Surface dynamics of Bi2CaSr2Cu2O8

Physica C 301 Ž1998. 55–71

Surface dynamics of Bi 2 CaSr2 Cu 2 O 8 U. Paltzer a , F.W. de Wette a

a,)

, U. Schroder ¨ b, E. Rampf

b

Department of Physics, UniÕersity of Texas, Austin, TX 78712-1081, USA b UniÕersitat ¨ Regensburg, D-93040 Regensburg, Germany Received 27 October 1997; accepted 30 January 1998

Abstract The high-Tc superconductor Bi 2 CaSr2 Cu 2 O 8 ŽBi 2:1:2:2. exhibits a superstructure in the bulk which is manifested at the surface. Helium atom scattering ŽHAS. from the Ž001. Bi–O surface has revealed unusual features of the surface phonons, such as the occurrence of apparently rather dispersionless surface modes and an only weak dependence of surface mode dispersion on direction in the surface. We report lattice dynamical calculations of a Ž001. slab of Bi 2:1:2:2 material, bounded by Bi–O surfaces. The calculations are based on shell model interactions, and take proper account of surface relaxation. We find that the above mentioned features of the surface phonon modes are direct consequences of the anisotropic surface structure. q 1998 Published by Elsevier Science B.V. All rights reserved. Keywords: High-Tc superconductor Bi 2 CaSr2 Cu 2 O 8 ; Surface phonon modes; Anisotropic surface structure

1. Introduction Ever since the discovery of the high-temperature superconductors ŽHTSC., their bulk phonon properties have been extensively studied by optical experiments w1–4x ŽRaman and infrared ŽIR.. and subsequently by inelastic neutron scattering w5x, in the hope to shed light on the role of phonons in the superconducting phenomena, and in general, to contribute to the overall physical characterization of these compounds. The phonon properties of thin films of HTSC have been studied by RHEELS w6x and inelastic helium atom scattering ŽHAS. w7x especially this latter technique has the potential of pro-

viding sensitive information about the surface localized phonons of the HTSCs. 1 In this paper, we report the first realistic calculation of surface localized phonons of a HTSCs compound; brief versions were published earlier w8,9x. For this study, we have chosen the Bi–OŽ001. surface of Bi 2 CaSr2 Cu 2 O 8 , which is a surface that can be easily produced and which is quite stable up to a temperature of about 400 K. The general interest and importance of phonons in HTSCs have been the motivation for our previous theoretical studies of the lattice dynamics and the phonon-related properties of these compounds. For a brief overview of this work and of our lattice dynam-

1

)

Corresponding author.

D. Schmicker, Doctoral dissertation, Max-Planck-Institut fur ¨ Stromungsforschung, Gottingen, 1993, unpublished. ¨ ¨

0921-4534r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 0 7 3 - 2

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U. Paltzer et al.r Physica C 301 (1998) 55–71

ical approach and its underlying principles, we refer to two recent publications w10,11x. In recent years, there has been an enormous increase in studies of the thin film properties of HTSC. From the viewpoint of lattice dynamics, the existence of surfaces of HTSC raises questions about the occurrence, importance and possible significance for superconductivity, of lattice vibrational modes which are localized at the surface, the so-called surface phonons. In this paper, we extend our shell model treatment of phonons in HTSC compounds to include surface phonons. This development can be broken down into three tasks: Ž1. formulation of additional conditions on the parameters of the interaction model so that the shell model can properly describe the ionic interactions in crystals with surfaces Žin our case: slab-shaped crystals.; Ž2. evaluation of the structural changes at the surface, i.e., the surface relaxation; and Ž3. evaluation of the lattice dynamics of the slab. These issues are dealt with in Sections 2–4, respectively.

2. Formalism The lattice dynamical model used in these calculations is the shell model ŽSM.. It takes into account long-range Coulomb interactions, short-range overlap interactions, which are represented by Born– Mayer potentials, and dynamic deformations of the electronic charge distributions in the form of ionic polarizabilities. In Refs. w10,11x, we have described the way in which the model parameters for specific compounds can be obtained from SMs of perovskites Žfor which inelastic neutron scattering data are available., and from SMs of previously treated HTSC compounds. Our previous studies of the lattice dynamics of bulk HTSC compounds have revealed that these materials exhibit quite delicate static and dynamic stability properties Žcf. Ref. w10x., so that requirements of static and dynamic stability of the crystal need to be strictly fulfilled, if the calculations are to be realistic and allow valid conclusions to be drawn about and from the phonon dynamics. This is particularly true for HTSC crystals with surfaces. The situation is the following: in the case of bulk crystals, a lattice dynamical model which is fitted to and

gives optimum agreement with experimental phonon dispersion data, may actually describe a crystal which is under compressional or dilatational stress Žsee Section 2.2.. While this ordinarily does not have any particular consequence for the dynamical description of the bulk Ži.e., infinite crystals., such a model may be totally inappropriate for crystals with surfaces. Since at a surface there is usually surface relaxation or reconstruction, a finite crystal described by a model involving unbalanced stresses will undergo non-converging dilatation or contraction when the surface is allowed to relax. To avoid such an unphysical situation, the dynamical model should not only lead to stable phonon dynamics Ži.e., real frequencies for wave vectors in the entire Brillouin zone ŽBZ.., but it should also satisfy the static equilibrium conditions of the crystal. 2.1. Static equilibrium conditions In our formalism, we use the static equilibrium conditions as formulated by Leibfried w12x according to which the energy of the unit cell is minimum, both with respect to the lattice vectors defining the unit cell, as well as with respect to the basis vectors defining the particle positions inside the unit cell. Writing the position vector of particle k in the unit cell l Ž l 1 l 2 l 3 . as r l sAPlqRŽ k . , k

Ž 1.

where AŽ Aab .Ž a , b s x, y, z . is the matrix defining the unit cell and R Ž k . is the basis vector of particle k ; the total static energy of the unit cell is Es

1 2

Ý l, m , kk

Vkk X Ž A P l q R Ž k . y R Ž k X . . .

Ž 2.

X

Here, Vkk X Ž r . is the potential of the interaction between the particles k and k X ; Vkk X contains Coulomb as well as short-range parts. Given Eq. Ž2., the Leibfried conditions are conditions I:

conditions II:

EE E Aab

s 0,

EE E Ra Ž k .

s 0;

Ž 3. Ž 4.

here, conditions I determine the size and shape of the

U. Paltzer et al.r Physica C 301 (1998) 55–71

57

Table 1 Measured and calculated Raman frequencies of Bi 2 CaSr2 Cu 2 O 8 in THz Symmetry

A1 g A1 g A1 g A1 g A1 g A1 g B1 g

Main vibrating atom

BiŽOŽ2., OŽ3.. Sr Cu OŽ1. OŽ3. OŽ2. OŽ1.

Experiment ŽRef. w13x.

Calculated

3.57 3.87 5.40 y 13.95 18.90 8.55

unit cell, and conditions II, the particle positions inside the unit cell. Our shell models are based on two-body central potentials, namely, Coulomb potentials for the long-range interactions and either Born–Mayer potentials w V BM Ž r . s a expŽybr .x or Buckingham potentials w V B Ž r . s a expŽybr . y cry6 x for the short-range interactions. The implementation of conditions I and II wEqs. Ž3. and Ž4.x for the Bi 2 CaSr2 Cu 2 O 8 ŽBi 2:1:2:2. structure leads to eight equilibrium conditions. ŽThese conditions and their derivation are given in Appendix A.. These eight conditions allow the determination of any eight model parameters.

This work

Ref. w14x

3.00 4.06 5.50 8.35 13.93 18.84 8.50

2.61 4.91 5.46 11.6 14.8 15.5 10.4

It should be kept in mind that fulfillment of the static equilibrium conditions does not automatically guarantee that the dynamics is stable, i.e., that no imaginary frequencies occur for any wave vector in the BZ. In fact, to achieve dynamic stability for Bi 2:1:2:2, it was necessary to add a small van der Waals component to the O–O interactions, by choosing the Buckingham potential Žcf. Table 2.. 2.2. Interaction model As a starting point for the interaction model to be used in the surface calculations, we use the SM

Table 2 Shell model parameters for Bi 2 CaSr2 Cu 2 O 8 . This set of parameters leads to fulfillment of the static equilibrium conditions Žcf. this table and Appendix A. Interaction

a ŽeV.

˚ y1 . b ŽA

Bi–OŽ2. Bi–OŽ3. a Bi–OŽ3. b Sr–OŽ1. Sr–OŽ2. Sr–OŽ3. Ca–OŽ1. Cu–OŽ1. Cu–OŽ2. OŽ1. –OŽ1. OŽ1. –OŽ2. OŽ2. –OŽ3. OŽ3. –OŽ3.

3000 3000 4339 3692 3010 3010 1670 1260 1832 1000 30 603 37 466 2859

3.00 3.00 3.00 2.93 2.93 2.93 3.10 3.35 3.35 3.00 3.00 3.00 3.00

a

˚6. c ŽeV A

Ion

Z Ž< e <.

Y Ž< e <.

k Ž e 2 rÕa .

Bi Sr Ca Cu OŽ1. OŽ2. c OŽ2. c OŽ3.

2.60 2.35 2.00 2.00 y1.99 y1.99 y1.99 y1.99

2.42 2.32 y0.50 3.22 y2.70 y2.70 y2.70 y2.70

1117 100 1387 975 165 700 Ž k 5 . 2146 Ž k H . 775

5303 4364

For Bi and OŽ3. in the same plane. For Bi and OŽ3. in different planes. c For the OŽ2. in the Sr–OŽ2. planes, we assume anisotropic polarizability with force constants k 5 parallel to the Cu–OŽ2. –Bi directions and k H perpendicular to these directions. b

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U. Paltzer et al.r Physica C 301 (1998) 55–71

developed by Prade et al. w14x for bulk dynamics calculations of Bi 2:1:2:2. To check whether this model represents a stress-free crystal required for surface calculations, the model was subjected to seven of the eight equilibrium conditions ŽA12– A18.. 2 It turned out that the model as constituted did not fulfill these conditions, 3 meaning that its parameter set is unsuitable for surface calculations. The next task therefore was to make judicious adjustments in a number of suitable parameters in order to obtain fulfillment of the seven equilibrium condi˚ Õ .. This procedure tions Žto the order of 10y6 e 2 Ar2 in principle determines seven parameters; but of course, parameters which do not appear in the equilibrium conditions are available for further adjustment. Next, we compared calculated optical frequencies ŽRaman and infrared. with the measurements w13,6x ŽTable 1.; some minor adjustments in some of the polarizabilities were made to improve the agreement. The main change was made to raise the maximum frequency to match the highest measured infrared frequency at 80 meV, which was also measured with high-resolution electron-energy-loss spectroscopy w6x ŽHREELS.. The shell model parameters obtained in this fashion are given in Table 2. 2.2.1. Structure For our calculations, we use the body centered tetragonal Žbct. structure of Bi 2:1:2:2, described by Tarascon et al. w15x. While the proper primitive cell would be a bct cell, the authors note that the structure can be described as neutral Bi 2 CaSr2 Cu 2 O 8 ˚ thick, parallel to Ž001.. These slabs are slabs 13 A weakly bonded together by long Bi–O bonds Ž2.71 ˚ . in the z-direction. In Fig. 1, we depict such a A neutral slab cell, which is tetragonal. This cell is particularly important in the description of the Ž001.-oriented slabs which we will consider below;

2 Because of possible screening in the a,b Ž x, y . planes, one cannot expect the Madelung sums in these planes, which appear in ŽA11., to represent the real situation. The condition ŽA11. has therefore been left out of account. 3 The model may simply not be optimal, or non-fulfillment may indicate that there are contributions from non-central interactions. The latter may well be the case for HTSC compounds, but whatever the cause, it renders the model unsuitable for surface calculations.

Fig. 1. Rectangular cell for the Bi 2:1:2:2 bct structure. ŽThe particle numbers, as used in the equilibrium conditions ŽA11. – ŽA18., are indicated..

we call it the primitiÕe slab cell. From the figure, the short-range interaction pairs of Table 1 can be identified.

3. Bulk dynamics Given the structure and the interaction model, the phonons of bulk Bi 2:1:2:2 can be calculated. We mentioned above that we require stable phonon dynamics, i.e., that the vibrational frequencies are positiÕe eÕerywhere in the bulk BZ. This is something that has to be checked every time a new parameter set is used in dynamical calculations. In the present case, the parameter set, obtained as described above, satisfied this condition of dynamic stability. In Fig. 2, we present the bulk phonon dispersion curves of Bi 2:1:2:2 along the symmetry directions of the bct BZ. Since the bulk unit cell contains 15 particles, there are 45 dispersion curves.

U. Paltzer et al.r Physica C 301 (1998) 55–71

59

Fig. 2. Calculated bulk phonon dispersion curves of Bi 2:1:2:2 along symmetry directions of the bct BZ Žindicated in the lower figure..

To our knowledge, no inelastic neutron scattering experiments have been performed for Bi 2:1:2:2. Therefore, comparisons of our calculations with experiments are possible only for measurements of optical mode frequencies and sound velocities. 3.1. Optical modes In Table 3, we compare the calculated Raman A1 g frequencies Žpresent calculation and those of

Ref. w14x. with those measured by Liu et al. w13x. Notice that all calculated frequencies of this work Žexcept the lowest. are within 5% of the measured ones. Also, notice the substantial improvement over the predictions of Ref. w14x. The 16% discrepancy in the present prediction of the lowest A1 g mode can possibly be accounted for by the fact that in that mode, the main vibrations are carried out by the Bi, OŽ2. and OŽ3. ions, which in our model occupy the

U. Paltzer et al.r Physica C 301 (1998) 55–71

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Table 3 Calculated and measured bulk sound velocities of Bi 2 CaSr2 Cu 2 O 8 in mrs ŽBZ stands for bulk Brillouin zone. Bulk direction BZ designation Žcf. Fig. 5.

²100: G –G

²110: G –X

²001: G –Z

Õl Õt 1 Õt 2

7368 Ž x . 4585 Ž y . 2280 Ž z .

7913 Ž x, y . 4012 Ž x,y y . 2239 Ž z .

6912 Ž z .

Average sound velocities

Calculated

Ref. w16x ŽT s 295 K.

Ref. w17x ŽT s 74 K.

Õl Õt 1 Õt 2

7158 4177 3109

3225 2029 2295

2846 1735

Õ l is the longitudinal sound velocity; Õt 1 and Õt 2 are the transverse sound velocities. The experimental sound velocities are from ultrasonic measurements.

ideal lattice positions Žcf. Fig. 1.. However, structure studies Žcf. Refs. w18–22x. have found that the superstructure in Bi 2:1:2:2 is associated with shifts in the Bi and OŽ3. positions in the Bi–O planes, and that additional oxygen ions can be located in these planes or between neighboring Bi–O planes. These deviations from the ideal structure could certainly account for the noted discrepancy.

4. (001)-Slabs of Bi 2:1:2:2

3.2. Bulk sound Õelocities In Table 4, we list the calculated bulk sound velocities together with experimental results obtained from ultrasonic measurements w17,16x. We see that the agreement between calculated and measured sound velocities, and the latter among themselves, is quite poor. There can be many reasons for these discrepancies, but we will not speculate on them here; it would require an extensive investigation, quite outside the scope of this work. The construction of any lattice dynamical model involves certain compromises. The model we used for Bi 2:1:2:2 was Table 4 Inward relaxations of the ions in the four outer layers at the Bi–OŽ001. surface of Bi 2 CaSr2 Cu 2 O 8 Layer 1 2 3 4

Ion Bi OŽ2. Cu Ca

Core 0.1092 0.1744 0.1826 0.0475

designed primarily to describe the oÕerall dynamical behavior of the HTSC, and in fact, it represents certain aspects of the dynamics quite well Že.g., the accurately measured Raman A1 g modes.; however, no attempt was made to account for the sound velocities.

Shell

Ion

Core

Shell

0.0987 0.0870 0.1121 0.0476

OŽ3. Sr OŽ1.

0.0921 0.0717 0.0588

0.0926 0.0478 0.0527

˚ The numbers give the shifts from the bulk positions in A.

As was mentioned above, the bulk Bi 2:1:2:2 can be considered to consist of neutral ‘slabs’ Žcf. Fig. 1. which are weakly bound between adjacent Bi–O layers, and this is where the structure cleaves most easily. Thus, upon cleavage Ž001.-oriented Bi–O surfaces are created, and they turn out to be quite stable Žup to 400 K.. For our surface calculations, we therefore consider Bi 2:1:2:2 slabs bounded by two Ž001. Bi–O surfaces, i.e., one on either side of the slab. The basic unit of the slab unit cell Žwhich extends between the two surfaces. is the primitive slab cell of Fig. 1. The thinnest possible slab contains just one primitive slab cell. In this slab, the Bi–O surfaces are mirror images of each other, with the mirror plane being the plane of the Ca ions. Unfortunately, such a slab is too thin for identifying the real surface modes. In fact, the thinnest slab which has mirror symmetry and for which the surface modes can be confidently identified has a thickness of five primitive slab cells. All the surface dynamical calculations have been performed on this ‘five cell slab’.

U. Paltzer et al.r Physica C 301 (1998) 55–71

61

the Bi 2:1:2:2 slab, bounded by two Bi–O planes. This relaxation has to be evaluated before the slab dynamical calculations can be carried out. Both calculations are carried out with the shell model of Section 2.2, which satisfies the equilibrium conditions in the z-direction and gives rise to stable bulk dynamics. In the relaxation calculation, we follow the procedure used by Reiger et al. w23x for the perovskites KZnF3 and KMnF3 . This is an iterative procedure in which the forces on the cores and the shells of the ions are evaluated independently. In the present calculations, the net forces on the particle are reduced, in seven iterations, by a factor of about 10y9 of their values in the unrelaxed slab. ˚ Fig. 3. Calculated surface relaxation of the Bi 2:1:2:2 slab in A.

4.1. Surface relaxation When a crystal slab is formed by cutting a slice out of bulk material, the forces on the particles at or near the surface are no longer vanishingly small Žas they are in the bulk., and as a consequence, the particles will move to new equilibrium positions; that is, the surface will reconstruct or relax. In the latter case, the shifts in the positions are only in the direction perpendicular to the surface Ž z-direction.; this happens when in the Ž x, y . planes all particle positions are points of inversion. This is the case for

4.1.1. Results The relaxation shifts of the cores and shells in the z-direction, of the four outer layers ŽBi–O, Sr–O, Cu–O and Ca; cf. Fig. 1., are shown in Fig. 3 and given in Table 5; all shifts are inwards, i.e., towards the bulk. The shifts of the cores of Bi, OŽ2. and Cu, which are lined up along the c-direction, are such that the Bi–OŽ2. distance is increased by 3.2% from its bulk value, while the OŽ2. –Cu distance remains the same. The intra-layer offset in the Cu–OŽ1. layer is reversed and increased by a factor of 10, resulting in a decrease in the distance between the Cu- and Ca-planes of 8.2%, while the distance between the OŽ1.- and Ca-planes remains unchanged.

Table 5 Sound velocities on the Bi–O surface of Bi 2 CaSr2 Cu 2 O 8 in mrs Calculated Surface direction

²100:

²110:

SBZ designation Žcf. Fig. 5.

GyX

GyM

Õl Õt 5 Õt H ŽRayleigh.

6649 Ž x . 4844 Ž y . 1286 Ž z .

6759 Ž x, y . 4052 Ž x,y y . 1594 Ž z .

Experimental Rayleigh sound velocities Ž Õ R s Õt H., measured with Brillouin scattering Reference ÕR w24x w25x w26x

1525 " 150 1540 " 66 1540–1570 1290–1320

SBZ direction isotropic ²100:, ²010:, ²110: ²100: ²010:

Õ l is the longitudinal velocity; Õt 5 is the transverse velocity with particle motion parallel to the surface; Õt H is the transverse velocity with particle motion perpendicular to the surface; this is the Rayleigh sound velocity Ž Õ R ..

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U. Paltzer et al.r Physica C 301 (1998) 55–71

It is interesting to note that in the bulk the Bi–O ˚ above plane is rumpled, i.e., the O-ions lie 0.22 A the plane of the Bi-ions Žassociated with this rumple is a dipole moment of the plane.. As a result of the relaxation of the free Bi–O surface layer, this rumple ˚ Finally, the relaxis increased by 10% to 0.24 A. ations at the bottom surface of the slab are the mirror images of those at the top surface. 4.2. Slab dynamics To obtain an overview of the dynamics of the slab, and in order to identify the surface modes, we

first performed a calculation on a slab of which the unit cell consists of three primitive slab cells instead of five. The slab has a plane of inversion and is sufficiently thick to identify the surface modes. Since the slab has translational symmetry in the x, y plane only, there exist only two-dimensional Ž2D. wave vectors q 5 , which are to be chosen in the primitive 2D Brillouin zone ŽBZ.. In Fig. 4, we show the slab dispersion curves for wave vectors q 5 along the symmetry directions G y X, X y M and M y G of the 2D BZ ŽSBZ.. For this slab, the unit cell contains 45 independent particles and, therefore, there are 135

Fig. 4. Calculated phonon dispersion curves of a relaxed ‘3-cell’ Bi 2:1:2:2 slab, along the main symmetry directions of the surface BZ ŽSBZ, indicated in the lower figure..

U. Paltzer et al.r Physica C 301 (1998) 55–71

Ž3 = 45. dispersion curves, a number of which are degenerate along the symmetry directions G y X, X y M and M y G . The full-drawn lines represent the surface Žlocalized. modes. For surface modes with frequencies above 8 THz, it is primarily the OŽ3.-ion in the top layer that vibrates, whereas, for surface modes below 6 THz, it is primarily the Bi-ion. The symbols indicate the dominant vibrational character of the modes: SP5 indicates vibration in the surface in the direction of the wave vector q 5 Žlongitudinal vibration., SPH vibration along the surface normal, and SH vibration in the surface, perpendicular to q 5 ; SPH and SH are transverse vibrations Žfor a full discussion of surface mode vibrations, we refer to Ref. w23,31x.. We note that the slab dynamics does not show any soft modes, which indicates that the ideal surface, that is used in the calculation, is stable under the interaction model; it shows no tendency to reconstruct, or form a superstructure. This is in contrast to what is found experimentally. It will turn out that the experimental surface mode dispersion curves can only be understood if the proper anisotropic superstructure is taken into account. 4.2.1. Sound Õelocities on the Bi–O(001) surface In Table 4, we present the results for the calculated sound velocities at the Bi–O surface. The results show anisotropy for the three velocities: the longitudinal velocity Õ l ; the transverse velocity Õt 5 , with particle motion parallel to the surface; and the transverse velocity Õt H , with particle motion perpendicular to the surface; this is the Rayleigh mode velocity Õ R . In the bottom part of the table, we list the values of the Rayleigh velocity Õ R , measured with Brillouin scattering. Unexpectedly, the first experimental entry w24x shows an isotropic Rayleigh velocity Õ R , while the second entry w25x shows the same Õ R for the for equivalent directions ²100: and ²010:, but also for the inequivalent direction ²110: Žcf. Fig. 5.. Finally, the last entry w26x shows different Õ R ’s for the equivalent directions ²100: and ²010:. We should keep in mind, however, that our statement about the equivalence of ²100: and ²010:, and the inequivalence of these two with ²110:, applies only to the ideal surface. But as we pointed out above, the physical Bi–O surface may be far from ideal.

63

Fig. 5. Ža. Simple cell Ž a1 , a 2 ., 62=62 ŽR458. cell Ž A1 , A 2 ., and 5-fold superstructure cell Ž A1 , 5 A 2 .. Žb. Surface Brillouin zones Žheavy outlines., defined by Ž B1 , B2 ., and Ž B1 , 15 B2 ., respectively. In the latter case, four extended zones are also shown. Žc. SBZ of the 19-fold commensurate superstructure Žheavy outline., and 18 extended zones.

It is interesting to note that, aside from the noted discrepancies between calculated and measured Rayleigh velocities, their Õalues are in good agreement. We attribute this to the fact that in the calculation, the surfaces were very carefully relaxed so that our shell model apparently describes the dynamics of Bi 2:1:2:2 quite well, even at the surface. This is very gratifying because neither bulk nor surface sound velocities were taken into account in determining the shell model parameters. 4.3. Helium atom scattering (HAS) measurements The structure and the dynamics of the Bi–OŽ001. surface of Bi 2:1:2:2 have been investigated by elastic and inelastic helium atom scattering by Skofronick and Toennies w7x and Schmicker 1. For the discussion of the measurements, we first need to define the relevant surface unit cells and Brillouin zones. In Fig. 5a, we show a primitiÕe 2D surface unit cell defined by the basis vectors a1 , a 2 , which lie in the ²100: and ²010: directions. However, since the superstructure is in the ²110: direction, it is more convenient to work with the non-

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U. Paltzer et al.r Physica C 301 (1998) 55–71

primitive 62 = 62 ŽR458. cell defined by A 1 , A 2 . Finally, the cell for the commensurate 5-fold superstructure is defined by the vectors A 1 and 5 A 2 . In Fig. 5b, we depict the surface Brillouin zone ŽSBZ. of the cell A 1 , A 2 , defined by the reciprocal lattice vectors B1 , B2 , and the SBZ of the 5-fold superstructure, defined by B1 and Ž1r5. B2 . For later use, we have indicated in the big B1 , B2 zone, four extended zones of the 5-fold superstructure zone. Fig. 5c shows the SBZ Žheavy outline. for a commensurate 19-fold superstructure in the ²110: direction, as well as 18 extended zones. 4.3.1. Helium atom scattering [27] (HAS) This is, by virtue of its unique surface sensitivity, non-destructive interaction and high resolution, an ideal tool for the study of the 2D structures of surfaces, and of surface vibrations with energies up to 30–50 meV. 4.3.2. Diffraction In Fig. 6, we show the diffraction patterns along the ²110:, ²100: and ²110: directions, measured ˚ y1 and surface with incident He wave vector of 5.7 A temperature Ts s 310 K. The closely spaced peaks in

Fig. 6. Angular distributions of helium atom scattering along the ²110:, ²100: and ²110: direction at the surface. Note the approximate 5-fold superstructure in the ²110: direction.

Fig. 7. Time-of-flight ŽTOF. HAS spectra of the Bi–OŽ001. surface in the ²110: direction; surface temperature Ts s 310 K, ˚ y1 , incident angles u i s 408, 418, incident wave vector k i s 5.8 A 428, 438. R, O1 , O 2 , O 3 and O4 are the original designations of the measured surface modes.

the ²110: direction reveal a 4.76-fold incommensurate superstructure compared to the ²110: direction. This incommensurate Žquasi. 5-fold superstructure had been detected earlier by scanning tunneling spectroscopy w18x and by EELS w19x; it is the manifestation at the surface of the bulk superstructure found for Bi 2:1:2:2 w20–22x. It will turn out that the experimentally measured surface vibrations of Bi– OŽ001. can only be understood if this anisotropic superstructure is properly taken into account in the dynamical calculations. 4.3.3. Inelastic scattering Time-of-flight ŽTOF. HAS spectra for the Bi– OŽ001. surface were obtained for the ²110: and ²100: directions Ži.e., along G y 12 M and G y X in Fig. 5b,c.; some of the spectra along G y 12 M for ˚ y1 and surincident helium wave vector k i s 5.8 A face temperature Ts s 310 K are shown in Fig. 7. The designations of the peaks in these spectra: R, O 1 , O 2 , O 3 , and O4 indicate, respectively, the Rayleigh mode, which is the acoustic SPH surface mode, and what were initially believed w7x 1 to be optical surface modes. The results of all the TOF

U. Paltzer et al.r Physica C 301 (1998) 55–71

measurements for wave vectors along G y 12 M Ž²110:. as well as along G y X Ž²100:. are presented together in Fig. 8. In the figure, the wave vectors along G y X were shortened by a factor 1r62 to facilitate the comparison and bring out the following characteristics: Ž1. the modes O 1 , O 2 , O 3 , O4 appear to be quite dispersionless Žparticularly O 2 ., and Ž2., the dispersion curves O 2 , O 3 , O4 along the ²110: and ²100: directions are quite similar to each other. Neither of these characteristics are generally found for customary surfaces, and in fact, were not found in our simple slab calculation of Bi–OŽ001. shown in Fig. 4. It will be seen below that the HAS measurements cannot be understood unless the superstructure is taken into account in the calculation. As a first step in this direction, we change from the simple 2D unit cell defined by a1 , a 2 , to the 62 = 62 ŽR458. cell defined by A 1 , A 2 Žcf. Fig. 5a., and we use the latter to perform a slab calculation for wave vectors along G y 12 M. The results are shown in Fig. 9: in the l.h. panel, the dispersion curve of the Rayleigh mode ŽSPH . and in the r.h. panel the longitudinal mode SP5 . These are the two surface modes in which the particles vibrate in the scattering plane, and which are therefore accessible by the helium scattering. We notice that only a few of the experimental points lie on or near the calculated dispersion curves, so that these calculations do not account for the measurements.

Fig. 8. Measured surface phonon dispersion data: I, ²110: direction; =, ²100: direction.

65

Fig. 9. Calculated dispersion curves for the Bi 2:1:2:2 slab, calculated on the basis of the isotropic 2D unit cell Ž A1 , A 2 .. The dotted curves are the bulk dispersion curves; the I indicate the experimental points. The solid curve in the l.h. panel is the SPH ŽRayleigh. mode; in the r.h. panel the SP5 mode.

4.3.4. The incommensurate 5-fold superstructure The next step is to incorporate the superstructure into the calculation. This should be done by using the anisotropic 2D unit cell defined by A 1 , 5 A 2 Žcf. Fig. 5a.. However, this makes the calculation too large for our present computational assets, because, while in the calculation with the 2D cell A 1 , A 2 , we had 150 particles in the slab unit cell, for the 5-fold superstructure we have 750 particles in the slab unit cell. Fortunately, the symmetry of the problem can be exploited to avoid such a large calculation. To explain this, we refer to Fig. 5b. The SBZ of 5-fold supercell is the rectangular heavy outline around G , bisecting the vectors "B1 ," 15 B2 . In the heavy outlined square of Fig. 5b Žthe BZ of the cell A 1 , A 2 . are also drawn two extended BZs of 5-fold structure on either side of the central BZ. By symmetry, the central lines of these extended cells Ždashed lines. are equivalent with the line G y 12 M, because they are related by the reciprocal lattice vectors " 15 B2 and " 52 B2 . The upshot is that the results of a calculation based on the anisotropic unit cell Ž A 1 , 5 A 2 . for wave vectors along G y 12 M in the SBZ Ž B1 , 51 B2 ., can also be obtained as the sum of the results of a much simpler calculation based on the

66

U. Paltzer et al.r Physica C 301 (1998) 55–71

Fig. 10. Folding of the Rayleigh dispersion sheet in the upper r.h. quadrant of the SBZ of Fig. 5b Žfor discussion, see text..

symmetric unit cell Ž A 1 , A 2 ., for wave vectors along G y 12 M, and those pointing to lines equivalent to G y 12 M, in the symmetric SBZ Ž B1 , B2 .. This is made pictorial in Fig. 10, where we show a 3D wave vector-energy diagram of the upper right-hand quadrant of the SBZ of Fig. 5b. The diagonal curved lines on the front faces of the cube-like structure are the intersections of these faces with the Rayleigh mode dispersion sheet drawn inside the cube. Similarly, the dashed-dotted Žslightly. curved lines are the intersections of vertical planes erected on the central lines of the extended zones at 0.2 and 0.4 along ²110:, with the Rayleigh sheet. If these dasheddotted curves are projected back on the left front face, which is the energy-wave vector plane through G y 12 M, we have the results that would be obtained for the true 5-fold superstructure calculation, for wave vectors along G y 12 M. In other words, three calculations for the A 1 , A 2-structure yield the same result as one calculation for the five times larger A 1 , 5 A 2 structure. 4.3.5. The commensurate 19-fold superstructure The 5-fold incommensurate superstructure was used to approximate the 4.76-fold incommensurate superstructure, detected by the helium diffraction. If we approximate the latter by a 4.75-fold superstructure, then a 19-fold superstructure would be commensurate Ž4 = 4.75 s 19.. Since a lattice dynamical calculation has to be performed with a truly periodic

cell, the proper 2D unit cell for the slab calculation would be the A1 , 19A 2-cell. In Fig. 5c, we show the SBZ of this cell Žheavy outline., as well as nine extended zones on either side of it. The same arguments, that applied to reducing the 5-fold superstructure calculation to three simple calculations, can be applied to reduce the 19-fold superstructure calculation; it can be reduced to a calculation for the A 1 , A 2-cell, for wave vectors along G y 12 M and along the central lines of the extended cells in the r.h. upper quadrant of Fig. 5c, followed by projection of the dispersion curves so obtained on the energywave vector plane through G y 12 M Ži.e., the left front panel of the ‘cube’ in Fig. 10.. The surface phonon dispersion curves which are obtained in this fashion are shown in Fig. 11; the Rayleigh branches ŽSPH . on the left, and the longitudinal mode branches ŽSP5 . on the right; the experimental points are indicated by the open squares. It is seen that these SPH and SP5 dispersion curves jointly cover the entire area in the wave vector-phonon energy plane where the experimental points are located. So, compared to the results of Fig. 9, we have made important progress. Nevertheless, some important facts still need to be explained, namely, why the experimental points are arranged the way they are, for instance, in a narrow band around 6 MeV, and one in the range 9–10 meV. To gain further insight, we have to

Fig. 11. L.h. panel: multiple Rayleigh dispersion curves SPH obtained for wave vectors along G y 12 M from the process of folding, for the case of the commensurate 19-fold superstructure. R.h. panel: similar for the mode SP5 .

U. Paltzer et al.r Physica C 301 (1998) 55–71

consider the so-called scan curÕes of the experiments in an extended-zone representation of the experimental points. A scan curve represents the wave vector-energy relation, for which, for given incident angle u i and momentum k i of the helium atoms, the kinematical conditions of energy- and crystal momentum conservation are fulfilled Žfor details, see Ref. w28x.. In Fig. 12 are plotted the scan curves Ždotted lines. for the experimental scattering condi˚ y1 and tions, namely incident wave vector k i s 5.8 A incident angles Žwith surface normal. u i between 388 and 448; the experimental points are also indicated. If, in Fig. 12, we had also displayed the dispersion curves of Fig. 11, the intersections of the latter with

67

the scan curves would indicate possible helium–phonon interactions. However, whether such interactions will lead to actual measurements depends on the strengths of the differential reflection coefficients Ž scattering cross-sections. along the scan curves. The scattering cross-sections depend on phonon dispersion, temperature Žphonon occupation., helium– surface interaction etc. Žfor details, we refer to Refs. w27,28x.. We have calculated the cross-sections as derived in the distorted wave Born approximation w27x, using the dispersion curves of Fig. 11. The peaks in the cross-sections with strength G 50 Žarbitrary units. have been indicated on the scan curves by qŽthe strongest peak in the cross-section is 400

Fig. 12. Scan curves Ž k i s 5.8Ay1 , u i between 388 and 448. in an extended zone scheme. Measured phonon energies are indicated by I; B are replicas of these points in the extended zones. Peaks of the differential reflection coefficients are indicated by q. Upper l.h. corner: differential reflection coefficient along the u i s 40.58 scan curve.

68

U. Paltzer et al.r Physica C 301 (1998) 55–71

in these units.; the actual measurement points are indicated by the solid squares Žthe open squares are the replicas of these points in the extended zones.. An analysis of the scattering cross-sections shows that the measured surface modes with energies - 6 meV Ži.e., the modes labeled R, O 1 , and O 2 in Fig. 8. are mainly due to the SPH vibrations of the Rayleigh mode branches Žl.h. panel of Fig. 11., whereas the modes with energies G 6.5 meV ŽO 3 and O4 in Fig. 8. are manifestations of the perpendicular components of the SP5 modes Žr.h. panel of Fig. 11.. The comparison of the calculated peak positions with the measurements can be summarized as follows: Ž1. the acoustic branches below 5 meV show the correct dispersion and a number of measured points are reproduced correctly; Ž2. as the energy increases, the dispersion becomes flatter and the dispersion of O 2 Žf 6 meV. is represented quite nicely; Ž3. the calculations do not show any scattering strength in the region of about 7 to 9 meV where there are also no measurements; Ž4. the high lying SP5 mode O4 is predicted only partially, but this is due to the strong diminution of the reflection coefficients for larger energies and wave vectors. We regard the successes of points 1–3 as extremely gratifying, considering the fact that our shell model was constructed with relatively little experimental input about bulk Bi 2:1:2:2; Žthere exist, for instance, no experimental phonon dispersion curves.. We emphasize that taking account of the anisotropic superstructure was absolutely essential for getting this good agreement with experiment. In this connection, we should stress that the superstructure could only be taken into account in a purely geometric way. But in the real crystal, the superstructure is caused by changes in the interactions which we do not know, and which result in position shifts Žmainly of the oxygen ions in the surface., which we also do not know Žbut about which there exist reasoned speculations; cf. Ref. w13x.. So a better knowledge of the precise positions of the ions in the superstructure might result in still better agreement between measurement and calculation. On the other hand, the existing discrepancies might also be due to variations in bulk and surface structure characterizations. For instance, certain variations in the incommensurate superstructure periodicity at the surface, varying from 4 to 6 have been detected with STM ŽRef. w19x..

Since the intensities of the inelastic peaks are relatively low, and that of the elastic peak relatively high Žcf. Fig. 7., one can conclude that there existed a fair amount of disorder in the surface, which could have been caused by the simultaneous presence of superstructures of different periodicity. Nevertheless, it would be interesting to see which improvements could be made if the actual ionic positions in the surface were known. 4.3.6. Similarity of G y M and G y X dispersions The discussion up to this point has dealt with the interpretation of surface phonon data along the direction G y 12 M Ž²110:.. From Fig. 8, it was seen that the measurements Ž=. taken along the direction G y X resemble very closely those taken along G y 12 M ŽI.. This similarity can be explained as a consequence of the anisotropic superstructure. To see this, we recall the argument about the equivalence of a full 19-fold superstructure calculation for wave vectors along G y 12 M, and the simpler calculation based on the 2D A 1 , A 2 unit cell, for wave vectors from G to the central lines of 18 extended cells in Fig. 5c, as well as along G y 12 M itself. The direction G y X cuts all these central lines, and for wave vectors from G to each of these intersections the results are the same as for the projections of these wave vectors on the direction G y 12 M. Therefore, since dispersion curves are continuous and smooth, the curves along G y X will resemble those along G y 12 M Žfor wave vectors, a factor 1r62 shorter. more and more as the density of the intersection points gets closer, that is, as the commensurate periodicity length in the ²110: direction becomes larger. In the limit of infinite commensurate periodicity length, which means the absence of periodicity in that direction, the dispersion curves in all other directions are the same, except for a scaling factor Žsin f .y1 for the wave vectors, where f is the angle between a given direction and the direction of infinite periodicity length.

5. Summary We have been able to interpret the unusual features of the HAS measurements of the surface phonons on the Bi–OŽ001. surface of Bi 2 Ca-

U. Paltzer et al.r Physica C 301 (1998) 55–71

Sr2 Cu 2 O 8 in a lattice dynamical calculation which is based on the shell model. Since no neutron scattering data are available for Bi 2:1:2:2, the parameters of this shell model were obtained on very general grounds Žfrom perovskites and other high-Tc compounds., with only the values of the maximum bulk frequency, and the frequencies of the Raman modes as experimental input. After refining the model by subjecting it to the bulk equilibrium conditions, and after allowing the surface to relax, the dynamical calculations were performed on a slab bounded by Bi–OŽ001. surfaces. It was shown that the peculiarities of the surface phonon spectra, namely, the appearance of quasi-dispersionless modes, and the similarity of the phonon spectra in different directions are direct consequences of the anisotropic superstructure on the surface. It is probably no accident that this effect has manifested itself for one of the high-Tc superconducting compounds ŽHTSC., because these are by nature rather complex materials with delicate structural properties. However, these effects should occur in all compounds that share these same characteristics. In conclusion, we point out the similarity of the dynamics of a surface with a pronounced anisotropic structure, as discussed here, with that of high Miller index surfaces Žvicinal surface., which have been studied in the past w29,30x. These particular studies deal with model calculations for high index surfaces, in which the actual anisotropic 2D surface unit cells could be used in the calculations, so that the folding procedure, employed in the present calculations, was not necessary.

The authors acknowledge support by the Robert A. Welch Foundation, the Deutsche Akademische Austauschdienst, FORSUPRA, and by a NATO travel grant. conditions

for

Our shell models are based on central potentials: Coulomb potential

Zi Z j r

,

Ž A1.

Born–Mayer potential yb i j r Vi BM , j Ž r . s ai j e

Ž A2.

and Buckingham potential Vi Bj Ž r . s a i j eyb i j r y

ci j

Ž A3.

r6.

For central potentials the equilibrium conditions I and II wEqs. Ž3. and Ž4.x take the form I:

fkk X , a Ž
Ý

X

lk k Ž l k /0 k X .

Ž A4. II:

fkk X , a Ž
Ý X

lk Ž l k /0 k X .

Ž A5. where

fkk X , a Ž r . s

E Vkk X Ž r . E ra

.

Ž A6.

The sums ŽA4. and ŽA5. are evaluated for the Coulomb, as well as the short-range potentials. The Coulomb contributions to conditions I and II wEqs. Ž3. and Ž4.x can be written as

E E Coul E Aab

Acknowledgements

Appendix A. Equilibrium Bi 2 CaSr 2 Cu 2 O 8

Vi Coul Ž r. s j

69

1 X s y e 2 Ý Zk Zk X Cakbk , 2 kk X

E E Coul E Ra Ž k .

Ž A7.

X

s ye 2 Zk Ý Zk X Fak k .

Ž A8.

kX

X

kk The X expressions for the Coulomb sums Cab and kk Fa as obtained with the Ewald method are given in Appendix B of Ref. w11x. The above derivation for the long-range Coulomb contributions holds for arbitrary structures, but of course each structure has its own set of Coulomb sums Žas in the case of the Madelung constant.. In order to derive expressions for the complete equilib-

U. Paltzer et al.r Physica C 301 (1998) 55–71

70

rium conditions, including the contributions of the short-range interactions, we have to specifically refer to the Bi 2:1:2:2 structure. We refer to Fig. 1 for the numbering of the particles in the unit cell. In our model, we take into account the short-range interactions between the neighbor pairs: Bi–O, Sr–O, Ca– O, Cu–O, and O–O. Using the definitions: B Ž k ,kX . s y

1 E Vksr, k X r

Er

rs rk , k

Ž in e 2r2 Õa . ,

2 y 434.5342 ZOŽ1. y 109.2163ZCa ZOŽ2.

y 75.3528 ZCu ZOŽ2. y 151.8499ZOŽ1. ZOŽ2. 2 q 31.9551ZOŽ2. y 131.1558 ZCa ZSr

y 99.1065ZCu ZSr 2 y 201.6307ZOŽ1. ZSr q 39.0355ZOŽ2. ZSr q 7.0271ZSr

y 72.2093ZCa Z Bi q 4.1783ZCu Z Bi q 6.4534ZOŽ1. Z Bi

X

Ž A9. rz Ž k , k X . s < R z Ž k . y R z Ž k X . <

Ž in A˚ . ,

Ž A10.

we obtain from Eqs. ŽA4. and ŽA5. the following eight equilibrium conditions. Condition I Žcf. Eq. Ž3.. determine the size of the unit cell;

q 143.0814ZOŽ2. Z Bi q 118.1739ZSr Z Bi 2 q 111.1212 Z Bi y 70.1854ZCa ZOŽ3.

q 8.4889ZCu ZOŽ3. q 15.0423ZOŽ1. ZOŽ3. q 147.3763ZOŽ2. ZOŽ3. q 122.4674ZSr ZOŽ3. 2 q 226.5367Z Bi ZOŽ3. q 115.4156ZOŽ3. s 0.

Ž A12.

a0  2 B Ž 1,2 . q 4 B Ž 2,3 . q 2 B Ž 2,4 . q 2 B Ž 2,5 . q2 B Ž 2,6 . q 4 B Ž 5,6 . q 4 B Ž 6,7 . q 4 B Ž 7,8 . 2 q2 B Ž 7,10 . 4 y 68.3574ZCa y 230.9877ZCa ZCu 2 y128.6475ZCu y 447.9153ZCa ZOŽ1.

Conditions II Žcf. Eq. Ž4.. determine the particle positions inside the unit cell; yB Ž 2,15X . rz Ž 2,15X . y 2 B Ž 1,2 . rz Ž 1,2 . q 2 B Ž 2,4 . rz Ž 2,4 . q 2 B Ž 2,5 . rz Ž 2,5 .

2 y865.2815ZCu ZOŽ1. y 668.7720ZOŽ1.

q 2 B Ž 2,6 . rz Ž 2,6 . q ZOŽ1. 137.2934ZCa

q61.7227ZCa ZOŽ2. q 79.1666ZCu ZOŽ2.

q106.3091ZCu q 218.5817ZOŽ1.

2 q104.7922 ZOŽ1. ZOŽ2. y 8.8394ZOŽ2.

y78.5411ZOŽ2. y 71.1368 ZSr

q19.0563ZCa ZSr y 61.8356ZCu ZSr

y82.8142 Z Bi y 82.8250ZOŽ3. 4 s 0,

y102.0104ZOŽ1. ZSr y 140.8207ZOŽ2. ZSr y14.5680ZSr2 q 222.1984ZCa Z Bi

Ž A13.

y4 B Ž 2,4 . rz Ž 2,4 . q B Ž 4,6 . rz Ž 4,6 .

q373.1219ZCu Z Bi q 745.9006ZOŽ1. Z Bi

q ZCu  116.4604ZCa q 111.8932 ZCu

q82.5104ZOŽ2. Z Bi q 141.0647ZSr Z Bi

q220.5182 ZOŽ1. y 111.7262 ZOŽ2.

2 y141.3480Z Bi q 230.9676ZCa ZOŽ3.

y51.2899ZSr y 84.0460Z Bi

q388.3462 ZCu ZOŽ3. q 780.3071ZOŽ1. ZOŽ3.

y83.0836ZOŽ3. 4 s 0,

Ž A14.

q68.1169ZOŽ2. ZOŽ3. q 175.0862 ZSr ZOŽ3. 2 y 431.1255Z Bi ZOŽ3. y 169.8412 ZOŽ3. s 0,

y4 B Ž 2,5 . rz Ž 2,5 . q 4 B Ž 5,6 . rz Ž 5,6 .

Ž A11. 4 B Ž 2,15X . rz Ž 2,15X . q 8 B Ž 1,2 . rz Ž 1,2 . 2 2 q 31.9605ZCa y 114.1588 ZCa ZCu y 114.1044ZCu

y 311.3836ZCa ZOŽ1. y 427.0124ZCu ZOŽ1.

q B Ž 5,7 . rz Ž 5,7 . q ZSr  111.0373ZCa q182.6822 ZCu q 407.0329ZOŽ1. y36.3321ZOŽ2. q 23.4849ZSr y160.6915Z Bi y 175.5533ZOŽ3. 4 s 0,

Ž A15.

U. Paltzer et al.r Physica C 301 (1998) 55–71

y4 B Ž 2,6 . rz Ž 2,6 . y B Ž 4,6 . rz Ž 4,6 . y 4 B Ž 5,6 . rz Ž 5,6 . q 4 B Ž 6,7 . rz Ž 6,7 . q B Ž 6,8 . rz Ž 6,8 . q ZOŽ2. 94.5588 ZCa q220.2234ZCu q 374.9161ZOŽ1. q0.0053ZOŽ2. q 59.7899ZSr y216.7182 Z Bi y 180.8371ZOŽ3. 4 s 0,

Ž A16.

yB Ž 5,7 . rz Ž 5,7 . y 4 B Ž 6,7 . rz Ž 6,7 . y 4 B Ž 7,8 . rz Ž 7,8 . q B Ž 7,9 . rz Ž 7,9 . q 4 B Ž 7,10 . rz Ž 7,10 . q ZOŽ3. 34.5435ZCa q68.6512 ZCu q 138.0363ZOŽ1. q57.8025ZOŽ2. q 76.4320ZSr y100.5129Z Bi y 115.7234ZOŽ3. 4 s 0,

Ž A17.

yB Ž 6,8 . rz Ž 6,8 . q 4 B Ž 7,8 . rz Ž 7,8 . q B Ž 8,10 . rz Ž 8,10 . q Z Bi  40.0037ZCa q80.6817ZCu q 160.1055ZOŽ1. q105.2376ZOŽ2. q 72.3826ZSr y108.1211Z Bi y 147.1440ZOŽ3. 4 s 0.

Ž A18.

In these expressions, the charge neutrality of the unit cell, Ýk Zk s 0, is implicitly taken into account. The coefficients in the terms containing the Zs are the Coulomb coefficients. The B Ž i, j .s are the transverse force constants of the short-range interactions of the ion pairs Ž i, j ., namely, Bi–O, Sr–O, Ca–O, Cu–O and O–O Žcf. Table 2..

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