several authors. The orthorhombic phase exhibits 39 translational modes: three acoustic, 21 IR-active, and 15 Raman-active. three modes less occur in the tetragonal phase, because one oxygen atom (O ( 4 ) ) is missing compared to the orthorhombic phase: one silent, three acoustic, 17 IR-active, and 15 Raman-active. Among these several are doubly degenerate Eg and Eu modes, which reduces the number of observable lines in the spectra. Deviations from the ideal compositions, as for the intermediate oxygen stoichiometries, change the local symmetry of the surrounding sites, so that the strict symmetry selection rules are no longer effective. This can give rise to otherwise forbidden lines in the R a m a n and IR spectra.

B2u Azu+ E~ 5A2~+6E. 4Alg+ Bl~+ 5Eg

silent acoustic IR Raman

3.2. R a m a n spectroscopy 3.2. I. Light scattering f r o m impurity phases. Before discussing the present state of R a m a n scattering on the pure c o m p o u n d YBa2Cu307 the influence o f impurity phases will be considered. This is important, because they exhibit various lines. Their intensity can be considerable as they are insulators and have a larger penetration depth for the light than the opaque superconductors. They can therefore contribute to the spectra of superconducting samples even if only small traces o f them occur. Beside the results of micro-Raman measurements on YBa2Cu306.s Hemley and Mao [76] also obtained the R a m a n spectrum o f the so-called green phase Y2BaCuO5 which in the early times of Y-based high Tc superconductors appeared as a by-product of the not-well-known synthesis of this material. This green phase together with Y2Cu205 (another green phase) and BaCuO2 obscured the results of many R a m a n investigations on the superconductors for some time. Several groups have measured separately the R a m a n spectra of these phases and also of BaO, BaCO3, Y203, and CuO, which were the starting substances for the synthesis, in order to distinguish between the spectra of the superconducting phase and the impurities. The positions of the R a m a n peaks

R. Feile I Lattice vibrations in high-To superconductors

from YzBaCuOs, Y2Cu2Os, and BaCuO2 as obtained by Hemley et al. [ 76 ], Udagawa et al. [77 ], Rosen et al. [78], Morioka et al. [94], Feile [250], Mascarenhas et al. [ 116 ], Bhadra et al. [ 117 ], Popovic et al. [125,154], and Pham-Thi et al. [164] are summarized in fig. 3. The spectra of the other materials were shown by Rosen et al. [78], Mascarenhas et al. [ 116 ], and Bhadra et al. [ 117 ]: CuO exhibits weak peaks at 294, 342, and 629 cm-~, and BaO at 193,219, 232, 252 cm -1. BaCO3 has lines at 142, 160, 186, 233, and 692 cm -~, Y203 has a strong peak at 380 cm-~ and weak lines at 130, 163, 468, and 590 cm -~, and Ba(OH)2 as a possible corrosion product of these materials has a broad band of unresolved peaks around 211 c m - 1. Liu et al. [ 110] have found resonant Raman scattering due to the blue exciton in Cu20 in Ar-annealed YBa2Cu306. It is observable especially at low temperatures and when a laser wavelength is chosen in the blue-green region. Cu20 shows strong lines at 155 and 658 cm -~, and additional weaker lines at 217, 310, 515, and 818 c m - L A detailed study of BaCuO2 in YBa2Cu307 has been performed by Rosen et al. [149]. They mixed the pure superconductor with 10% of this impurity, Y2Ba Cu 05

IHemley etaJ al a[ Thlel Udagawa el al Rosen et Monoka et QI MQscQrenhOs e¢ PopOvlC el QI RhQm el

I

I I

II

I

II

Iiii

I

[I II

I I

I

200

300

I

/.00

500

3.2.2. Influence of oxygen stoichiometry

Y2 CU2 05

Roseootal

Mos:are.hQs e~i Elhadro el al Popovic et Q~ P~.~ rh, ~I~l

I I I I I I I I I I I

I I

II I I I I l I

,oo

I l I I I

I Jl I I

2oo Remon

I I I I I

I I I I I I I II I I I I I I I I I I I I I l

360

~ao

shift

I J I I I I

s~o

I I I I I I I I I I

I I I

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.........

M~scetenhos el ol et ol

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I

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I I

0

ioo

I I

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Ramanshift

,,, I I

I

~oo

and obtained those spectra interpreted as spectra from the superconducting phase by several groups. The characteristic lines around 640 cm-~ vanished if annealing experiments were performed with the mixture to reduce the oxygen contents. After annealing in oxygen atmosphere the spectra recovered their original shape. Similar changes in the Raman spectra of BaCuO2 on annealing have been obtained by Kakihana et al. [251]. They did not find this mode in an oxygen annealed SmBa2Cu307 sample. However, it appeared in their experiments when samples had a reduced oxygen content. Because of the changes in the spectra of BaCuO2 in annealing experiments the lines around 640 cm-~ have been misinterpreted as O(4) chain vibrations [103,252,253]. Among the impurity phases the two green phases have been studied in more detail. Popovic et al. [ 125,154 ] performed a group theoretical analysis of both compounds and obtained for the symmetry of the F-point phonons for the D~6 / P b n m structure of Y2BaCuOs: 16 Ag+ 16 Big+ 11 B2g+ 11 B3g--]- 11 Au+ 11 B~u+ 16 B2u-F16 B3u, of which 3 are acoustic, 11 silent, 40 IR-active, and 54 Raman-active. For the C9v/Pna2j structure of Y2Cu205 they obtained 27 A~+27 A2+27 B~+27 B2 modes of which 3 are acoustic, 27 are pure Raman modes, and 78 are both Raman and IR-active. The large number of Raman allowed lines in these two compounds is consistent with the experiment. They can easily give rise to spurious lines in the spectra of superconductors.

600

(cm -~)

Roman shift

[

I II I I[ I I

I I

I 100

I I

I I

II

7

l

I

5~o

I I

,, I I

I I

6~o

(cm -~)

Fig. 3. Raman lines from lattice vibrations in Y2BaCuOs, YzCu2Os, and BaCuO2 obtained by different authors. Small lines indicate weak Raman lines, fat lines give strong ones.

Slightly different frequencies of those phonons attributed to the superconducting phase have been reported by various groups. The differences were assumed to be due to variations in the oxygen content of the samples depending on the preparation conditions. Stavola et al. [ 80] reported first studies on YBa2Cu3Ox for a few oxygen stoichiometries between the stability limits x - 6 and x = 7 of the tetragonal semiconductor and the orthogonal superconductor, respectively. They found the phonon line at ~ 500 cm-~ to decrease and the ratio of the intensities of this line and the ~ 340 c m - ~ phonon to decrease with the oxygen deficiency. The relative increase of the 340 c m - J phonon line in the semiconducting phase was interpreted to reflect the reduced

8

R. Feile / Lattice vibrations in high-To superconductors

shielding of this phonon in the CuO2 planes. Several other groups have also investigated the influence of the oxygen content on the Raman spectra. Some of them presented data from samples which had an oxygen stoichiometry in the vicinity of the stability limits x = 6 and x = 7 (Yamanaka et al. [ 93 ], Morioka et al. [94], Cardona et al. [99], Krol et al. [ 102], Nakashima et al. [ 105], Bhadra et al. [ 117], Thomsen et al. [ 123], and Zhang et al. [ 129] ), and near the tetragonal-orthorhombic transition at x ~ 6.5. (Burns et al. [9], and Kulakovskii et al. [ 135 ] ). Detailed studies in the whole concentration range have been performed by Kourouklis et al. [ 101 ], Kirillov et al. [ 107 ], Thomsen et al. [ 108 ], Hangyo et al. [ 114], Kuzrnany et al. [ 128,136], and Macfarlane et al. [ 152,163 ]. The results of these investigations are summarized in fig. 4. It shows an almost linear dependence of the ~ 500 cm-~ line on the oxygen contents: o9(x) = O9o+o9'x, with O9o=312_+60 cm - t and O9'=27+_9 cm -l. Other phonons exhibited similar influence on oxygen stoichiometry. Kirillov et al. [107], Thomsen et al. [108], Macfarlane et al. [152,163], and Krol et al. [ 175], who also have observed the same stiffening

500 •

c

•vD

v

/,90

'D

o"

o:

c o o ~ ¢13_

*

/,80

YBa2 Cu30 x

o []

/`70

':'

o

L

J

6.0

6.5

7.0

x

Fig. 4. Influence of the oxygen content on the ~ 500 cm-~ p h o n o n in YBa2Cu30,.. The data points are results of various groups: o Stavolta et al. [80 ], • Kourouklis et al. [ 101 ], A Hangyo et al. [ 114 ], [] Kirillov et al. [ 107 ], ~> T h o m s e n et al. [ 108 ], • Kuzm a n i e t al. [ 1 2 8 ] , a n d ~7 Macfarlane et al. [152].

of the 700 c m - 1phonon with increasing oxygen content, found the 440 crn- l phonon to become softer. The excitation at 335 crn-1 exhibited only a very small softening of less than 3 c m - 1between x = 6 and X-----7.

This behaviour has been used to determine the oxygen contents of samples which were not suited for chemical analysis because of the small amount of material available, as in the case of thin polycrystalline films (Yang et al. [ 167], Liu et al. [ 170] ) or for single crystals which may have an oxygen contents at the surface different from the bulk. This method has, however, a limited applicability because the 500 cm-1 phonon is only observable for distinct scattering geometries. Especially thin epitaxial films on SrTiO3 which are preferentially oriented with the c-axis perpendicular to the film surface (see section 3.2.5 ) exhibit this line very weakly. The 335 cm phonon, which is the dominating line in these films, cannot be used in this respect because of its small dependence on oxygen stoichiometry. An explanation for the change of the phonon frequencies with oxygen contents has not been given yet. The increase of t h e frequencies with higher oxygen concentration for some phonons may result from the contraction of the unit cell. Raman experiments under pressure by Syassen et al. [ 140 ] yielded the pressure coefficients dog/dP of 1.3___0.2, 3.2_+0.2, 4.4+0.3, and 5.5_+0.2 c m - I / G P a for the 145, 338, 440, and 504 cm-~ phonon lines, respectively, and supported this picture. An additional mode, however, splitting off the 338 cm-~ line exhibited a negative pressure dependence. This and the softening of the 440 c m - ~ phonon on an increase of the oxygen contents shows that the behaviour of these materials is not completely described by the anharmonicity of nearest neighbor potentials, and a sensitive balance of different interactions is probably important for their properties. In spite of these complications, the spectroscopy of the lattice vibrations of YBa2Cu307 is a sensitive tool to study minute effects due to changes in lattice parameters and interactions as will be shown in sections 3.2.5 and 3.2.6.

3.2.3. Substitution of elements in Y B a 2 C u 3 0 7 From the substitution of different elements in the lattice of the superconductor one can, in principle,

R. Feile /Lattice vibrations in high-T,, superconductors

identify certain phonon modes by their frequency shift due to the mass change, provided the binding forces remain constant. The situation becomes more complicated if also the chemical binding conditions are altered, e.g. by the substitution with elements of different valence. This will modify the structure of the compounds and also change the effective force constants because of the anharmonicity of the potentials. The experiments summarized below report on the substitution of all components, either with different isotopes, or with neighboring elements. Only a few reports were given on the effect of Basubstitution on the Raman spectra. Liu et al. [87] mention a shift of the 500 c m - 1 line to higher frequencies by 30 c m - ~in a Sr substituted sample which was not superconducting and probably of a different structure. Leising et al. [ 137] observed a downshift of the 500 cm-~ line and the evolution of a broad structure around 600 c m - ~on substituting Ca for Ba. The replacement of Y has been performed by several groups. Only minor changes in the phonon frequencies are expected to be observed as the Y site is not Raman active and frequency shifts may only occur via anharmonic effects. Morioka et al. [94] observed shifts from 500 to 510 cm -~ and from 338 to 314 cm -~ for a Nd compound and from 400 to 433 c m - 1 for Nd and Ho. The same shifts have been observed by Kourouklis et al. [ 101 ] who used Eu and Gd as substitutes, Krol et al. [ 102 ] who also exchanged Y by Gd, Liu et al. [87] who used Sm and Eu, and Thomsen et al. [113] who investigated mixed compounds of 50% Ho and Sin. Ho based samples were also measured by Cardona et al. [99 ] who summarized the results of these rare earth substitutes with supplementary experiments on Nd, Dy, Er and T m compounds in a subsequent paper [ 109 ]. The results were similar to those obtained by Bhadra et al. [ 117]. McMillan et al. [253] investigated the Eu-compound and found the 335 and 500 cm-1 lines of the Y-compound shifted to 310 and 512 c m - 1, respectively. They observed also several additional lines which probably were due to impurities. SmBa2Cu307 exhibited a similar spectrum as determined by Kakihana et al. [251]. Thomsen et al. [254] investigated single crystals of nonsuperconducting PrBa2Cu307 (see section 3.2.4). All phonon frequencies observed in Raman and IR-experiments increased continuously with the increase of the ionic

9

radii of the Y-substitutes, except the 340 cm-1 line which softened. A similar softening may also be deduced from the data of Cardona et al. for the 440 cm-1 although this line is less well defined than the other lines. Substituting the Y-ion by one of larger ionic radius expands the lattice less than expected from the excess volume (see e.g. Alario et al. [255 ] ). Especially along the c-direction bonds become shorter, so that the stretching vibrations increase in frequency due to anharmonicities, as, e.g. the 500 cm-1 line. The opposite may occur in the case of bending modes, as for the in-phase and out-of-phase 0 ( 2 ) / 0 ( 3 ) vibrations, where one expects a softening to occur because of larger nearest neigbor distances resulting from the net expansion of the CuO2 planes. The insensitivity of the superconducting transition temperature to the exchange of Y by rare earth ions with large magnetic moments indicates that the overlap of the 4f-electrons on these sites with the superconducting charge carriers is negligible. In contrast to the Ba and Y-sites, a replacement of Cu has a strong influence on the superconducting properties as it changes the electronic states of both the CuO2 planes and CuO chains. Which of the two structural units are preferentially exchanged may be inferred from the changes in the vibrational spectra. With Raman spectroscopy the vibration of the Cu(2)-ion (the one within the CuO2 planes) along the c-direction is directly observable will therefore give the information to answer this question. Its frequency, however, is quite low due to the relatively large Cu mass, and the vibration is often obscured by the high elastic background intensity near the laser line, Therefore a frequency-shift of the Cu-vibration could not be observed in the experiments by Kirby et al. [ 106 ], Zhang et al. [ 111 ], Beale et al. [ 112 ] and Kuzmany et al. [ 136 ] who substituted Cu by A1, V, Sn and Zn, respectively. The substitution showed up only in a broadening of the 500 cm-1 line representing the c-directed vibration of the O( 1 ), which bridges the two Cu-sites. The broadening has been interpreted as reflecting the disorder on the Cu-sites. The substitution of Cu by the neighboring elements in the periodic table, Ni and Co, has been investigated by a few groups. Iqbal et al. [86] found the 500 c m - l line to be split into two bands, one unshifted and the other at 536 c m - l . This was attrib-

1o

R. Feile /Lattice vibrations in high-T,, superconductors

uted to a change of the distance of the O ( 1 )-ion to Ni or Co-ions partially replacing the Cu (2) sites. This upward shift has not been observed by other authors and may either be due to the simultaneous partial substitution of Y by Sc in the work of Iqbal et al., or it reflects problems with the sample preparation, which may be seen from lines due to impurity phases also occurring in the spectra. The experiments of Morioka et al. [156] revealed no pronounced changes in the spectra of Ni substituted samples. For Co the 500 cm-~ line was found to be shifted to a lower frequency, and the 115 cm-L line had vanished. Hillebrecht et al. [ 172] studied the substitution of Cu by Co in more detail. They confirmed that it was not an oxygen deficiency which shifted the 500 cm-~ phonon line down to 475 crn-~ in these samples. As the modes at 335 and 440 cm -1 involving the CuO2 planes did not change in frequency the authors concluded that the Cu( 1 )-sites had been substituted by Co. These authors found this interpretation supported by the fact that the 150 c m - ~ line, which they attributed to the Cu(2) vibration, was also uneffected by the substitution. After the experiments of Cooper et al. [ 122 ], however, this mode has probably to be assigned as the Ba mode (see section 3.2.4) which should not be affected by the Cu ( 2 ) - C o exchange. In addition to the substitution with Ni and Co Hangyo et al. [ 178] replaced Cu by Fe. The results for Fe were similar to Co: the 500 cm-~ line shifted to lower frequency, but remained unchanged for Ni. In contrast, the 440 crn-~ excitation was only affected by Cu-Ni substitution and not for Co and Fe. From these facts Hangyo et al. [ 178] concluded that Fe and Co replace Cu ( 1 ) sites and Ni the Cu(2) sites. In the experiments described above a substitution of ions in YBa2Cu3Ov by different elements has been performed. An interesting series of experiments has been carried out on isotope exchanged compounds. These had two aims: firstly, an influence of the isotope exchange on the superconducting transition temperature would support BCS-like pairing [ 75 ] as the cause for the superconductivity in the new materials (see section 5). Secondly, the isotopic exchange should give the most definite effects on the Rarnan spectra as it leaves the chemical bonds unaltered and introduces a mass change only. A relative large decrease in frequency due to the

change in mass (6%) is expected for the substitution of ~60 by ~80. Thus the oxygen vibrations can clearly be identified in these experiments from the mass dependent frequency shifts. First measurements reported by Batlogg et al. [ 79 ] and Kourouklis et al. [ 101 ] identified the 500 and 440 cm-~ lines as oxygen modes (and also the BaCuO2 impurity modes at 600 and 640 c m - ~which also exhibit oxygen vibrations). The simultaneous shift of modes involving different oxygen ions confirmed that isotopic exchange is homogeneous, which is important for the search of the isotope effect on Tc (see section 5). Kulakovskii et al. [135] presented experiments on isotope exchanged single crystals which also exhibited the isotope shift of the 335 cm-~ peak. Different stages of ~sO substitution have been investigated by Cardona et al. [ 174 ] who found the experimental shifts in agreement with the results of lattice dynamics calculations. The best result obtained there for oxygen substitution was six O-ions out of seven. A 100% ~80 sample has been measured by Thomsen et al. [179] who had obtained the fully exchanged material by preparing the sample from the metals in a pure ~sO atmosphere. From all these experiments it became clear that the 335, 440 and 500 c m - ~lines in the Raman spectra exhibit oxygen vibrations and that the two other lines at 118 and 150 c m - J are the Cu and Ba modes, respectively. 3.2.4. Single crystal investigations

Raman experiments on polycrystalline material yielded several modes which, as discussed in former sections, could be identified as Raman active lattice vibrations of the superconductor YBazCu3OT: ~ 118, ~ 145, ~ 335, ~ 440 and ~ 500 c m - ~. These lines can be assigned to certain normal modes with the knowledge of their symmetry and with some additional assumptions about the binding forces. The experimental information about the symmetry of the modes comes from Raman spectroscopy on single crystals in different scattering geometries. This yields the different components of the Raman tensor ~j. Already in the first paper about Raman measurements on YBa2Cu307 Hemley et al. [76]. presented micro Rarnan results from small crystallites embedded in a sintered sample. They observed lines at 142, 483 and 585 c m - ~in polarized spectra which yielded

R. Feile /Lattice vibrations in high-Tosuperconductors the components of the Raman tensor C~xx, cezc and a_-c, respectively. In depolarized spectra with crossed polarization of incident and scattered light they observed a vibration at 335 c m - ' which contributed to the ax,, component. Krol et al. [ 102 ] obtained results from millimetersized crystals of YBa2CuaOx and GdBa2Cu3Ox. Both substances gave similar results as expected from the previous section. The lines at 140, 435 and 500 cm-1 have been observed in polarized (xx,zz), (zz) and (zz) spectra, respectively. Different from Hemley et al. [76] the excitation at 335 cm -1 has been found in polarized (xx) spectra. The reason for this discrepancy will be discussed below. An additional weak line at 230 cm-1 was observed in (xx) spectra. These modes were ascribed to the symmetric stretching of the Ba planes (140 c m - ' ) , Cu(2) axial symmetric stretching ( 230 c m - 1), O ( 2 ) / O ( 3 ) out-of-phase and in-phase bond bending (335 and 440 cm-1, respectively), and the O ( l ) oxygen symmetric stretching (500 c m - l ) vibrations. The peaks above 500 c m - l were assumed to arise from disorder scattering because of nonstoichiometry which allows q ¢ 0 scattering to be observed in the Raman experiments as local defect modes a n d / o r the phonon density of states. The difference in the frequency of the 500 c m - 1 mode compared to the line at 483 cm -1 observed by Hemley et al. [ 76 ] was attributed to a different oxygen content in the samples (see section 3.2.2). Similar results have been obtained by Bhadra et al. [ 117 ] who, however, measured additional lines due to impurities. Kulakovskii et al. [ 119 ] found in (zz) spectra the line at 118 c m - l and assigned it to the Ba vibration. Therefore the 145 cm -1 vibration must have been the vibration of the other heavy mass in the lattice, the C u ( 2 ) ion. The two other (zz) modes at 445 and 500 c m - ~were assigned similar to the preceding authors. A different interpretation was given for the 585 cm-1 excitation: B2g/B3g valence vibration of the 0 ( 2 ) / 0 ( 3 ) ions along x / y direction, for the 335 c m - l phonon: bending modes of the O (2), O (3) and the O( 1 ) ions, and a B3g bending mode of the O( 1 ) ion perpendicular to the O ( 4 ) - C u ( 1 ) - 0 ( 4 ) chains for the mode observed at 220 cm -~. A similar assignment as in ref. [ 119 ] has been given by Feile et al. [ 133,160 ] on the basis of lattice dynamics calculations (see section 4). In this model,

11

however, the splitting of symmetric and antisymmetric O ( 2 ) / O (3) vibrations is much smaller than it is observed for the 335 and 440 cm-~ lines. This may be due to the Coulomb forces between the 0 ( 2 ) / 0 ( 3 ) ions which have been omitted in these calculations. Cooper et al. [122] observed the lines at ~ 116, ~ 150, ~ 340, ~ 440, and ~ 500 c m - 1 and identified them as to have A s symmetry. In contrast to the preceding authors they assigned the 150 cm-1 excitation to the Ba vibration because of its narrow lineshape. This was attributed to the insulating character of the Ba bond to its neighbors. The 116 c m - 1 has been interpreted as the Cu (2) vibration along the caxis in the CuO2 planes as inferred from the characteristic Fano-like lineshape of this phonon similar to the 335 cm -1 line (see section 3.2.6). Liu et al. [124,142] gave a slightly different assignment of the 340 cm-1 line. This excitation is observed in the tetragonal phase too, having Big symmetry there which describes an antisymmetric bending mode of the 0 ( 2 ) / 0 ( 3 ) ions along the caxis. As the CuO2 planes in the orthorhombic phase are almost undistorted this assignment may also be transferred to the orthorhombic phase. It reflects the special symmetry of the phonon, and it may even be taken litterally: The mode should be observable in depolarized scattering with crossed polarizations along the x' and y' axes which are rotated by 45 deg from the x and y axis in the a-b plane. This was clearly shown by Liu et al. [ 124,142 ] and explains why this line had appeared in the experiments of Hemley et al. in crossed polarization and was assigned as an o<~vmode. The same symmetry selection rules are valid in the nonsuperconducting compound PrBazCu307 as shown by Thomsen et al. [254]. Burns et al. [146,22] concluded also from their experiments on 180 substituted single crystals that the 340 cm-1 involved only oxygen motion and assigned it to the asymmetric O ( 2 ) / O (3) vibration. Also several other authors reported on results from single crystals: Aleksandrov et al. [ 126 ], Hadjiev et al. [127,134], Mihailovic et al. [138], McCarty et al. [168], Denisov et al. [171] and Mirlin et al. [ 256 ]. They all gave the same assignments of the observed phonon modes. Hadjiev et al. [ 134 ] found, in addition to the other lines, a mode at 59 c m - ' and ascribed it to a Ba B2g/B3g mode.

12

R. Feile / Lattice vibrations in high-Tosuperconductors

The experiments mentioned before have been performed on single crystals which normally exhibit microscopic twin domains. The two domains characterized by the direction of the Cu ( 1 ) - O (4) chains are formed at the tetragonal-orthorhombic phase transition and have common ( l l 0 ) and (110) planes. The angle of almost 90 deg between the twins is determined by the small orthorhombicity a - b ~ 0. Because of the twinning no distinction between a and b axis was possible and the components of the Raman tensor axx and ayy could not be determined independently in experiments mentioned so far. Thomsen et al. [ 151 ], however, reported Raman spectra from a single crystal which had two large domains of the same orientation separated by a narrow domain of the other orientation. The different intensities of the observed phonon lines for the two orientations have been attributed to the contribution of the O (4)-ions along the chains to the polarizability which, in addition, determine the strength of the disorder induced scattering from IR-modes at 220 and 576 cm-~. 3.2.5. Investigation o f YBa2Cus07 films The primary interests in thin films of the superconductor for Raman spectroscopy was due to their flat surface which gave less elastically scattered light than polycrystalline sintered pellets. Especially films grown epitaxially on single crystal substrates have mirror-like surfaces. Secondly, the films have been assumed to be more homogeneous than the sintered samples so that their Raman spectra obtained from a depth of a few 100 A should clearly exhibit the vibrational lines of the superconducting phase. The resuits showed that with the knowledge from experiments on polycrystalline sinter material and single crystals of YBa2Cu307 Raman experiments can be used to correlate the superconducting properties of thin films of this material with the oxygen contents, the occurrence of impurities, and the morphology of the films. Lyons et al. [ 97 ] were the first who presented Raman spectra from films of the superconductors. Because of the low elastically scattered light intensity they were able to measure low frequency Raman shifts and found first hints for excitations across the superconducting gap below Tc (see section 3.2.7). Excitonic bands were searched for by Bozovic et

al. [ 104] in Raman and reflectivity (mid-IR through ultraviolet) spectra of thin films. Their absence was interpreted as to show that such excitations do not mediate the pairing interact on in these superconductors. A detailed investigation of thin films of the superconducting material has been performed by Feile et al. [133,147,148]. They found a broad band of excitations in an as-prepared film which exhibited its amorphicity. After the annealing at elevated temperatures under oxygen atmosphere the film crystallized and became superconducting. All of the allowed Raman lines have been observed showing the polycrystalline nature of the film. Additional lines at 220 and 600 cm-1 could be identified as disorderinduced IR-active vibrations. The best results have been obtained from films sputtered onto SrTiO3 substrate which was held at elevated temperatures during the sputtering. They exhibited a preferred orientation of the crystallites in the film with the c-axis of the film perpendicular to the substrate surface. Subsequent experiments [ 250 ] showed that the films were not only oriented with respect to the c-axis, their orientation was also strongly correlated in the a-b plane across the whole surface of the film. This clearly demonstrated the epitaxial growth of the film on the substrate which can explain the high critical current densities of ~ 106 A/ cm 2 obtained. The influence of the preparation conditions on the superconducting properties of evaporated polycrystalline films was investigated by Berberich et al. [ 131 ]. They observed indications of the green phase and a shifted 500 cm-1 line due to oxygen deficit in lower quality films consistent with the results discussed in previous sections. Films with a sharp superconducting transition exhibited clearly lines of BaCuO2. This may indicate that BaCuO2 may serve as a flux for the growth of better films as in the case of single crystals. The same may have held for polycrystalline sintered samples which exhibited good superconducting properties and which contained these impurity lines in the spectra. This has first led to the conclusion that these vibrations were due to the good superconducting phase. Chrzanowski et al. [ 150], Yang et al. [ 167], and Liu et al. [ 170] have also investigated superconducting films via Raman spectroscopy. They found

R. Feile / Lattice vibrations in high- Tc superconductors

the same influence of oxygen stoichiometry to the vibrational line at 500 c m - l as in polycrystalline sintered pellets. Although the spectroscopic and the superconducting properties of the films are similar to those of single crystals the temperature dependence of the spectra differs as found by Feile et al. [182] (see next section). 3.2.6. Temperature effects

Since the advent of the new superconductors several new theoretical models have been discussed beside the classical phonon-mediated BCS-theory [ 75 ] to explain this phenomenon and the high transition temperatures. Among these theories the resonating valence bond (RVB) model by Anderson et al. [257,258], the spin-bag model by Schrieffer et al. [259], and the magnetic frustration model by Aharony et al. [260] introduce the pairing by a strong Mort-Hubbard correlation between the charge carriers and do without lattice vibrations. Other models apart from classical strong coupling theories (see, e.g. Weber and Mattheis [261-263 ] ) include the phonons as, e.g. Chakraverty et al. [264,265 ] and de Jongh [266] who discuss the occurrence of bound charge pairs by (bi) polarons which are formed because of the electron-phonon (e-p) interaction. An enhanced e-p interaction can be introduced by defects in the lattice or by quenched large vibrational amplitudes as shown by Phillips [ 267,268 ]. If the electron-phonon interaction plays a role in the occurrence of the high Tc superconductivity it should influence the temperature dependence of the phonon spectrum. For conventional superconductors phonon softening has been observed in the A 15compounds (see, e.g. Schweiss et al. [269] ) as a precursor of the martensitic phase transition near the superconducting transition. Neutron scattering experiments by, e.g. Shapiro et al. [270] revealed phonon shifts in Nb on cooling the sample below Tc accompanied with characteristic changes in the phonon linewidth. Raman experiments on these materials by Wipfet al. [271 ], Schicktanz et al. [272] and Dierker et al. [273 ] (see also M.V. Klein [2 ~/4] ) have successfully observed phonons in these metallic samples. They found asymmetric lineshapes for some phonons reflecting the strong electron-phonon interaction in these systems. In addition Dierker et al.

13

[273] observed electronic Raman scattering by excited quasiparticles in NbaSn and V3Si. Part of the strong interest in vibrational spectroscopy in the high Tc materials was the aim to see whether similar effects could be found there as well to give information on electron-phonon interaction and structural phase transitions in these materials. Macfarlane et al. [ 81 ] observed the frequency of the 502 cm-1 line to increase up to 506 cm-1 on cooling from 300 K to 4 K. Also the 337 c m - ~ line first increased in frequency, but then started softening below 100 K. These results have been confirmed by Wittlin et al. [95 ] for the Y-compound, but not for samples where Y was replaced by Ho, although in the IR-reflectivity experiments the softening of the 280 and 318 cmphonons have clearly been observed for the Ho, Gd, Sm and Eu compounds. Also other rare earth substituted samples (Dy, Er, T m ) exhibited the softening as shown by Cardona et al. [ 109 ]. No softening has been observed in samples of low superconducting quality (Wittlin et al. [95], Thomsen et al. [113] ), but, surprisingly, the phonon softening could be observed by IR-measurements on the same samples. This is perhaps due to different probing depth of visible light and IR-radiation in accordance with a possible reduction of the oxygen stoichiometry within a thin surface layer. Feile et al. [ 147,148] studied the temperature dependence of all phonon lines in the Raman spectrum above 130 cm-~ and confirmed that the anomalous phonon softening below Tc of Raman active vibrations is restricted to the 335 c m - l phonon. The same amount for the overall softening of about 5 cm-l (far from the transition temperature) was obtained by different authors. An exception was the behavior of thin epitaxial films studied by Feile et al. [ 182 ]. These films exhibited a reduced softening which was attributed to strain effects due to the epitaxy as it did not occur in polycrystalline films. The sharpness of the onset of the softening below Tc (see Thomsen et al. [151]) may depend on the sample quality. A broadening of the transition by laser heating might be possible in some cases, but it could be excluded for some experiments on films [182]. No softening has been found by Cooper et al. [ 122 ] for the 115 c m - ~ line, ascribed for the Cu (2) vi-

14

R. Feile / Lattice vibrations in high-Tosuperconductors

bration. They have studied this phonon and the one at 340 cm-~ in detail. Both lines have been interpreted as to be confined to the CuO2 planes from the typical asymmetric Fano-like lineshape which indicated the strong e - p interaction of these two vibrations with an electronic continuum within the conducting planes. No such asymmetric lineshape and no softening of the phonon equivalent to the 340 c m - ~ line is found in the nonsuperconducting compound PrBa2Cu307 (Thomsen et al. [254] ). The e - p interaction was found to be enhanced below Tc by Cooper et al. [ 122] and Feile et al. [ 182] as concluded from an increased linewidth of the 335 cm-~ line just below the transition. It was interpreted as being due to the opening of a gap in the quasi-particle excitations with an increased density of states near the gap energy which can give rise to a resonantly enhanced interaction with a phonon of the same energy. Such an increase of the linewidth has also been observed for phonons in classical superconductors by Shapiro et al. [270] in neutron scattering experiments. It is normally followed by an exponential decrease, as in the case of the attenuation of sound waves below T~, if the gap-energy exceeds the phonon frequency. The finite and almost constant linewidth in YBa2Cu307 could indicate either a distribution of gaps or certain excitations inside the gap. From these results a gap may be deduced of the order of 2A~ 330 c m - l giving 2A/kBT~~ 5.2, a value indicating a strong coupling mechanism for the pairing interaction. Zeyher et al. [275] have calculated the renormalization of the phonon frequency due to the transition to the superconducting state. Assuming htop> 2A for the phonon frequency Ogp they concluded from the softening of these Raman and IRactive phonons that YBa2Cu307 is a strong coupling superconductor. The relative size of 2A and htop may be deduced from the results of Cooper et al. [ 183 ] who observed the electronic contribution to the Raman signal to exhibit considerable anisotropy (see section 3.2.7). The larger value of the energy at the maximum of intensity in the B~g electronic contribution compared with the 335 c m - J phonon might indicate the reversed relation hogp< 2/1. The smaller gap in Ag symmetry yields htnp ~ 2/1.

A direct proof that the softening of the 335 c m phonon is characteristic for the superconducting state has been given by Ruf et al. [ 180]. They observed that the phonon softening below Tc was canceled if a magnetic field of several (0-12.7) tesla was applied which exceeded the upper critical field H¢2 and destroyed the superconductivity. The low temperature experiments described in this section clearly showed the electron-phonon interaction by the asymmetric lineshape of the 335 cm-1 phonon and its softening below T~. This strong e-p interaction probably arises as the pseudo tetragonal Big symmetry of this phonon is the same as the symmetry of the dx2_y2-band of holes in the CuO2-planes which are responsible for the superconductivity in the oxyde superconductors. Therefore these experiments clearly indicate that the electron-phonon interaction plays an important role in these materials, and the theoretical models will have to account for this. For completeness, temperature effects above roomtemperature should also be mentioned. McCarty et al. [168] reported results from a single crystal of YBa2Cu307_a for temperatures up to 700 K. They observed a softening of the 500 c m - ~ line similar to the softening on the reduction of the oxygen contents, the 335 cm-~ line remained almost constant. The oxygen vacancies at higher temperatures produced disorder induced scattering from an IR-active phonon with a peak at 570 c m - ~ (see section 3.3.3). 3.2. 7. The superconducting gap in Rarnan spectroscopy

An alternative way beside IR-measurements and tunneling experiments to observe the gap structure of superconductors can be taken via Raman spectroscopy as it has been demonstrated for the AI5 compounds by Dierker et al. [ 273 ] and Hackl [ 276 ]. First attempts to observe the gap in the high Tc superconductors in a Raman experiment have been reported by Osipyan et al. [ 139 ], Bashenov et al. [ 82 ] and Lyons et al. [97]. A gap structure at 2/1~300 cm-~ has been reported in polycrystalline material by these authors. This gap was also found in experiments on a single crystal by Thomsen et al. [ 143,151 ]. A detailed Raman study of the excitations across the gap have been reported by Cooper et al. [ 122]. They showed that the almost constant

R. Feile / Lattice vibrationsin high-T~superconductors background in the energy-range up to 800 cm-1 due to scattering from electronic excitations was reduced below 300 cm-~ when the single crystal sample became superconducting. In addition, the intensity piled up around 450 c m - i, marking the energy gap. Different from the results from the A 15 compounds, however, the scattering intensity did not vanish below the gap energy. This was interpreted as excited states within the gap [ 122 ]. More details of the gap structure were presented in subsequent papers by Hackl et al. [176] and Cooper et al. [183] who both found a pronounced gap anisotropy. The maximum of the cross section in Alg scattering geometry was at considerably lower energy than in Big geometry, giving values for 2Ao/ kBTc of 3.4 and 5.5, respectively [176]. Ao was the mean of a Gaussian distribution of gaps used by Hackl et al. to simulate the observed gap function which did not vanish at low energies, as mentioned before. The same anisotropy given by the ratio of both gap values was obtained by Cooper et al. [ 183 ] although their maxima of the measured gap function were at 22% higher energies. These differences may have been due to sample quality. Cooper et al. interpreted the occurrence of the gap anisotropy to reflect the inverse effective-mass tens o r / z - I whose isotropic part is responsible for the Atg component of the spectra, and whose B~ component is associated with Fermi-surface regions with a large difference of the inverse mass tensors ( / t - l ) xx_ (/z- 1)yy. They speculate that the large B l~ gap is related to the CuO chains. A striking fact is that the gap energy does not exhibit any temperature dependence, as soon as it can be observed below T~ is has its full T o O limit value.

3.3. Infrared spectroscopy 3.3. I. Scattering from impurity phases The influence of impurity phases has also been studied by IR-spectroscopy, yet not in that detail as by Raman spectroscopy. Popovic et al. [ 125,154 ] have measured the infrared spectra of the two green phases in addition to the Raman spectra. They observed 28 IR-active modes in YzBaCuOs, and 14 in YzCuzOs. Similar results from Y2BaCuO5 have been obtained by Udagawa et al. [77] and Jandl et al. [227]. The IR-spectra of the other impurities were

15

measured by Min-Guang et al. [ 215 ], Dongming et al. [177], and Muraleedharan et al. [239]: BaCO3 exhibited a dominant line at 693 c m - 1, CuO at 484, 534, and 582 cm -1, and Y203 at 419, 466, and 563 cm -l. Especially BaCuO2 was sometimes identified in infrared spectra of superconductors [239].

3.3.2. IR-active phonons in YBaeCu30z Because of the centrosymmetric structure of the high temperature superconductors which are considered here IR-spectroscopy gives information about the lattice vibrations in these materials complementary to Raman spectroscopy. As already discussed for LazfuO4 and the related compounds IR-data exhibit two contributions, the vibrational and the electronic part. For a separation of these two parts different procedures have been applied as discussed in section 2.3. For the electronic contribution a Drude-model has been used to describe the decrease of the reflectivity at certain frequencies. Different values have been obtained for the plasma frequency by different groups ranging from 0.1.eV [209] to 4.2 eV [233]. A detailed discussion of these values is beyond the scope of this article. The results for the phonon frequencies obtained by different groups (Kirtley et al. [ 197], Bonn et al. [198,214], Thomas et al. [199], Sulewski et al. [277], Genzel et al. [202], Bozovic et al. [89], Kamaras et al. [2041, Collins et al. [205 ], Sugai [206 ], Wittlin et al. [95], Taliani et al. [208], Perkowitz et al. [209], Cardona et al. [99,109] Noh et al. [52], Thomsen et al. [ 108 ], Crawford et al. [ 216 ], Briiesch et al. [7], Onari et al. [218], Ose et al. [221 ], Klamut et al. [223]) are summarized in fig. 5. Obviously several phonons are well defined, especially between 150 and 350 cm-1 and around 560 cm-1. The mean values of these phonons are 152+3, 191 +8, 277+7, 312+6, and 565+ 14 cm -1. Phonons below 150 cm -1, between 400 and 550 cm -1, and above 600 c m - t have been reported by only a few groups and are less reliable. The number of observed lattice vibrations is in most cases much less than predicted from the structure and from symmetry considerations. This is probably due to the high electrical conductivity within the CuO2-planes which screens the phonons confined to these planes.

16

R. Feile / Lattice vibrations in high-T,, superconductors

ir phonons in YBa 2 0U307_ 6 (6<0.2)

,,l

Kirtley et ol

I

Bonn e t al.

i

I

i

,

fr I I Ill

ir phonons in YBa 2 Cu306. 8 (5< 0.2) I t

II

I

Burns et ctl

Thomas et al

t

Sulewski el at Genzal et al

Kamo.ras et al

Kamaras

Ill

I i

Collins el el

I

Sugoi

I

I

Onori et a l

Burns et cfl

I I

Ii I Illl

Taliani ¢t al

i

I

I

Kuzmany et at Crawford et el

i

J,

Jandl ¢t al

I I

TO

LO

a/b

Bonn et al. Thomsan et al Cardona et a l

i,I

Crawford et c,I

I

Onari at el Osa et al Klamut at el.

iI

880--

I I

[ i

I I I III 1 I i, 113o 20O

Ii,

!

I I

I I

I

I

I I

I i

I i

400 500 phonon frequency (cm -1 ) 300

11, I

II '

I

I iii

II i i I I IiI

I

I I

IJl iI

]l

I IE

I

,ll", "I,II', 200

300

400

phonon frequency

500

600

(cm -~)

Fig. 6. Results of infrared spectroscopy on lattice vibrations in YBazCu306 obtained by different authors. TO and LO indicate transverse and longitudinal optical phonons, A and E denote phonons Au and Eu symmetry, respectively.

L'

III I I I I I II ,I

TO LO

,

I I

I I,

II I I

I

BriJesch ¢t al

c

I i

I

II

100

Co.rdona at al Nob at a l

i

I 'r',!l,, 1!' III

I I

Wittiin et o.L

I

TO LO

fhornsen et o1.

i

Perkowilz at al

et al

[, i

iI iI II

Sugai

!, ,!

i!

Bozovic etal

I ii

S t a v o l a ¢t a l

I i

600

Fig. 5. Results of infrared spectroscopy on lattice vibrations in YBa2Cu307 obtained by different authors. Noh et al. have distinguished transverse (TO) and longitudinal (LO) optical phonons, a, b, and c indicate the polarization of the light.

3.3.3. Influence of oxygen stoichiometry on IRspectra Removing oxygen from this material, e.g. by high temperature annealing reduces the conductivity and eventually creates the semiconducting material. Thus the screening within the CuO2-1ayers is reduced and more lattice vibrations become visible. The results of Stavola et al. [80], Bums et al. [91,219], Kamaras et al. [204], Sugai [206], Onari et al. [218], Thornsen et al. [108,123], Kuzmany et al. [136], Crawford et al. [216], and Jandl et al. [238] are summarized in fig. 6 showing the main lines at 107 + 2, 118_+4, 151_+6, 191_+3, 215_+4, 252+5, 356-+4, 597 +_9, and 642 + 12 c m - ~in YBazCu306. A few additional lines have also been found by some authors, e.g. the number of phonon lines given by Jandl et al. [238] exceeded the number of 11 IR-active modes

for that structure possibly indicating the presence of impurity phases. A correlation of x = 6 and x = 7 spectra has been proposed by Thomsen et al. [ 123 ] for a few vibrations (shift from x = 6 to x = 7 , respectively): The two lines around 151 and 191 c m - l remain merely constant, they represent an in-phase motion of O ( 1 ) - C u ( 1 ) and an Y-vibration parallel to the c-direction, respectively. Large shifts occur for the Y-vibration parallel to the CuO2 planes (215--,277 cm -1) and 0 ( 2 ) / 0 ( 3 ) bond bending modes within the conducting planes (252--,312 c m - l ) . The line at 356 cm - l for x = 6 has no counterpart in the superconducting material. It may represent O ( 1 ) vibrations parallel to the a/b-plane. The situation for the higher frequency modes is quite unclear. On one hand, if the mode at 565 cm - l ( x = 7 ) either is the O ( 1 ) vibration along the c-direction or an 0 ( 2 ) / 0 ( 3 ) in-plane motion then it should still be observable in the semiconducting phase. The next line for x = 6 at 597 c m - t is rather high in frequency, an it is not quite understandable how this large frequency shift should occur, On the other hand, if the 565 c m - t were an 0 ( 4 ) vibration, it should vanish as the oxygen is removed. Then the 597 cm-l-line may be due to impurity phases (see section 3.3.1 ) as in the case of Raman active modes observed in this frequency range. This may hold also for those lines

R. Feile /Lattice vibrations in high-Tosuperconductors which have been observed by a few groups only. 3. 3.4. Element substitution Several experiments have been performed on samples in which some of the ionic specimen in YBa2Cu307 have been replaced by others of different mass and ionic radius. Due to the mass change a direct influence on the frequency is expected for those vibrations in which the replaced ions are involved. The change of the ionic radii as, e.g. by rare earth substitution for Y will effect all phonon frequencies more or less depending on the anharmonicity of the potentials. As in the case of Raman experiments this is of help for the assignment of observed modes. The exchange of oxygen by the heavier isotope 180 should have a relative large effect on the modes involving oxygen ions. Such experiments have been performed by Benitez et al. [231 ], Crawford et al. [216], and Cardona et al. [174] on samples which had been produced by 160-180 gas exchange at elevated temperatures and by Thomsen et al. [ 179] who prepared a sample from the metals and exclusively ~80. In this way the vibrations at 276, 312 and 570 cm -j have been identified as oxygen modes. Their shifts on 180 substitution were found to be about those shifts obtained from lattice dynamics calculations for O ( 1 ) vibrations along x and y and for the O (4) bond stretching vibration along the CuO chains, respectively (Cardona et al. [ 174 ], Thomsen et al. [179]). Replacing Y by a rare earth ion leaves the structure of the superconductor almost unchanged and also the superconducting properties are not influenced (the exceptions are Ce and Pr). As described in section 3.1 vibrations on the Y site can be observed by IR-spectroscopy. Tm, Eu, Ho, Sin, Gd compounds have been investigated by Cardona et al. [ 174,99 ], Dy and mixed compounds of H o - S m and Sm-Y by Wittlin et al. [213 ], Er, Nd, Sm, Gd by Crawford et al. [216, 232], Pr by Klamut et al. [223], Gd by Perkowitz et al. [209], Er, Ho by Onari et al. [ 218 ], and Thomsen et al. performed experiments on Gd, Sm, and Ho substituted samples [ 123,113 ] and on mixed-crystal compounds of Y Ho, Y-Sm, and Ho-Sm. In these investigation the 192 c m - ' vibration in the Y-compound shifted to lower energy according to the mass change on sub-

17

stitution. Thus this line is clearly identified as an Y vibration. Small frequency shifts have also been observed for other phonons by the authors given above. This is due to changes in the lattice constants resulting from the different ionic radii as discussed in the case of the Raman results (see section 3.2.3). Substitution of other ionic species for the Ba and Cu sites which introduces stronger disturbances in the lattice as shown by Raman experiments has not been investigated by IR-spectroscopy. 3.3.5. The superconducting gap and phonon softening Whereas the Raman spectra only change slightly when the samples become superconducting, the infrared reflection spectra display pronounced differences between normal and superconducting state. For frequencies below the gap 2A the material becomes perfectly conducting as Cooper pairs cannot be broken any more. For BCS superconductors this hold for T - 0 , at higher temperatures ( T < To) the interaction with excited quasiparticles will give some deviation from the 100% reflectivity. If the infrared light can couple to lattice vibrations additional loss is introduced reducing the ideal reflectivity in the superconducting state. Thus IR-active phonons at frequencies below 2zJ(T) appear as dips in the IRspectra whereas they appear as peaks above Tc (the phonons above 2zl(T) are always peaks). The value for the gap energy is usually obtained from the ratio R s / R , of the reflectivities below and above To. The energy where this ratio crosses Rs/Rn= 1 gives in principle 2zt(T). The strong interference of electronic and vibrational contributions to the spectra makes this procedure slightly uncertain especially near Tc when phonons near the gap change their lineshape drastically. Most of the authors (Bazhenov et al. [82], Kirtley et al. [197], Bonn et al. [198], Thomas et al. [ 199], Genzel et al. [202], Perkowitz et al. [209], Sugai et al. [211 ], Abel et al. [220], and van Bentum et al. [ 222 ] ) report a gap around the mean value 2A=227 + 24 c m - 1 for YBazCu307. This gives 2A = 3.5kB T¢ the usual weak coupling BCS relation. Smaller gaps have been obtained by Lee et al. [217] ( < 2 0 0 c m - l ) , Collins et al. [205] and Noh et al. [52] (170 c m - l ) , Ye et al. [53] (130 c m - l ) , Balashov et al. [185] (120 c m - l ) , and Gershzon et al. [207] (40 cm-~). Larger gaps have

18

R. Feile / Lattice vibrations in high-Tosuperconductors

been reported by Ose et al. [221 ] and by Collins et al. [205] on an oriented film (300 cm -~ ), Saito et al. [210] (350 cm -~ ), and Schlesinger et al. [212] (500 c m - ~). The different values for the gap could result from small shifts ofphonons near the gap which influences the reflectivity ratio in that region. A further possibility is that the gap is reduced together with the decreased superconducting transition temperature due to different oxygen contents. The largest gaps have been observed in oriented samples (single crystal or epitaxial films). The different 23 values may reflect the anisotropy of the gap. In order to avoid the interference of phononic and electronic contribution Wittlin et al. [213 ] and Thomsen et al. [113] have investigated a mixed sample (Yo.~Hoo.sBa2Cu307) which exhibited only broad phonon structure and less distortion of the gap. The gap observed in this compound was 220 cm -~. Abel et al. [220] have used a different procedure in order to measure the temperature dependence of the gap: They normalized the spectra at the temperature T < Tc to the T = 0 spectra. Thus small changes of the gap can be observed clearly as the phonon lines are almost unshifted below T < To. By this Abel et al. [220] found the gap to follow the BCS prediction with 2A(0) = 190 cm -~. The gap obtained from IR-spectroscopy is smaller than the one found in Raman experiments. The reason for this is not clear, it may depend on the analysis of the data. Possibly the interference of electronic and vibrational contributions around the 300 c m - ~region in the IR-spectra may obscure the onset of quasi-particle excitations. The influence of the superconducting transition on the phonon spectrum as it is observed by Raman spectroscopy (see section 3.2.6) has also been found in IR-experiments. Bonn et al. [198] observed a softening of 4 cm -~ for the 279 and 310 cm -~ phonon below Tc. This was also measured in experiments by Wittlin et al. [95], Cardona et al. [ 109], Thomsen et al. [113], Ose et al. [221 ], and Klamut et al. [223]. These results indicate that both phonons are also confined to the CuO2-planes and involve 0 ( 2 ) / 0 ( 3 ) bond bending vibrations as in the case of the soft Raman active mode at 335 cm-~.

3.4. High frequency scattering

In the preceding sections only excitations below 1000 cm-~ have been considered. We now summarize Raman spectra with larger frequency shifts. First results were reported by Mihailovic et al. [ 88 ] who found an intense Raman signal between 100 and 4000 cm -~ when the polycrystalline YBa2Cu307 samples were cooled below room temperature. The maximum of the intensity distribution was around 1300 c m - ~. The results were interpreted as to be due to scattering from quasiparticle states above the superconducting groundstate, but they have not been found by other authors. Broad spectra with much higher energies have been obtained by other groups. Lyons et al. [ 18 ], and Sugai et al. [21 ] report Raman spectra from La2CuO4 with energy shifts as large as 4500 cm -~. Similar spectra were found in YBa2Cu3Ov_~ by Lyons et al. [ 120 ], and Krol et al. [ 181 ]. These excitations were only observed in the non-superconducting materials which reveal strong correlated spin fluctuations of a two-dimensional nature and antiferromagnetism by neutron scattering [278 ].Thus these excitations were assumed to be of magnetic origin. A detailed analysis on single crystals [ 120,18 ] revealed similarities with light scattering results from two-magnon excitations in K2NiF4. On the basis of the magnon dispersion as given by Parkinson [279,280] the exchange interaction constant for the spins was determined to be J ~ 1000 cm -1 and J ~ 9 5 0 cm -~ for the lanthanum and yttrium compounds, respectively. In addition to the broad intensity distribution several sharp peaks were obtained in the range up to 1500 c m - t for distinct scattering geometries. These peaks have been attributed to excitonically enhanced second order scattering from the lattice vibrations [68].

4. Lattice dynamics Lattice dynamics calculations for the high-To superconductors have been performed for mainly two purposes. The first was to calculate the electronphonon interaction and its influence on the increased transition temperatures for (La, Sr)2CuO4 and later on also for YBazCu307 [261-263]. Sec-

R. Feile / Lattice vibrations in high-To superconductors

ondly, the increasing number of experiments on the phonon system needed an assignment of the observed vibrational excitations. The first calculations in the context of electronphonon interaction were carried out by Weber [ 261 ] on LaaCuO4. These have been performed within the framework of the nonorthogonal tight-binding theory of lattice dynamics and were based on electron energy-band structure calculations ofMattheis [ 262 ]. In this theory the "bare" phonon frequencies calculated with the help of a force constant model are renormalized due to the interaction with the conduction electrons. The results for the F-point phonons are included in table IV. These calculations also revealed possible lattice instabilities of this structure near the zone boundary. An estimation of the transition temperature of L a 2 C u O 4 indicated conventional pairing for superconductivity in this system. Similar calculations on YBazCu307 [263], in contrast, could not yield the high transition temperatures of this compound, and it was concluded that an additional ( a n d / o r alternative) mechanism is required there. Beside these studies of the electron-phonon interaction several authors have calculated the phonon frequencies for a comparison with experimental results. B r u n e t al. [ 12 ] presented the first calculation in this context on La2CuO4 using a stretch force constant model. The values for these force constants (table III) were determined on the basis of chemical bonding and adjusted in order to match the calculated frequencies (table IV) with the measured Raman spectra. The crystallographic unit cell determined by Jorgens et al. [64] also entered this calculation. For Y B a 2 C u 3 0 7 the calculation of phonon frequencies on the basis of a force constant model was first reported by Stavola et al. [ 80 ]. In this model the structure of the unit cell [243] and force constants for the short-range interaction were used (see table V). The latter were adjusted in order to match the calculated frequencies (see table VI) with the observed Raman and infrared spectra. The authors have performed a symmetry analysis of the possible modes. Comparing these with those from the experiments the authors supported the structure proposed

19

Table III The short-range force constants used by different authors (Weber 1261 ], Brunet al. [ 12], Mase et al. [285], Feile [250], and Prade et al. [286]) for the lattice dynamics calculation of La2CuO4. The labeling of the atoms is as per Jorgensen et al. [ 64 ]. Force constants (N m-~ )

CuO(1) CuO(2) La-O(1) La-O(2) La-O(2) La-La La-Cu O(1)-O(1) 0(2)-0(2) O(1)-O(2)

ref. [261]

ref. [12]

ref. [285]

ref. [250]

ref. [286]

144

85 20 160 105 50 30 10 20 7 4

18.4 9.2 4.9 27.6 7.4 7.4 4.3 15.3 7.4 12.3

131 97 90 110 90 7 7 30 14 35

92 40

by Beech et al. [243 ]. In addition they calculated the phonon density of states (PDOS). Bates and Eldridge [281 ] used a similar valencebond force model with stretching and bending force constants estimated from vibrational frequencies of CuO and BaYzO4. (The stretching force constants of this reference are included in table V. ) The results of these calculations for YBa2Cu307 gave reasonable agreement with the observed spectra with slightly too large frequencies (see table VI). They also showed figures of the normal coordinates for the Ag and Blu modes which were assumed to produce the strongest features in the Raman and IR spectra, respectively. The zone centre phonons of YBa2Cu307 were also calculated by Gupta [282] who also used a force constant model with central and angular force constants (for the central force constants see table V). In his first paper [282] Gupta used a tetragonal structure with the O(4)-sites and the tetragonal equivalent 0 ( 5 )-sites partially occupied (this may serve as a high temperature model). The results presented without an assignment of the symmetry of the phonons are included in table VI. In a subsequent paper [283 ] Gupta improved these calculations and presented results for tetragonal YBa2Cu306 and orthorhombic YBa2Cu307. In addition, dispersion curves were given for both compounds which revealed phonon anomalies. Within the framework of this model the author also published a calculation of

20

R, Feile / Lattice vibrations in high- T~ superconductors

Table IV Phonon frequencies of La2CuO4 as calculated by different authors (Weber [261 ], Brun et al. [ 12], Mase et al. [285 ], Feile [250], Prade et al. [ 286 ], and Cohen et al. [ 289 ] ) using the force constants given in table III. Frequencies (cm-~ ) Optical mode

ref. [261 ] a)

ref. [ 12]

E~ A~g Eg A~g

126 137 168 203

149 229 370 425

Eu A2u Eu A2u Eu A2u Eu

442 621

132 140 340 406 507 620 646

B2u

ref. [285] Raman 180 250 In~ared 89 170 220 300 Inactive

586

ref. [250]

ref. [286]

ref. [289] b~

126 185 516 605

105 218 333 553

i53.99 303.48 336.60 676.23

142 212 472 495 502 637 687

120(135) 163(169) 179(256) 305(498) 263(483) 549(569) 547(617)

i184.86 200.27 48.29 370.19 373.93 847.58 964.17

526

206

98.64

a) Symmetries are not given. b~ i denotes unstable modes. the zone centre p h o n o n s o f Bi2Sr2CaCu208 [284]. Mase et al. [285 ] gave a detailed group theoretical t r e a t m e n t o f the lattice v i b r a t i o n s in YBa2Cu307 a n d La2CuO4 in a d d i t i o n to the lattice d y n a m i c s calculations a n d explained the n o r m a l m o d e s in these systems. The small optical p h o n o n energies in the dispersion curves for La2CuO4 resulted from the rather small force constants as c o m p a r e d to those used by other authors, as the force constants were o b t a i n e d from matching the slope o f the acoustic p h o n o n branches to observed s o u n d velocities. A lattice d y n a m i c s m o d e l with first a n d second nearest neighbor force constants was used by M c M u l l a n et al. [253]. They calculated the influence of Eu-substitution in YBazCu307 on the p h o n o n frequencies a n d also the effect o f the exchange 0f160 by the isotope 180. Feile et al. [ 133] used a similar m o d e l for a calculation o f the lattice d y n a m i c s o f YBa2Cu307. The p h o n o n frequencies given there for the R a m a n active p h o n o n s were slightly too high because o f the m i s i n t e r p r e t a t i o n o f the d i s o r d e r i n d u c e d IR-active m o d e at 600 e m - 1 . The p h o n o n frequencies obt a i n e d with m o d i f i e d force constants in o r d e r to m a t c h the 500 c m - t are i n d u c e d in table VI. As the

m a i n interest in this work was to get the n o r m a l coordinates no efforts have been u n d e r t a k e n to get further frequencies right. The n o r m a l m o d e s for the Ram a n - a c t i v e Ag vibrations were shown in ref. [ 173 ]. These calculations have also been p e r f o r m e d on La2CuO4 [250] using the same force constants as for YBa2Cu3Ov for the equivalent bonds, a n d also the same force constants o f the Y-bonds were used for La-bonds. The results are included in table IV. In order to give an impression o f the lattice vibrations the n o r m a l coordinates o f the vibrations in both comp o u n d s are presented in figs. 7 and 8. The m a i n p r o b l e m o f all calculations m e n t i o n e d above is that a direct correspondence between calculated and observed energies is rather vague, because a pure force constant model, which m a y well be a p p l i e d to covalent crystals, is p r o b a b l y too simple for the substances u n d e r consideration here. It can for instance not account for the T O - L O splitting o f optical phonons because the C o u l o m b interaction is neglected. The n o r m a l coordinates, however, are less sensitive to the different models because they mainly reflect the s y m m e t r y o f the lattice. Briiesch a n d Bfihrer [7 ] have included the longrange Coulomb interaction o f the ions in YBa2Cu307

R. Feile / Lattice vibrations in high- Tc superconductors

21

Table V The short-range force constants used by different authors (Stavola et al. [80], Mase et al. [285], Bates and Eldridge [281 ], Gupta [283 ], McMullan et al. [253 ], Feile et al. [ 173 ], Briiesch and Biiherer [7 ], and Kress et al. 1287], (numbers were not given by Chaplot [288 ] ) for the lattice dynamics calculation of YBa2Cu307. The labeling of the atoms is as per Beech et al. [243 ], Force constants (N m - ~)

Cu(1)-O(1) Cu(2)-O(2) Cu(2)-O(3) Cu(1)-O(4) Cu(2)-O(4) Ba-O( 1 ) Ba-O(2) Ba-O (3) Ba-O(4) Y-O(2 ) Y-O(3) Cu(2)-Cu(2) Ba-Cu ( 1 ) Ba-Cu (2) Cu-Y Ba-Y Y-Y Ba-Ba O( 1 ) - 0 ( 4 ) O( 1 ) - 0 ( 2 / 3 ) 0(2)-0(3)

re£ [80]"7

reE [285]

reE [281] b)

refi [253]

refi [173]

reE [7]

120 110 110 110 40

36.8 18.4 18.4 17.2 17.2 7.4 9.2 7.4 7.4 7.4 7.4

160 140 140 140 110 100 80 80 80 100 110 50

140 130 130 130 90 92 87 87 87 106 106 75 72 76 79

131 97 83 90

105 50 50 50 26 15 15 15 15 22 22

24.5 12.3 i5.3

33 21 21 24 90 90 20 7 7 7 28 7 7 35 14 30

reE [287] b)

383

76

374

re£ [283] 100 100 100 50 100 50 50 50 25 50 50

16 16 13 18 16 18 85 30 41

a~ Further not specified force constants range from 8-30. b~ Bending force constants given there are not included in this table.

within a rigid-ion model. The short-range part, in addition, was described by valence-forces (table V). The Raman and IR-active phonons given by these authors are listed in table VI. The short-range parameters were adopted from related compounds as Cu20, BaO and SrTiO3 and were modified to fit the known spectra. As in ref. [133] the misinterpretation of the high frequency lines in the reported Raman spectra led to too large calculated phonon energies in this paper. In a subsequent publication [ 8 ] this was corrected. The authors calculated the phonon density of states and compared it to neutron scattering results and specific heat measurements. In addition, they discussed some zone boundary modes which were described as axial breathing modes and copper dimerization modes. A different route for the calculation of phonon frequencies of the high-Tc superconductors was chosen by Prade et al. [286] for LazCuO4 and Kress et al.

[287] for Y B a 2 C u 3 0 7. Within the framework of a shell model which treats the displacement-induced electronic charge density deformations the shortrange forces were represented by repulsive BornMayer potentials Vij=a;jexp(-bor)

.

(1)

Different force constants for the same atomic pairs at different distances were obtained quite naturally from these potentials. The number of parameters for the short-range interaction could thus be reduced. The values for these parameters together with those for the long-range interaction given by Coulomb forces between the ions and those which describe the ionic properties were determined by fits to known phonon dispersion curves of related compounds. This model is very sensitive to structural parameters [64,243 ] and the balance of short and long-range interactions. Therefore the structural and the potential

22

R. Feile /Lattice vibrations in high-T,, superconductors

Table VI Phonon frequencies of YBa2Cu307 as calculated by different authors (Stavola et al. [80 ], Mase et al. [ 285 ], Bates and Eldridge [ 281 ], Gupta [ 283 ], McMullan et al. [ 253 ], Feile et al. [ 173 ], Briiesch and Biihrer [ 7 ], Kress et al. [ 287 ], and Chaplot [ 288 ] ) using the force constants given in table V. The values in brackets give the frequencies of the corresponding LO-phonons. frequencies (cm-] ) Optical mode

ref. [80]

Ag

125 193 346 352 496 88 159 314 339 599 86 160 260 339 6OO

B2g

B3g

BLu

B2u

B3u

158 185 222 269 284 326 539 101 144 196 209 261 345 6O0 103 167 197 308 344 593 6O0

ref. [285]

ref. [281]

ref. [253]

40 69 164 177 305

145 218 424 448 555 104 139 480 586 636 99 140 559 583 635

202 328 436 457 516 151 230 458 482 594 133 229 463 486 594

38 54 85 111 214 323 339

117 200 254 424 474 575 628 104 180 200 557 590 626 642 102 168 203 453 513 594 647

244 263 327 436 465 495 567 166 233 276 470 523 596 597 170 243 272 407 468 520 595

ref. [173] Raman 115 183 412 421 500 77 139 352 420 568 76 139 293 426 588 Infrared 134 171 210 348 383 420 548 72 129 164 339 452 562 577 63 83 162 218 294 458 596

ref. [7]

ref. [287]

ref. [283] a~

ref. [288]

119 157 497 507 600 68 136 218 287 668 78 140 282 390 679

116 157 355 378 508 73 142 356 429 564 92 137 412 496 544

86 92 101 120 131 140 158 158 164 169 170 173 173 181 218

114 142 317 348 633 61 120 211 326 492 48 126 234 268 498

151 167 179 281 640

95(122) 155(184) 199(209) 312(312) 363(417) 509(529) 573(577) 103(104) 127(140) 191(193) 350(358) 368(449) 545(546) 573(573) 81(81) 121(121) 168(171) 198(204) 356(367) 369(416) 565(565)

404 424 438 442 445 455 457 488 488 491 493 498 50l 523 644 663 692 704 704 705 707

66(86) 92(97) 137(140) 145(208) 208(347) 427(499) 550(622) 84(84) 135(137) 170(179) 233(257) 276(414) 415(526) 527(559) 16(28) 94(96) 159(159) 182(194) 269(270) 280(528) 529(572)

148 167 261 297 682

165 217 289 671

a) Phonon frequencies given without a symmetry assignment.

p a r a m e t e r s h a d t o b e m o d i f i e d in o r d e r t o get s t a t i c and dynamic stability of the systems. Some force constants derived from these parameters are inc l u d e d i n t a b l e s III a n d V. It c a n b e s e e n t h a t t h e y are c o n s i d e r a b l y l a r g e r t h a n t h o s e u s e d i n o t h e r

m o d e l s . T h e r e s u l t s f r o m t h e s e l a t t i c e d y n a m i c s calc u l a t i o n s a r e i n c l u d e d i n t a b l e s I V a n d VI. I n a d dition to these results the authors determined the phonon dispersion curves and calculated the phonon d e n s i t y o f s t a t e s w h i c h r e p r o d u c e d t h e o v e r a l l fea-

23

R. Feile / Lattice vibrations in high- T¢ superconductors

F,

Rl,

685

185

516

126

526

OLa o Cu

.

6-~7

4cJ5

.

.

.

212

.

~-

687

~

582

~

472

,)

'

142

Fig. 7. Normal coordinates of La2CuO4. The symmetry of the modes and frequencies (cm -t ) as given by ref. [250] are indicated. For details see text. tures of inelastic neutron scattering experiments [ 5,6,9 ]. The frequencies for YBa2Cu307given in ref. [287 ] differed slightly from those in an earlier paper [ 124 ] as a result of improved calculations. It should be mentioned that although the number of parameters for short and long-range forces together with those describing the ionic system is similar or even larger than for the other models they may give some more insight into the physics involved. An open question are the large differences between the parameters of YBa2Cu307 [287] and La2CuO4 [286]. The normal coordinates of the optical phonons in YBa2Cu307 and YBa2Cu306have been published in the papers of Liu et al. [124] and Thomsen et al. [123], respectively, as results of the calculations mentioned above [ 287 ]. The influence of oxygen isotopic substitution on the phonon frequencies was calculated within this model and compared with experimental IR and Ra-

man results [ 179 ]. It was shown that the calculated frequency shifts coincide well with the observed ones. The same route for a calculation of phonon dispersion curves for YBa2Cu307 was taken by Chaplot [288]. He also introduced Coulomb and BornMayer potentials. In order to keep the number of parameters small the ions were regarded as rigid (rigidion model). The effect of polarisation of the ions was included in effective charges for them. As in refs. [286,287 ] and instability of the lattice was indicated by imaginary frequencies. The results for the optical phonon frequencies are listed in table VI for comparison. In the paper of Cohen et al. [289] the authors present detailed results on phonon frequencies and normal coordinates of La2CuO4 from an ab initio potential-induced-breathing (PIB) model. The phonon dispersion curves given there display an instability of the tetragonal lattice at the X-point as observed in neutron scattering experiments by Birge-

24

R. Feile /Lattice vibrations in high-To superconductors

Ag

~~

y

0

Ba

0

Cu

599

421

412

18~

1~5

~.:: f "

0o

g~g (626 ]

588 (556)

II

548

429

425 (426)

~8~

23~ (Z52)

~48

L~9 (1Z9)

219

76 (77)

171

t~4

("

-

b

(B2u) 595 (577)

458 (562)

254 (452)

218 (Z~[5)

L52 (154)

63 (t 2'])

5~ (72)

Fig. 8. Normal coordinates of YBa2Cu30~. The upper part gives the Raman-active modes, the lower one the IR-modes. The symmetry of the modes and the frequencies ( c m - ~) as given by refs. [ 173,250] are indicated. In the B28 and B:u modes the ions move along the xdirection. The B3g and B3u modes exhibit equivalent motions along the y-direction. For details see text.

R. Feile/Lattice vibrationsin high-Tosuperconductors neau et al. [69] and BOni et al. [70]. Even for the orthorhombic structure Cohen et al. predict a lower symmetry structure which should be more stable.

5. Isotopic effect The isotope effect in classical superconductors [290,291 ] gave strong indications for phonon mediated electron coupling in these systems. This could be well explained within the BCS theory [ 75 ] where the superconducting transition temperature T~ was related to the Debye-frequency ~OD of the lattice by

kBT~=l.14hO)Dexp(

N(O1)V )

(2)

with N(O)V as the effective electron-phonon coupling. The Debye-frequency is proportional to the inverse square root of the ionic mass m(coDocm -~ with o~= ½) in simple phonon theories. Investigations of the isotope effect in classical superconductors yielded o~= ½ for the ratio c~=

ATJT~ Am/m

(3)

supporting the validity of the BCS-theory in these systems, and the pairing was attributed to electronphonon interaction. The first experiments on the isotope effect in the high Tc superconductor YBa2Cu307 were reported by Batlogg et al. [16] and Bourne et al. [292] who both found ce~ 0. Slightly larger values of about 0.1 where obtained by Leary et al. [293 ] and zur Loye et al. [294]. Further measurements by Benitez et al. [231 ] and Morris et al. [295] yielded again values for a below 0.02. A crucial point of those experiments is the oxygen stoichiometry of the YBa2Cu307, as T~ decreases with decreasing oxygen content (Cava et al. [249]). A shift in Tc can therefore be due to a deviating oxygen stoichiometry in isotope exchanged samples instead of the isotope shift. This explains the different values reported for c~. From the (almost) vanishing isotope effect in their experiment Batlogg et al. [79] concluded that a non-phonon pairing mechanism is responsible for the high transition temperature in this compound. A further suggestion for the vanishing isotope ef-

25

fect was the following: a preferential exchange of the oxygen isotopes on other oxygen sites than those within CuO2-planes might have occurred. Then no influence on Tc is expected under the assumption that mainly these planes play the dominant role for the superconductivity of these materials. The test for oxygen substitution also in the CuO2 planes was performed by Raman spectroscopy. Already Batlogg et al. [79 ] presented Raman spectra which showed that all phonon modes above 335 cmconnected with oxygen vibrations shifted to lower frequencies upon ~80 substitution. This was confirmed by Cardona et al. [ 174 ] who, in addition, observed the changes of IR-active oxygen modes. The shifts were in agreement with predictions of lattice dynamics calculations. Similar results were obtained by Thomsen et al. [ 179 ] who investigated samples exclusively prepared with 180. The values for a from both references [ 174,179 ] were again below 0.1. Isotope shifts for other elements than oxygen were searched for by Lin et al. [296] who prepared samples with 63Cu and 65Cu. They observed no isotope shift. In (La, Sr)2CuO4 the isotope shift was investigated by Batlogg et al. [ 16 ], Faltens et al. [297 ], and Bourne et al. [298]. the values for ( 0 . 1 4 < a < 0 . 3 5 ) were discussed in the context of strong coupling theories [299,300] which can yield smaller isotope shifts compared to the BCS theory [301 ]. The authors concluded that pairing in (La, Sr)zCuO 4 may be phonon mediated. A calculation of the isotope effect within strong coupling BCS theory in both high Tc superconductors was performed by the Wette et al. [ 302 ]. As the other authors mentioned before, they found that this theory is consistent with the results from (La, Sr)2CuO4 hut not with those from YBa2Cu307. In the discussion of the isotope effect it has often been neglected that values of o~distinclty smaller than ½ are also observed in conventional superconductors, e.g. in d-metals. An explanation for this deviation was given by Garland [301 ] within a two band model. Strong coupling theories, e.g. from McMillan [299] yielded also values c~# ½, and anharmonicities and two-level-systems have been made responsible for deviations from the ideal value for a, and even a < 0 was obtained by Drechsler et al. [303305]. Similarly, Phillips [267,306,268] estimated

26

R. Feile / Lattice vibrations in high-T,, superconductors

values of a = 0 . 2 for (La, Sr)2Cu04 and a = O for YBa2Cu307 on the basis of models which included strong anharmonicities.

Acknowledgement The author gratefully acknowledges m a n y helpful discussions with P. Leiderer, who also carefully read the manuscript.

6. Summary The optical spectroscopy on the two series of high Tc superconducting materials (La, Sr)2CuO4 and YBa2Cu307 has added essential pieces for a solution of the puzzle of high Tc superconductivity. Features in the spectra which have been identified as the gap in the quasi-particle density of states have been observed in IR and R a m a n experiments. Its size from IR-results is comparable with weak coupling BCS predictions based on the k n o w n transition temperatures. An anisotropy of the gap is observed in R a m a n experiments. The values for the gap from these experiments indicate, however, strong coupling superconductivity. The symmetry of the gap function for the large gap is the same as of one Ram a n active vibrational mode. This p h o n o n has an asymmetric lineshape because of its strong interaction with an electronic c o n t i n u u m . The electronp h o n o n can be made responsible for the softening of this p h o n o n below Tc as shown theoretically. F r o m these spectroscopic results one could come to the conclusion that the superconductivity of high Tc materials La2CuO4 a n d YBazCu307 is at least partially due to the e l e c t r o n - p h o n o n interaction. Even the small isotope shifts in T~ may be accounted for in theories including lattice vibrations. Due to the large n u m b e r of IR and R a m a n studies the optical p h o n o n spectra are now known quite well. Many of the observed p h o n o n s have been reproduced by lattice dynamics calculations. The spectroscopy and particularly the R a m a n work has contributed considerably to the characterization of these materials. Several impurities have been identified with large sensitivity, The future work (already started) on the new Biand Tl-based superconductors will have to show whether the p h e n o m e n a observed in the "old" compounds as reviewed in this article are u n i q u e for all the oxide superconductors.

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28

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