Layer and bulk phonon modes in the YBa2Cu3O6 compound

PHYSICA ELSEVIER

Physica C 245 (1995) 48-56

Layer and bulk phonon modes in the YBa2Cu306compound Yu.E. Kitaev a L.V. Laisheva a M.F. Limonov a,*, H. Lichtenstern b, A.P. Mirgorodsky c, R.A. Evarestov d a A.F. loffe Physical-Technical Institute, Politekhnicheskaya 26, 194021 St.-Petersburg, Russian Federation b Institutfiir Angewandte Physik, Universitiit Hamburg, Jungiusstr. 11, 20355 Hamburg, Germany e Institute for Silicate Chemistry, Odoevskogo 24 / 2, 199155 St.-Petersburg, Russian Federation St.-Petersburg State University, Universitetskaya nab. 7/9, 199034 St-Petersburg, Russian Federation Received 1 September 1994

Abstract

The layer description of the phonon subsystem of the YBa2Cu306 lattice has been performed. A group-theory analysis and model dynamical calculations were carried out both for isolated layers and for the bulk YBa2CuaO6 structure. The correspondence between layer and bulk modes has been established. It has been shown that acoustic modes of isolated layers give rise to acoustic bulk modes Fac and interlayer bulk modes Finter, and optical vibrations of isolated layers induce intralayer bulk modes Fintra. The set of vibrations in the Brillouin zone center is given by the expression F = 3Fac + 15Finte r + 18rint~a.

1. Introduction

In this paper, we apply the layer description of phonon subsystems of perovskite-like superconductors [1-3] to the YBa2Cu306 compound. This method has proved to be successful in studying the vibrational properties of the YBa2Cu307 crystal [1]. It implies the joint consideration of the group-theory results, lattice-dynamical calculations, and experimental data. In Ref. [1], for group-theory analysis, we adopted a separation of the bulk crystal into individual layers, and this particular choice of indi-

* Corresponding author.

vidual layers was justified by dynamical calculations of bulk phonon frequencies and by comparison with optical and neutron scattering data thereafter. In the present paper we carry out the dynamical calculations for bulk YBa2Cu306 crystals and for individual layers which form this structure. We employ the following scheme. First, we separate the YBa2Cu306 bulk crystal into individual layers as shown in Fig. 1. Secondly, we analyze the phonon symmetry and perform model dynamical calculations of both individual layers and the bulk crystal. And thirdly, we establish a correspondence between the layer and bulk vibrations. Thus we confirm the validity of the layer approach for the description of phonon subsystems of the perovskitelike compounds under consideration.

0921-4534/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved

SSDI 0 9 2 1 - 4 5 3 4 ( 9 5 ) 0 0 0 7 8 - X

Yu.E. Kitaev et aL / Physica C 245 (1995) 48-56

49

2. Group theory analysis

Table 1 Phonon symmetry in Cu(1) and Y layers

2.1. Layer phonon symmetry

Cu(1)

Similarly to the separation of YBa2Cu30 7 into individual layers [1], we consider four types of layers in YBa2Cu306 compounds: Y-, Cu(1)-, Cu(Oc,)2-, and BaOBa-layers (Fig. 1). The symmetry of each layer is described by one of the 80 diperiodic space groups (DG) in three dimensions [4]. Thus, the bulk primitive unit cell includes pairs of Cu(Oc,) 2 and BaOaa layers having the symmetry of DG55(P4mu) as well as one Y-layer and one Cu(1)-layer with symmetry DG61(P4/mmm). The arrangements of atoms over the Wyckoff positions in these layers are presented in Tables 1 and 2, respectively. The phonon symmetry of isolated layers was determined using the method of induced band representations of space groups, whose detailed description could be found elsewhere [5]. Analyzing the layer phonon symmetry, we should take into account that the layer space group DG55 is a subgroup of the corresponding triperiodic group G = C]v(P4mm) and DG61 is a subgroup of G = D41h(P4/mmm). Induced band representations of a diperiodic group DG can be obtained from induced

Y Cu(Ocu)z BaOaa Cul BaOBa cu(Oc.)2

Y

P4/mmm(DG61)

F

M

X

(00)

(~) ~

(o½)

D4h

D4h

D2h

1

2

3

4

5

6

7

Cu(1)

-

la (000) C4v

a2u(Z) e,(x, y)

35-

35-

23-,4-

Y

lc (½½0) C4v

a2u(z) e,(x, y)

35-

25-

3+ 1+,2 +

band representations of a corresponding triperiodic space group G [5]. The results of the group-theory analysis of phonon symmetry in the Y, Cu(1), BaOBa and CU(Ocu)2 layers constituting the Y B a E C U 3 0 6 crystal are given in Tables 1 and 2. Tables 1 and 2 have the following structure. The first two columns contain the arrangements of atoms over the Wyckoff positions in the primitive unit cell. The Wyckoff positions are given in column 3 together with their coordinates and site symmetry groups. Column 4 contains the irreducible representations of site symmetry groups according to which the local atomic displacements (x, y, z) are transformed. The remaining columns give the labels of the induced representations in the k-basis, which determine the symmetry of normal vibrations. In the

Table 2 Phonon symmetry in Cu(Oc.) 2 and BaOaa layers

CU(Ocu)2

Y

BaO

P4mm

(DG55)

F

M

X

(oo) (½1) (o½)

~lt

a

F

C4v

C4v

C2v

1

2

3

4

5

6

7

Cu

OBa

la (00z) C4v

al(z) e(x, y)

1

1

1

5

5

3,4

(00~) Z

Fig. 1. Layer separation of YBa2Cu306 bulk crystal, correspondence between layer and bulk modes and schematic bulk mode dispersion curves along the F --* Z direction.Acoustic layer modes generate acoustic (a) and optical interlayer (b) bulk modes; optical layer modes generate optical intralayer (c) modes. The hatching indicates the structure fragments that participate in the corresponding vibration - individual atoms or layers.

-

Oc.

Ba

-

( ~1 z1) C4v

lb

al(z)

1

1

3

e(x, y)

5

5

1,2

2c (0½z) C4v

al(z) bz(y) bl(x)

1,2 5 5

5 1,2 3, 4

1,3 1,4 2, 3

Yu.E. Kitaev et al. /Physica C 245 (1995) 48-56

50

headings of these columns, the symmetry points in the Brillouin zone (BZ) are given together with their coordinates in the reciprocal lattice and point groups of the wave vectors. The labels of the layer group irreducible representations correspond to those of the related triperiodic group G. In these tables the labeling of space-group irreducible representations is that of Ref. [6], the labeling of the irreducible representations of site symmetry groups is that of Ref. [7], and the notations of Wyckoff positions follow Ref. [8]. From Tables 1 and 2, we can easily write down the full vibrational representation at any symmetry point of the BZ. For example, at the F-point for the Y and Cu(1) layers we obtain

r = r 3 + f f =pzu +Eu,

(1)

for the BaOBa layer we have F = 2 F 1 + 2 F 5 = 2A a + 2E,

(2)

We shall call the layer modes which correspond to a motion of a layer as an entity acoustic modes, and the others optical ones. It should be noted that the monatomic layers (e.g., Y- or Cu(1)-layer) induce only three acoustic modes. If the number of nonequivalent atoms in the layer primitive unit cell is N i > 1, the 3(N i - 1) optical modes arise.

2.2. Bulk phonon symmetry By using the same procedure, we have obtained the phonon symmetry of the bulk YBa2Cu306 crystals. The results of the group-theory analysis are presented in Table 3. The structure of Table 3 is similar to that of Tables 1 and 2. From this table we can write down the full vibrational representations at the symmetry points of the bulk BZ. For example, for the F, M, and Z-points

and for the Cu(Ocu) 2 layer the vibrational representation is

r = 4 F ( + F2+ + 5 r ; + 6 r f + r4- + 7r5-,

r = 2Fl + I"2 + 3Fs = 2A1 + a~ + 3E.

M = 3M~- + M~- + M~" + 2M + + 4M~"

(3)

Normal vibrations of isolated layers could be divided into "acoustic" and "optical" ones. The main idea of this classification is shown in Fig. 1.

+M 1+3M 2 +4M 3 +M 4 +6M 5, Z = 5Z~-Z~ + 6Z~- + 5Z 3 + Z 4 + 6 Z 5 ,

(4)

(5) (6)

Table 3 Phonon symmetry in the YBa2Cu306 crystals O41h

F (000)

M

Z

A

X

R

11 Q~o)

(oo½)

(~)"~

(o½o)

(o~)"

O4h

D4h

D4h

D4h

D2h

D2h

1

2

3

4

5

6

7

8

9

Cu(1)

lb (00½)

a2u(z) eu(x, y)

35-

35-

1+ 5+

1+ 5+

23-, 4-

1+ 3 +, 4 +

aEu(Z) %(x, y)

35-

25-

35-

2 5

3+ 1+,2 +

3+ 1+,2 +

al(z) e(x, y)

1+, 35 +, 5 -

1+~ 3 -

1 -I-, 3 -

5+,5 -

5+,5

1+, 3 5 +, 5

1 +, 2 3 + , 3 - , 4 +, 4 -

1 +, 2 3 +, 3-, 4 +, 4-

1 +, 3 5-

2-, 4 + 5+,5 -

1+,3 5 +, 5 -

2-, 4 + 5 +, 5 -

3 +, 4 1+,1-,2 +,2-

3 +, 4 1 +, 1-, 2+, 2 -

1 +, 2 + , 3 - , 4 -

5+,5 1-, 2 - , 3 +, 4 + 1+, 2+, 3 - , 4 -

1 +, 2 +, 3 - , 4 5+,5 5+,5 -

5 +,51-,2-,3 +,4 + I +, 2 +, 3 - , 4 -

1+,2-,3 +,41-,2 +,3 +,41 +, 2 - , 3 - , 4 +

1 +, 2 - , 3+, 4 1-, 2 +, 3 +, 4 1 +, 2 - , 3 - , 4 +

O,h Y

lc (1½0)

D4h Cu Oaa

Ba

Ocu

2g (00z) C4v 2h

a l(z)

I 1 (27 z) C4v

e(x, y)

4i

al(z)

5 +'

(0½z) bl(x)

5L 5 -

C2v

5 +, 5 -

b2(y)

51

Yu.E. Kitaev et al. / Physica C 245 (1995) 48-56

Similar relations for the BaOBa layer are

where Fl+ = Alg,

Ff = Big,

F5+ = Eg,

Ff=Azu,

F4=B2u,

Ff=Eu.

rL = Fac + Fopt = (A1 + E) + (A 1 + E) (7)

rc = ri°,or + r,°tra = (Alg + Eg + Aeu + Eu)

2.3. Symmetry correspondence between layer and bulk modes

Since for the Y-1236 crystal the layer groups (DG55 and DG61) are subgroups of the space group of the bulk crystal (Dlh) the symmetry correspondence between layer and bulk modes can be obtained by an induction procedure [5]. The vibrations of the Y-layer transform into the following bulk modes

rf

r3+z

;

rf

rf+z;,

(8)

whereas those of the Cu(1) layer transform into

+ {(A,g + A2u) + (Eg + E,)}

(16)

and those for the Cu(Ocu)2 layer are

rL = rac + rop, = ( g , + E) + ( g , + B, + 2E) =*' Fc = Fi.ter + Fiutra = (Alg + Eg + A2u + Eu) + {(nlg + nzu) + (Big + Bzu) + 2(Eg + Eu) }.

(17)

The vibrations of the BaOaa layer transform into

By summing up the contributions of all the layers of the YBavCu306 crystal, we obtain the acoustic, interlayer and intralayer contributions to the full vibrational representation at the F-point of the BZ:

F 1 = F 1 + + F 3 + Z ~ - + Z 3,

(10)

rc = rac + rop, = ru~ + r i ° , . + ri.,ra

r s ~ F 5 + +1"5- +Z~- + Z ; ,

(11)

Vf = F 3 +Z~-;

1"5- = F f

+Z~-.

(9)

and those of the Cu(Ocu)2 layer transform into FI~F(+Ff

+ {2Alg ( xx, yy, zz) + 2Eg( xz, yz)

+ Z ~ - + Z 3,

(12)

+ F 4- +Z~- +Z~-,

(13)

F5 ~ F5+ + 1'5- + Z~- + Z ; .

(14)

F2 = E l

Taking into account the layered structure of YBa2Cu306, we can assume that the normal mode spectrum is divided into the acoustic and two types of optical modes, i.e. intralayer and interlayer ones. (Their genesis is shown in Fig. 1.) The acoustic bulk modes are formed by in-phase motions of all the layers, i.e. by in-phase combinations of layer acoustic modes having the same amplitudes. If the acoustic layer modes have different phases or amplitudes they form optical interlayer modes. In turn, the intralayer bulk modes originate from the layer optical modes. Thus, for Y- and Cu(1)-layers the symmetry correspondence between layer (F L) and bulk (F c ) modes could be written as F L = rac + Fopt = (A2u + Eu) + (0) ==~Fc = Finter + Fiutra = (A2u + E u ) + (0).

= {Azu(Z ) + Eu(X, Y)}

(15)

+ 3 A 2 , ( z ) + 3Eu(X, Y)}

yy, =) + al (x , yy) +3Eg(XZ, yz) + 2A2o(z ) +B2, + 3Eu(X, y)},

(18)

where nonzero components of the polarizability tensor for Raman-active modes and dipole orientations for IR-active ones are given in parentheses. From the above group-theory analysis we come to the following conclusions: (1) In the YBa2Cu306 crystal the bulk phonons are divided into three acoustic modes, 15 interlayer optical modes and 18 intralayer optical modes. This agrees with the general formula obtained in Ref. [1], r = 3rao + 3(NL - 1)Finter + 3 ( N A - NL)Fi.t~ a,

(19) where N L is the number of layers per primitive unit cell in the layered crystals and Na is the total number of atoms per primitive unit cell.

Yu.E. Kitaevet al./Physica C 245 (1995) 48-56

52

Table 4 Force constants and structure parameters for YBa2CuaO6 (for 3D lattice and for isolated layers) Atoms involved

Distance (.A) Value or angle (deg) crystal

$1

Cu(1)-Oaa

S2

Cu-Ocu(Ocu)

S3 S4 S5 S6 S7 S8 S9 B1 B2 Hi H2

Y-Ocu(O~u) Cu-Oaa Ba-Oaa Ba-Ocu(O~u) Ocu-Ocu(O~u) Y-Cu Ba-Cul Ocu-Cu-O~u OBa-Cu-Ocu(O~u) Ona-Cu(1)/Cu(1)-Oaa Ocu-Y/Y-Ocu

1.822 1.944 2.395 2.411 2.763 2.937 2.840 3.196 3.544 89.1 97.2

a Force constant units a r e : mdyn/,~.

Si -

1.9 1.3 0.55 0.4 0.25 0.22 0.22 0.1 0.2 0.45 0.2 -0.3 0.14

mdyn/A;

n i -

layer 0 1.3 0 0 0.25 0 0 0 0 0.45 0 0 0

mdyn A; and

H i -

(2) From Eqs. (8)-(14), it can be seen that the frequencies of the corresponding intralayer bulk modes at the F and Z points should be close, i.e. the intralayer modes should be almost dispersionless along the F ~ Z direction.

3. Dynamical calculations 3.1. Potential function of the model

YBa2Cu306 lattice:

We have performed the calculations of the YBa2Cu306 vibrational spectra by using the program C R Y M E X [9], which was earlier successfully applied to ionic-covalent oxides [10], and recently to YBa2Cu30 7 [1]. The potential function of the YBa2Cu306 lattice was described by a general valence-force field (GVFF). The structural data (see Table 4) were taken from Ref. [11]. The GVFF set of force constants consists of ten diagonal stretching constants Si, two diagonal bending c o n s t a n t s B i , and two off-diagonal constants H i describing stretch-stretch interactions. Based on the set determined for the YBazCu307 potential function [1], we obtained the values of YBa2Cu306 constants in the following way. First, all the constants describing interactions with chain

oxygen atoms were omitted (these atoms are absent in the YBa2Cu306 lattice). Further, the magnitudes of the S i constants were varied according to the variations of interatomic distances in the YBa2Cu306 lattice relative to YBa2Cu30 7. And, finally, the quantities B i and H i were slightly changed (by no more than 0.1) for a better agreement between the calculated frequency values at the BZ centre F(0, 0, 0) and the experimental data [12,13].

3.2. Comparison of the calculated and experimental data at the F(O, O, O)-point of the BZ Table 5 contains the calculated frequency values for the BZ center F(0, 0, 0) and for the high symmetry points Z(0, 0, "rrc) and M(~r/a, ~r/a, 0). The corresponding experimental data are limited to frequencies of the Alg- and Big-vibrations obtained from Raman measurements [12], and those of the A2u- and Eu-vibrations derived from IR-spectra [13]. To our knowledge no investigations of the Eg Raman-active vibrations have been performed. Recall that B2u-vibrations are not active in first-order optical spectra. Since the experimental and the calculated frequency values (see Table 5) agree well, and taking into account the same good agreement obtained for the YBa2Cu30 7 compound earlier [1,3], we may conclude that the proposed model of the potentional function for the YBa2Cu30 7 ~ lattice seems to be physically reasonable.

3.3. LO-TO splitting of lR-active vibrations When determining the magnitude of L O - T O splitting, we used the atomic charges calculated in Ref. [14] and a dielectric permeability of ~ = 4.5 [13]. According to Ref. [14] the magnitudes of charges in the YBa2Cu306 crystal lattice are the following: + 2.44 (Y), + 1.99 (Ba), + 0.97 (Cu(1)), + 1.57 (Cu), - 1.65 (Oct), - 1.97 (Oaa). The calculated results for the magnitudes VLO are given in Table 5. We can see that the calculated and experimental values are fairly well related. In particular, a small L O - T O splitting for low-frequency modes and a large one for high-frequency vibrations are reproduced by the calculations. For example, the largest experimental splitting is 96 cm -1 (A2u-vibra-

53

Yu.E. Kitaev et al. / Physica C 245 (1995) 48-56

.=. a

0

I

©

lb.

~

~

2

+

o

0

u

0

i m

;~

I

,~

0

~

0

13

i ~

;~

,~

I

0

U

0

+m

+~

+

m

~

.O-

0

0

;~+

T~ +~ "

;~'

o

o

o

t'N

~-1

t¢3

+

+

+

~a

'i~ +

+

+

8.

~

g~

~

~

oo

oo

ooso

, ,

"0

g

o

oo

Sooooos o

©

o

.. 0

+ ~

+

0

0

o ~

©

+

+

0

0

o

0

0

I

I

+

0

=

0

+

~

~

%

~

o+

~

~

+(D

+

I

I

..~+

0

0

0

+~

+ "~

~uu

""

o

o

!~

oo

~

+

+

I~ .

~'~

+

""

+

+

~-

0

Yu.E. Kitaev et al. /Physica C 245 (1995) 48-56

54

tions of the Ocu atoms) and the calculated one is 75 cm -1. There is only one obvious difference: the experimentally observed splitting by 31 cm-~ of the A2,-vibration at 156 cm -1 is not reproduced in the model. Besides, it can be suspected that the measured splitting is overestimated due to the line broadening caused by an inhomogeneity of the sample, since all remaining low-frequency modes have essentially lower values of the LO-TO splitting.

immobile layers of Y and Cu(1) atoms, and the E,-vibration (126 cm -1) is formed by the out-ofphase displacements of BaOBa and Cu(1) layers relative to weakly displaced Y and Cu(Ocu) 2 layers. At the same time, the higher-frequency 3Eg- and 3E,-modes have to be attributed to intralayer vibrations, since each mode is mainly determined by vibrations of oxygen atoms belonging to the same layer, whereas the displacement amplitudes of the other atoms are nearly one order of magnitude less. As to vibrations of Alg and A2, symmetries which are connected with the atomic displacements along the z-axis (i.e. perpendicular to the layer planes), their interlayer character is less pronounced than that of the Eg- and E,-modes. But the high-frequency modes (2Alg , Big , 2A2, , and Bz.) are well-defined intralayer modes. The dispersion of the vibrational branches over the BZ is another important criterion specifying the vibrations as intra- and interlayer ones. The calculated dispersion curves in the (~, ~, 0) and (0, 0, ~) directions are given in Fig. 2, the frequency values for M(aT/a, aT~a, O) and Z(0, 0, aT~C) points in Table 5. The symmetry correspondence between the vibrations at the BZ-centre, in the (0, 0, sc) direction and at the Z-point is as follows:

3.4. Intra- and interlayer optical vibrations Table 5 presents relative amplitudes of atomic displacements of the longwave optical vibrations. Note that in our lattice dynamical model the layered structure of the YBa2Cu306 lattice is specified by neither the choice of parameters nor by their numerical values. Nevertheless, the analysis of the vibrational patterns clearly indicates the existence of the intra- and interlayer modes in agreement with the group-theory analysis. This is more clearly exhibited in the case of atomic displacements in the layer planes, i.e. vibrations of Eg- and Eu-symmetries. In fact, two lowfrequency Eg-modes and three Eu-modes have to be considered as interlayer vibrations, since they are reduced to displacements of separated layers as an entity relative to each other (see Fig. 1), i.e. the atoms belonging to one layer displaced in-phase with equal (or close) amplitudes. For example, the Egmode (66 cm-1) is formed by the in-phase displacements of BaOBa and CU(Ocu)2 layers relative to

V [

A1

A2

4A,g + 6Az. ~ 10A a ~ 5Z~ (A,g) + 5Z3(A2.).

(2o) Big + Bzu ~ 2B1 ~ Z~-(Blg) + Z4 (B2.),

(21)

5Eg + 7E. --* 12E 1 ~ 6Z~-(Eg) + 6 Z ~ ( E . ) ;

(22)

B2

B1

(cm'l)t

,oo

B1

El

.....

300

.

.

2 o 0 ~ ~ ' -- . . -. . . -. .

0 /' F

AI

.........

(~o)

(~o)

.

.

.

.

.

.

.

.

- . . .~. : . ~ .2. . .

/ ~. . . .

(~o)

//

(~,o)

M .(oo~) (oo~) (oo~)Z

Fig. 2. Calculated dispersion curves corresponding at the BZ center to even (solid lines) and odd (dashed lines) modes in YBa2Cu306 along the (~:, ~:, 0) and (0, 0, ~) symmetry lines.

Yu.E. Kitaev et al. / Physica C 245 (1995) 48-56

and that for the (g, ~, 0) direction is the following: 4A1~ + 7E u llA 1 --, 3M~-(Alg ) + 2M~-(B2g ) + 6M~(Eu),

(23)

Big + 7E u ~ 8B 1 M~-(B1g ) + M~" (A2g) + 6M~(Eu), (24) 5Eg + B2u ~ 6A 2 --) 4M~- (Eg) + M~- (A,u) + M4(B2.),

(25) 5Eg + 6A 2u llB 2 ~ 4 M ~ ( E g ) +3M~-(Blu) + 4 M 3 ( A 2 ~ ) .

(26)

It follows from Table 5 and Fig. 2 that all the modes which were attributed to intralayer ones according to the analysis of the atomic displacement patterns are dispersionless in the (0, 0, ~) direction. This result is trivial because of a negligible contribution of interlayer interactions in these vibrations. At the same time, most of the vibration branches corresponding to interlayer modes have a pronounced dispersion along (0, 0, ~:). Note that in agreement with our calculations and with experimental data [15], the acoustic branch (Eu-symmetry in the BZ centre) propagating along the (g, ~:, 0)-direction demonstrates an unusual behaviour. It has two maxima inside the BZ and a minimum at the boundary (see Fig. 2). Such a dependence may testify to a decrease of the structural stability of the YBa2Cu306 lattice in the M(~r/a, ~r/a, 0) point. Our preliminary calculations of its dynamical properties under hydrostatic pressure have revealed a stress-induced softening of the lowest mode in this point. The results of these calculations will be presented in a separate article. Investigations of the phonon dispersion branches by means of inelastic neutron scattering in different perovskite-like compounds, namely YBa2Cu307 [16], La2CuO 4 [17], and Nd2CuO 4 [15], confirm our conclusion of the existence of two types of optical branches: low-frequency ones with a significant dispersion in the (0, 0, ~) direction, and high-frequency ones which are dispersionless along this axis.

55

An additional argument for the separation of optical vibrations into intra- and interlayer ones was derived from the calculation of mode frequencies within the isolated layer approach. For this purpose, all the force constants describing interlayer interactions were set to zero (see column "layer" in Table 4). The corresponding frequencies are given in Table 5 (column "layer"). One can see that the frequencies of the 2Alg + 2Eg + 3A2u + 3E u vibrations vanish while the rest (2Alg + B~g + 3Eg + 2A2u + B2, + 3E u) remain nonzero, being different from the corresponding bulk frequencies. So, the frequencies of the intralayer atomic vibrations perpendicular to the layer really changed, i.e. the influence of neighbouring layers on these modes is fairly strong. In contrast, the frequencies of in-plane vibrations (Egand Eu-symmetries) changed weakly, i.e. they are "ideal" intralayer modes. Thus, using three criteria, namely (i) the atomic displacement patterns of the optical vibrations, (ii) the dispersion curve behaviour in the direction perpendicular to the layers and, (iii) the frequencies of isolated layer vibrations, the optical frequencies of YBa2Cu306 may be classified as (2A~g + 2E~ + 2A2, + 3Eu) interlayer and (2A1~ + Big + 3Eg + 3A2~ + B2~ + 3E~) intralayer modes. This set exactly coincides with the results of the group-theory analysis in Section 2.

4. Conclusions

The above considerations, including the grouptheory analysis, the lattice dynamical calculations, and the experimental data reveal, that the layered structure of YBa2Cu306 crystals directly influences the dynamical properties of the phonon subsystem the frequency values, the atomic displacement patterns, and the behaviour of the dispersion branches. It was shown that the set of vibrations in the BZ centre can be given by E = 3Fac + 15F~,ter + 18F,,tr a. This result is in agreement with the general formula received in Ref. [1] for any layered crystal: F = 3Fa¢ + 3(N L - 1)Fi,te r + 3(NA - NL)Fintra; in the case of the YBa2Cu306 crystal N L = 6 and NA = 12. Thus, it is possible to divide the optical vibrations

56

Yu.E. Kitaev et al. /Physica C 245 (1995) 48-56

into the inter- and intralayer modes. The interlayer bulk vibrations originate from acoustic layer modes, whereas the intralayer modes are induced by optical vibration of isolated layers. Optical interlayer vibrations are characterized by the displacement of the layers as an entity relative to each other, have low frequencies and significant dispersion in all directions of the Brillouin zone, including F --->Z. Optical intralayer vibrations occupy the high-frequency spectral region, and mainly originate from the displacement of oxygen atoms belonging to identically composed layers. These are dispersionless in the F ~ Z direction. The results of the present work testify to the validity of the layer approach proposed in Ref. [1] for the description of the phonon subsystem of perovskite-like high-temperature superconductors and related compounds.

Acknowledgements We thank M.B. Smirnov for his help in carrying out the dynamical calculations. This work was supported by the Scientific Council on HTSC problems, Grant "ELPHON" of the Russian State program "High temperature superconductivity".

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