Chapter

A BRIEF REVIEW ON THE HIGH-Tc SUPERCONDUCTING COMPOUNDS AND AN INTRODUCTION TO THE THEORY

OF ELASTICITY AND THERMAL EXPANSION

1 .I Introduction

The discovery of superconductivity in La-Ba-CuO by Bednorz and

Muller (1) in 1986 near 30 K triggered a flood of research on high-temperature

superconductivity that has resulted in thousands of papers and innumerable

conferences in this field. A sequence of new materials which show

high-temperature superconductivity has subsequently been discovered. Although

each of these materials has its own individual chanckristics, they share a largely

common phenomenology (2).

The direct consequence of the high transition temperature Tc and low

carrier concentrations is that the Bardeen-Cooper-Schreiffer (BCS) coherence

length (3) 5 0 is very small. A representative value of 50 is 10A, which is of the

order of unit cell dimensions in these compounds. The small coherence length

gives a coherence volume so small as to contain only a few Cooper pairs, which

implies that the fluctuations can play a much larger role in these materials than in

the classical superconductors.

After the discovery of high temperature superconductivity in La-Ba-CuO

system, there have been many elemental substitutions and different processing

conditions on the structure and superconducting properties of this oxide. In 1987,

Wu el al. (4) discovered superconductivity above 90 K in the Y-Ba-CuO system.

This discovery triggered a fluny of crystallographic research unprecedented in the

history of materials science (5). This discovery was quickly reproduced and the

90 K phase was identified as YBa~Cu307 and studies of the effects of substitution

and processing conditions were initiated. The cycle has been repeated twice in

1988 with the independent discoveries of superconductivity near 100 K in the

Bi-Sr-Ca-CuO system by Hiroshi Maeda et al. (6) and in the TI-Ba-Ca-CuO system

by Zhengzhi Sheng and Allen Hermann (7,s).

Since the remarkable discovery of high-Tc superconductors, there has been

an enormous effort to elucidate their fundamental properties. Research on the

atomic structures of high-temperature superconductors has played a prominent role

In characterising these materials.

Early in 1987, researchers (9-1 1) found that substitution of Sr for Ba in the

La-Ba-CuO compound raises the transition temperature to approximately 40 K.

The superconducting Laz-,Sr,CuOd is derived from the stoichiometric compound

La2Cu04 by replacing ~ a ~ ' with srZ+ partially, and it has a tetragonal structure at

room temperature. The high-symmetry form which is tetragonal can be adopted by

La2Cu04 at temperatures above 500 K (12) and it is stabilised at lower

temperatures by the partial substitution of Sr or Ba for La (13). Among the high-Tc

cuprates, La2.,SrxCu04 has the simplest crystal structure with the single CuOz

planes separated by the L a 0 charge reservoirs (14). Ramirez (15) recently

observed that by substituting divalent strontium for trivalent lanthanum in La-based

214 compounds, the antiferrornagnetism is destroyed. La2..SrxCu04 compounds

have a structure of perovskite type, belonging to the I4/mmm space group (1 6). The

unit cell contains two formula units with CuOs octahedra occupying the body

centre and eight corner sites. Each copper atom in this T structure is co-ordinated

to four oxygens in a perfect corner-linked array of square planes. Superconducting

behaviour with a transition temperature Tc near 35 K is achieved in LazCu04

when it is doped with strontium on the lanthanum site to form La1.8,Sr0.15Cu04

(17).

Cuprate superconductors have in common a lower carrier density. Their

characteristic structural elements are chains, planes, pyramids or octahedrons

formed by Cu or Bi with unstable valence and oxygen atoms. These building

blocks are assembled by ions which act as spacers and dopants. For the critical

doping level leading to superconductivity, one charge carrier per formula unit

typically is delocalised in the Cu-0 bond (19).

Another class of high-Tc superconductors which aroused much interest are

the Nd-based 214 compounds. Tokura el al. (20) synthesised Ce doped NdzCu04

with an onset of critical temperature at about 25 K. Ndz.,Ce,CuOa was the first

electron superconductor with excess negative charge per unit cell. Its structure is

tetragonal I4lmrnm with Nd, Cu and 0 in sites identical to the T-phase (21).

Copper becomes strictly square-planar co-ordinated, with four Cu-0 bonds of

about 1.9 A" and there are no apical oxygens. The Nd atoms bond to eight oxygens

compared to nine oxygens for larger lanthanum in the T structure. The most

studied among these superconductors is Ndl,~~Ceu,15Cu04 which has a

superconducting transition temperature at 23 K. Intermediate valence state of

cerium ion plays an important role in its properties (22). The structure of

N ~ I . ~ ~ C ~ , , . ~ ~ C U O ~ is tetragonal with I4/mmm space group and the lattice parameters

a rea=b=0 ,395nmandc= 1.208nm(23).

Since the discovery of superconductivity in the bismuth-strontium-calcium-

copper oxides (6) , there have been a number of studies on their physical properties

due to their higher superconducting transition temperature. The Bi-2:2: 1 :2 phase

of bismuth cuprates has the ideal composition of Bi~Sr~CaCu2Og and has Tc of

90 K (24). The structure is nearly tetragonal with I4Jmmm space, group (21) and

the lattice parameters are a = 0.541 nm, b = 0.544 nm and c = 3.078 nm with CuO

layers in the ab-plane (18). The copper and oxygen are in sheets typical of the high

temperature superconductors and are spaced by cations and interleaved with Bi202

layers. Calclum adopts eight co-ordination, similar to the Y environment in

Y-Ba-CuO. There are no oxygens at this level and copper atoms have only five

nearest-neighbours in the square pyramidal co-ordination.

The elastic properties of these high-Tc superconductors are of interest for

both technology and basic research. Anisotropy in elasticity is a fundamental

property of the superconductlng crystals. We study primarily on La-based and

Nd-based 214 compounds and bismuth cuprates. We review here briefly the status

of the experimental measurement on the elastic properties which give additional

insight into the basic microscopic mechanism underlying the condensed state of

these materials.

For high-7c superconductors, majority of the ultrasonic investigations has

been carried out on ceramic materials with material defects such as voids, twins,

microcracks and texture, which lower the sound velocity, necessitating corrections

as large as a factor of 2 or 3 (25). La2,SrxCu04 show extreme acoustic mode

softening below room temperature (26, 27). Using the shear modulus as a probe

Bhattachaqa et al. (28) found evidence for phase transition at 95 K in

LalgSrt2CuO~~. The acoustic mode softening persists down to very low

temperatures and is accompanied by a large ultrasonic attenuation (26, 29, 30).

The structural phase transition is triggered by the softening of two degenerate

transverse optical phonons at the X-point of the first Brillouin zone boundary of the

body-centered tetragonal Bravais lattice (31). In a Lal.8Sro.zCu04 ceramic system,

elastic anomalies are detected at 30 K (27) and are ascribed to the structural phase

transition from the I41mmm phase to the P42fncm phase. Esquinazi el al. (32)

observed a minimum in the Young's modulus near 20 K and a change of slope in

attenuation at 44 K. The differences between reported observations in the phase

transition temperatures could possibly be due to difference in oxygen content, an

effect which has considerable influence on the structural phase transition (33) as

well as on its elastic properties (34).

A large discontinuous drop of acoistic wave velocity is observed near the

transition temperature Tc in Laz..Sr.&u04 (29,35-37). Attenuation normally shows

no special feature at Tc except a maximum below Tc (28, 37). For single crystals

of Laz,SrxCu04, with Sr concentration of 0.12 and with Li impurity, anomalies of

the sound velocity with frequency 50 MHz in the temperature range 60-1 50 K are

observed (39). These anomalies are attributed to the deformation of the soft

optical mode (39). Fukase el al. (40) observed that under magnetic fields, the shear

mode velocity is enhanced in the lower temperature region below Tc in

Lal.aSro.15Cu04 single crystals. Suzuki ef al. (41) measured the longitudinal and

transverse wave velocities over a wide temperature range from 1.8 to 300 K in

single crystals of La1.86Sr0.14C~04 and La1.81sr0.1~Cu04. They also observed a

structural phase transition from tetragonal to orthorhombic phase at 210 K with a

large softening of C6s mode and a small softening of C11 mode. Nohara et al. (42)

observed an enhancement in the elastic constants CII, CM and C66 below 10 K

which suggests the existence of another state different from the P4z/ncm phase in

La1.~5Sr015Cu04. Their finding is an advancement from the earlier suggestion

6

made by Fukase et al. (38) in Laz,SrxCu04 with 0.10 < x 5 0.12 that this new

phase is P4~1ncm itself Fujita and Suzuki (43) found evidence for the structural

phase transition in single crystals of L~I .sS~O.I~CUO~ by studying the anisotropic

response to various sicoustic modes. Hanaguri el af. (44) observed a jump and a

kink in the CU and C33 mode velocity respectively at the superconducting transition

temperature in La1~5Sr0.15Cu04. Nohara el al. (45) measured the longitudinal

elastic constants in single crystals of Lal.8sSr0.1~Cu04 across the superconducting

transition temperature under magnetic fields upto 14 T applied both parallel and

normal to the CuOz-plane. They (45) also found anomalies in elastic constants at

Tc indicating the anisotropic coupling between the high-Tc superconductivity and

the lattice. Mode softening (46) has been found in the transverse elastic constant

(CII-CIZ)/~ below 50 K in single crystalline Lal.saSr0.14Cu04 by ultrasonic

measurements in magnetic fields along the c-axis. Bums (47) showed that the

lattice motion due to shear is responsible for the structural phase transition in

Laz.Sr,CuO4 systems. Ultrasonic measurements (48) performed on single crystals

of Laz.,SrxCu04 with x = 0.09, 0.14 and 0.19, reveal that some elastic constants

increase below the superconducting transition temperature Tc.

Increase in superconducting transition temperature by applying uniaxial

pressure in L ~ I . U ~ S ~ O . ~ ~ C U O ~ has been accomplished recently (49, 50). Locquet

et a1. (51) achieved a doubling of the critical temperature Tc of the Lal,$r~,lCu04

by compressively straining thin films, though their conclusions (51) regarding this

finding have been criticised later (52).

Earlier works on the non-linear elastic properties of La-based 214 ceramic

high-Tc superconductors have been reviewed by many workers (53-56). The central

work on elasticity in single-crystalline Lal..saSro.l4Cu04 is by Migliori et al. (57, 58)

who determined the elastic constants by the method of resonant eigen frequencies.

Fanggao et al. (59) have measured three out of six non-vanishing elastic constants

of La18Sr~.zCuO4 using the ultrasonic technique. Bezuglyi et al. (60) have

determined the elastic constants of Lal.g.1Sro.l,CuO4 from sound velocity

measurements. Fil et al. (61) measured the changes in elastic moduli during the

tetragonal to orthorhombic phase transition in L a l . s ~ S r ~ . ~ E u O ~ single crystals.

The vast majority of high-Tc superconductors are of hole-doped variety.

In Ndl.~5C&l5CuOc one of the electron-doped high-Tc superconductors, the

doping level of Ce changes the physical properties (62). Murayama et al. (63)

reported that the resistively determined superconducting transition temperature of

Ndl.a5C~.lsCuOG, with Tc = 22 K remains unchanged under the application of

hydrostatic pressure upto 2.5 GPa. Bucher et al. (64, 65) also found Tc to be

insensitive to changes in oxygen content.

Al-Kheffaji et al. (66) showed that sound velocity measurements provide

evidence for mode softening in Lal.8Sro.tCuOs but not in Ndl.s~Ceo.1sCuO4.

Fanggao et al. (67) have observed a pronounced change of gradient of the

longitudinal mode velocity with temperature near 220 K in Ndl.8sCeo.l5Cu04.

Pressure dependence of the longtudinal and shear wave velocities in

Ndl.~~Ceo,l,Cu04 , was studied by Fanggao el al. (59). They found a linear

dependence of the ultrasonic wave velocities on pressure for N d l , a C ~ , l j C u O ~ ~

and that of the bulk modulus. Saint-Paul et al. (68) measured the sound velocity in

single crystals of Nd1s~Ceo.1sCuO4 along the z-axis. Three of the six independent

second-order elastic constants of Ndl.g5Cq.l5Cu04 have been measured by Fanggao

el al. (59) using the ultrasonic wave velocity measurements.

The Bi-2:2: 1:2 phase of bismuth cuprates, which has the ideal composition

of Bi2Sr2CaCuzOs has aroused much interest because of its potential applications

and high transition temperature. Groen and Zandbergen (69) have pointed out that

the lone-pair bond from the Bi p-orbital leads to weak binding between the

adjacent BiO layers. Olsen et al. (70) found that the superconductor

Bi2Sr~CaCuzO8 is the most compressible of all the systems with perovskite-like

structure and the volume compressibility K, obtained is 16 x 10" GP~-'. Liu et al.

(71) estimated the volume dependence of Tc from the measurements of pressure

dependence of the transition temperature of a single crystal of Bi-Sr-Ca-CuO. The

pressure dependence of transition temperature Tc has been extensively studied in

bismuth systems by many workers (72-75).

There exist a number of ultrasonic and other measurements on the elastic

properties of Bi~Sr2CaCu208 (54, 56). Ledbetter el al. (76) observed a nearly

regular behaviour of bulk modulus and shear modulus in the temperature range

from 5 to 295 K. Wang (77) showed that there exists a phase-like transition

characterised by a jump of lattice parameters between 100 and 150 K in

Bi~Sr2CaCuzO8. The absolute values of the elastic moduli of flux-line lattice of

Bi~Sr2CaCu~08 have been probed by Yoon el al. (78). Chen et al. (79) measured

the effect of elastic stress on the resistivity of Bi-2:2:1:2 whiskers in the a-

and c-directions. Lubenets et al. (80) observed that the elastic compliance in

Bi-Sr-Ca-CuO crystals is higher than that of the Y-based superconductors.

The ultrasonic longitudinal velocity measurements as a function of

temperature were performed by the pulse transmission technique in

potassium-doped (81) and sodium-doped (82) single crystals of Bi-2:2:1:2.

Aleksandrov et a/. (83) measured the surface wave velocities in the ab-plane and

found that the velocity along the a-axis is higher than that along b-axis in

Bi2Sr2CaCu208. In textured samples, higher values of velocities in the ab-plane

than along c-axis were also found for Bi-2:2:1:2 system (84). The data from high

pressure experiments yield a value of 61 GPa for the bulk modulus of Bi-2:2: 1:2

system (74).

Ultrasonic wave velocity measurements on single crystal Bi2SrzCaCu~08

over a wide temperature range revealed three softening minima between 100 and

250 K (85). Room temperature sound velocity measurements on the (001) and

(0 10) planes of BizSrzCaCuzO8 single crystals were performed using Brillouin light

scattering experiments by Boekholt et al. (86) to determine the elastic constants.

Fanggao et al. (87) studied the hydrostatic pressure dependence of longitudinal and

shear wave velocity in ceramic specimen of single crystal BizSrzCaCuzOs using

ultrasound technique. Wu et al. (88) determined the temperature dependence of

elastic constants in Bi~SrzCaCuzO8 crystals from sound velocity measurements.

The velocities of longitudinal and shear ultrasonic waves propagated in very dense,

highly-textured, ceramic BizSrzCaCu208 were measured (84, 89) as functions of

temperature and hydrostatic pressure.

For anisotropic media, elastic waves are neither longitudinal nor transverse

except for some special symmetry directions. We give below the theory of

elasticity of crystals, particularly applicable to high-Tc superconductors.

1.5 Theory of Eladklty

Cons~der an elastic medium where the co-ordinates of any point can be

denoted as (a,, az, a3). Choose a set of orthonormal vectors el, 4, 4 as the basis

vectors for the co-ordinate system and denote the k'h component of the stress acting

on the plane ei = 0 by a& where i and k are the component indices. Consider the

equilibrium of a small element centered at the point ai and bounded by the plane

ai + ?'id&. Let ui denote the elastic displacement of the point ai of the body and

p the density of this point. The equation of volume element can be derived by

considering the total force acting on the volume element. If we ignore the body

forces, the equations of motion for an elastic solid can be written as (the convention

that repeated indices indicate summation over the indices will be followed here).

where the stress tensor oik is given by

ok = @/&I

where 41 IS the crystal potential and E* are the components of the strain tensor given

by

o~ and EI are symmetric tensors of second rank. According to Hooke's law

or = C*I, ~ l r n (1.4)

The constants C i h form a fourth rank tensor with 81 components.

From equations ( I .2) and (1.4), we have

Hence the elastic constants C& are multiple strain derivatives of the state

functions and since the strains Q,,, are symmetric, the elastic constants possess

complete Voigt symmetry. Thus,

C*l, = Ckilrn = CM = Clmik (I,6)

These quantities are symmetric with respect to interchange of the subscripts.

It will be convenient to abbreviate the double subscript notation to the single

subscript Voigt notation running from 1 to 6, according to the following scheme:

11+1; 2242; 3 3 4 3 ; 2 3 4 ; 13+5 and 1 2 4 .

Hence the matrix of elastic constants Ciu, would contain a 6 x 6 array of

36 independent quantities in the most general case. This number is, however,

reduced to 2 1 by the requirements that the matrices be symmetric on interchange of

double indices. The number of independent elastic constants will be further

reduced by the symmetry operations of the respective crystal classes. The high-Tc

superconducting crystals La1.8Sr0.zCu0~ Ndl.s5Ceo,l~Cu04 and BizSrzCaCu208

belong to the tetragonal14fmmm class which have six independent elastic constants

(90). The elastic constant matrix for this class of compounds is given by

In the equation of motion for an elastic medium, the forces on an element of

volume, are given by the divergence of the stress field.

Using equations (I .3) and (1.4), the equation (1.1) can be written as

where & are the components of the amplitude of vibration, w is the angular

frequency and k 1s the wave vector corresponding to the wavelength h = 2xk. The

resulting equations of motion from equation (1.8) are

substituting k = kii , where 13 is the unit vector, we get

where Tijh = Ci,dp are the reduced elastic constants and v is the phase velocity

given by v = ok. The components of second rank tensor A are given by

Hence equation ( 1 . l I ) can be written as

This shows that u is the eigen vector of tensor A where eigen value is vz. Hence

v2 is the root of the equation

I A - ~ ~ I = 0 (1.14)

This IS the Christoffet equation. The theory of elastic waves generally

reduces to finding u and v for all plane waves propagating in an arbitrary direction

for crystals possessing different symmetries. In this situation, all terms in equation

(1.11) which involve differentiation with respect to co-ordinates other than that

along the propagation direction drop out.

A more fundamental significance to the elastic constants is implied by their

appearance as the second derivatives of elastic energy with respect to strains.

It should be noted here that the stored elastic energy is only a part of the complete

thermodynamic potential of the crystal, since it depends on many other variables.

Also, one can introduce elastic constants as a constitutive, local relation between

stress and strain for materials in which long-range atomic forces are unimportant.

1.6 Finite Strain Theory of Elasticity

Let the position co-ordinates of a material particle in the unstrained state be

( i = 1, 2 3). Let the w-ordinates of the material particle in the strained state be

xi. Consider two material particles locatd at ai and ai + dai. Let their co-ordinates

in the deformed state be xi and xi + dx,. The elements dx, are related to da, by the

equation

& dx, = L d a ,

8%

The convention that repeated indices indicate summation over the indices

will be followed here. ag is the Kronecker delta and &i, are the deformation

parameters. The Jacobian of the transformation

is taken to be positive for all real transfonirations. IfdV. is a volume element in the

natural state and dV, its volume after deformation

where po and p are the densities in the natural and strained states respectively. Let

the square of the length of arc from ai to ai + dai be dl0 in the unstrained state and

dl in the strained state. Then

= (3 dr. - 6*] dqd9 da, da,

where q,k are the Lagrangian strain components which are symmetric with respect

to the interchange of the indices j and k. In terms of &r,

Q. = %(E* +E. +Z&,,C.) (1.19)

The internal energy function U(S, ~ k ) for the material is a function of the

entropy S and Lagrangan strain components. U can be expanded in powers of the

strain parameters about the unstrained state as

The linear term in strain is absent because the unstrained state is one where

U is minimum. We shall define the elastic constants of different orders referred to

the unstrained state as (91)

and

Here the der~vatlves are to be evaluated at equilibrium configuration and constant

entropy C:,ki and CS,,,,, are the adiabatic elastic constants of second- and

third-orders respectively. They are tensors of fourth and sixth ranks. The number

of independent second-order elastic constants and third-order elastic constants for

different crystal classes are tabulated by Bhagavantam (90).

1.7 Quasi-harmonic Theory of Thermal Expansion

In the harmonic approximation, the atoms in a solid are assumed to oscillate

symmetrically about their equilibrium positions which remain unaltered irrespective

of the temperature. The thermal expansion of a solid, therefore, is a property

arising strictly due to the anharmonicity of %e lattice. In the quasi-harmonic

approximation, the oscill&ions are still assumed to be harmonic in nature but the

frequencies are taken to be functions of the strain components in the lattice.

The strained state of the lattice is specified filly by the six strain components

llrs (r, s = 1, 2, 3; ?In = q s r ) .

A normal mode with frequency o(q, j) makes a contribution F(q, j) to the

total vibrational free energy Fvib given by

F(q, j) = K~T[%X + log(1 - ex)] (1.23)

where X = Fim(q, j) 1 KBT, h = h/2x, h being the Plank's constant, KB is the

Boltzmann's constant, T is the absolute temperature and q is the wave vector of the

j" acoustic mode. The total vibrational free energy is, therefore

The thermal coefficients ah of the crystal are obtained as

Here, .I,,, = (%) d

or,,, are the components of the stress tensor, Sh are the compliance coefficients

relating q, and GI,,, and

Here, y, are the generalised Gnineisen parameters (GPs) of the normal mode

frequencies. In equation (1.25). the subscript a' means that all other ail, are to be

held constant while differentiating with respect to crlm and subscript a means that

2 -x -x 2 . all oh are held constant. a[o(q, j)T] = X e /(l-e ) 1s the Einstein specific heat

function. In the quasi-hannonic approximation, the GPs are assumed to be

constants independent of temperature.

It is more advantageous to choose such strains that do not alter the crystal

symmetry, instead of choosing any arbitrary strain, while defining GPs. For

Lal,~Sro,zCu04, N ~ I . ~ J C ~ ~ . I ~ C U O ~ and. Bi2SrzCaCuzOs, there are two principal

thermal expansion coefficients, namely a1 which is the linear expansion coefficient

parallel to the c-axis and a, which is the linear thermal expansion coefficient

perpendicular to the c-axis. Here, it is convenient to use the following strains for

the determination of the thermal expansion.

(i) A uniform longitudinal strain E" along the c-axis. Then all the q, are zero,

except q33 = E" = d log c, where c is the axial length.

(~ i ) A uniform areal strain E' in the basal plane perpendicular to the c-axis.

Then q11 = q 2 ~ = , where A is the area of the basal plane of A

the crystal. All other q, vanish.

From equation (l.25), we now obtain

Va33 = Val = C [2s13yf(q,j) + S33 Y"(q,j)I K B ~ ( o (q,j)) 4.1

and Val, = Vazz = Va,

= c [(%I +slz)y1(s, j) + su ~"(4 , j)l K B O ( ~ (q, j)) (1.28) 4,)

The effective Grimeisen functions are defined as

= [ ( c S , +csZ)U, +cs3arl V/CP

-

Y ,,(TI = [ 2 ~ s , a , +cs,ail VICP (1.29)

The C: are the adiabatic elastic constants, Cp is the specific heat at constant

pressure and V is the volume of the crystal.

Comparing equations (1.29) with (1.28), .we get

- C v "(a j) o[o(q, r ,(T) =

4.1

z .[a (4. j). T] 9.1

The expression (1.30) give the temperature dependence of effective

Grimeisen function.

In the low temperature limit, only the low frequency acoustic modes make a

contribution to the specific heat. The number of such normal modes in the jm

18

acoustic branch is proportional to vf (8,4), where vj(8, 4) is the velocity of the j"

acoustic mode travelling in the direction (8, 4). The GP y(q, j) depends only on the

branch index j and the direction (8,4). It is independent of the magnitude of the

wave vector q. The effective lattice Griineisen functions ?,(T) and T,(T)

approach the limits defined below, at low temperatures.

where y ; (0, 4) and y x0,4) are the GPs for the acoustic modes propagating in the

direction (8, 9). Calculations of the low temperature limit of ?,(T) and 7 ,(T) are

possible knowing the pressure derivatives of the second-order elastic constants

(SOEC) or the third-order elastic constants (TOEC) of the crystal. The evaluation

of the low temperature limits y,(0) and 7 ,(O) for La1.sSr0.tCuO4, N ~ I . s ~ C Q . I ~ C U O ~

and BitSrzCaCuzO~ is given in Chapter 6 of this thesis

References

J. G. Bednorz and K. A. Miiller, Z. Phys. B 64 (1 986) 189.

R. Beyers and T. M. Shaw, "Solid State Physics," Vol. 42, Academic Press,

San Diego, 1989.

J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 108 (1957) 1175.

M. K. Wu, J. R. Ashbum, C. J. Tomg, P. H. Hor, R. L. Meng, L. Gao,

2. J. Huang, Y. Q. Wang and C. W. Chu, Phys. Rev. Lett. 58 (1987) 408.

R. M. Hazen, "The Breakthrough: The Race for the Superconductor,"

Summit, New York, 1988.

H. Maeda, Y. Tanaka, M. Fukutomi and T. Asano, Jpn. J. Appl. Phys.

27 (1988) L209.

2. Z. Sheng and A. M. Hermann, Nature 332 (1988) 55.

Z. Z. Sheng and A. M. Hermann, Name 332 (1988) 138.

K. Kishio, K. Kitizawa, S. Kanbe, I. Yasuda, N. Sugii, H. Takagi,

S. Uchida, K. Fueki, S. Tanaka, Chem. Lett. Jpn. (1987) 429.

J . M. Tarascon, L. H. Greene, W. R. Mc Kinnon, G. W. Hull and

T. Geballe, Science 235 (1987) 1373.

R. B. Vad Dover, R. Cava, B. Batlogg and E. Rietman, Phys. Rev. B

35 (1987) 5337.

M. Onoda, S. Shamoto, M. Sato and S. Hosoya, in: S. A. Wolf and

V. Z. Kresin (ed.), "Novel Superconductivity," Plenum, New York, 1987,

p. 919.

D. E. Cox, S. C. Moss, R. L. Meng, P. H. Hor and C. W. Chu, J. Muter.

Res. 3 ( 1 988) 1327.

A. Fujimori, A. Ino, T. Mizokawa, T. Kimura, K. Gshio, M. Takaba,

K. Tamasaku, H. Eisalu and S. Uchidau, J. Phys. Chem. Solids 59 (1998)

1892.

A. P. Ramirez, Nature 399 (1999) 527.

R. Cava, A. Santoro, D. W. Johnson Jr. and W. W. Rhodes, Phys. Rev. B

35 (1987) 6716.

J. D. Jorgensen, Physics Today (June 1991), p. 36.

R. M. Hazen, C. T. Prewitt, R. J. Angel, N. L. Ross, L. W. Finger,

C. G. Hadidiacos, D. R. Veblen, P. J. Heaney, P. H. Hor, R. L. Meng,

Y. Y. Sun, Y. Q. Wang, Y. Y. Xue, Z. J. Huang, L. Gao, J. Bechtold and

C. W. Chu, Phys. Rev. Lett. 60 (1988) 1174.

D. M. Ginsberg, in: D. M. Ginsberg (ed.), "Physical Properties of High

Temperature Superconductors 11," World Scientific, Singapore, 1990.

Y. Tokura, H. Takagi and S. Uchida, Nature 337 (1989) 345.

R. M. Hazen, in: D. M. Ginsberg (ed.), "Physical Properties of High

Temperature Superconductors 11," World Scientific, Singapore, 1990.

Chang Fanggao, A. Al-Kheffaji, P. J. Ford, D. A. Ladds, J. Freestone,

B. Chapman, L1. Manosa, D. P. Almond and G. A. Saunders, Supercond.

Sci. Technol. 3 (1 990) 422.

F. Izumi, Y. Matsui, H. Takagi, S. Uchida, Y. Tokura and H. Asano,

Physica C 158 (1989) 433.

.4. Maeda, M. Hase, I. Tsukada, K. Noda, S. Takebayashi and

K. Uchinokura, Phys. Rev. B 41 (1990) 6418.

H. Ledbetter and M. Lei, J. Mater. Res. 5 (1990) 241.

K. Fossheim, T. Laegreid, E. Sandvold, F. Vassenden, K. A. Miiller and

J. G. Bednorz, Solid State Commun. 63 (1987) 53 1.

A. Al-Kheffaji, J. Freestone, D. P. Almond, G. A. Saunders and J. Wang,

J. Phys. Condens. Matter 1 (1989) 5993.

S. Bhattacharya, M. J. Higgins, D. C. Johnston, A. J. Jacobson, J. P. Stokes,

D. P. Goshorn and J. T. Lewandowski, Phys. Rev. Lett. 60 (1 988) 1 18 1.

S. Bhanacharya, M. J. Higgins, D. C. Johnston, A. J. Jacobson, J. P. Stokes,

J. T. Lewandowski and D. P. Goshom, Phys. Rev. B 37 (1988) 5901.

W. K. Lee, M. Lew and A. S. Nowick, Phys. Rev. B 41 (1990) 149.

R. J. Birgeneau, C. Y. Chen, D. R. Gabbe, H. P. Jenssen, M. A. Kastner,

C. J . Peters, P. J. Picone, Tineke Thio, T. R. Thurston, H. L. Tuller,

J. D. Axe, P. Boni and G. Shirane, Phys. Rev. Len. 59 (1987) 1329.

P. Esquinazi, J. Luzuriaga, C. Duran, D. A. Esparza and C. D'Ovidio,

Phys. Rev. B 36 (1 987) 23 16.

Wu Ting, K. Fossheim and T. Laegreid, Solid. State Commun.

75 (1990) 727.

Q. Wang, G. A. Saunders, D. P. Almond, M. Cankurtaran and

K. C. Goretta, Phys. Rev. B 52 (1995) 371 1 .

D. J. Bishop, P. L. Gammel, A. P. Ramirez, R. J. Cava, B. Batlogg and

E. A. Rietman, Phys. Rev. B 35 (1987) 8788.

X. D. Xiang, J. W. Brill, L. E. DeLong, L. C. Bourne, A. Zettl, J. C. Jones

and L. A. Rice, Solid State Commun. 65 (1988) 1073.

L. C. Bourne, A. Zettl, K. J. Chang, M. L. Cohen, A. M. Stacy and W. K.

Ham, Phys. Rev. B 35 (1987) 8785.

T. Fukase, T. Nomoto, T. Hanaguri, T. Goto and Y. Koike, Physica C

165-166 (1990) 1289.

P. V. Zavaritskii, V. I. Markarov, V. S. Klochko, V. N. Molchanov, G. A.

Tamazyan and A. A. Yurgens, Sov. Phys. JETP 73 (1 99 1) 1099.

T. Fukase, T. Hanaguri, M. Kamata, K. Ishizuka, T. Suuki, T. Goto and

T. Sosaki, SCI. Rep. HITU A42 (1996) 327.

T. Suzuki, T. Fujita, M. Nohara, Y. Maeno, I. Tanaka and H. Kojima,

"Mechanisms of Superconductivity," JJAP Ser. 7, 1992, p. 219.

M. Nohara, T. Suzuki, Y. Maeno, T. Fujita, I. Tanaka and H. Kojima,

in: Y. Iye and H. Yasuoka (eds.), "The Physics and Chemistry of Oxide

Superconductors" Springer Proceedings in Physics, Vol. 60,

Springer-Verlag, Berlin, Heidelberg, 1992, p. 21 3.

T. Fujita and T. Suzuki, in: S. Mackarna and M. Sato (ed.), "Physics of

High Temperature Superconductors," Springer Series in Solid-State

Sciences, Vol. 106, Springer-Verlag, Berlin, Heidelberg, 1992, p. 333.

T. Hanaguri, R. Toda, T. Fukase, I. Tanaka and H. Kojima, Physica C

185-189 (1991) 1395.

M. Nohara, T. Suzuki, Y. Maeno, T. Fujita, I. Tanaka and H. Kojima,

Physica C 185-189 (1991) 1397.

T. Suzuki, M. Nohara, Y. Maeno, T. Fujita, 1. Tanaka and H. Kojima,

J. Supercond. 7 (1 994) 4 1 9.

S. J . Bums, in: W. C. Johnson, J. M. Howe, D. E. Laughlin and

W. A. Soffa (eds.), "Proc. Int. Conf. Solid-Solid Phase Transformations,"

TMS-Miner. Metals and and Mater. Soc., PA, USA, 1994.

M. Nohara, T. Suzuki, Y. Maeno, T. Fujita, I. Tanaka and H. Kojima,

Physica (,' 235-240 (1 994) 1249.

H. Sato and M. Naito, Physica C274 (1997) 221

J.-P. Locquet and E. J. Williams, Acta Polonica A 92 (1997) 69.

J.-P. Locquet, J. Perret, J. Fompeyrine, E. Mkhler, J. W. Seo and

G. Van Tendeloo, Nature 394 (1998) 453.

Lurens Jansen and Ruud Block, Nature 399 (1999) 1 14.

Ming Lei, Ph. D. thesis, University of Colorado, Boulder, 1991.

J. Dorninec, Supercond. Sci. Technol. 2 (1989) 91; J. Dominec, Supercond,

Sci. l'echnol. 6 (1 993) 153.

Wu Ting and K. Fossheim, Int. J. Mod. Phys. B 8 (1993) 275.

S. V. Lubenets, V. D. Natsik and L. S. Fomenko, Low Temp. Phys.

21 (1995) 367.

A. Migliori, W. M. Visscher, S. Wong, S. E. Brown, I. Tanaka, H. Kojima

and P. B. Allen, Phys. Rev. Lett. 64 (1990) 2458.

A. Migliori, J . L. Sarrao, W. M. Visscher, T. M. Bell, Ming Lei, Z. Fisk and

R. G. Leisure, Physica B 183 (1993) 1..

59. Chang Fanggao, M. Cankurtaran, G. A. Saunders, A. Al-Kheffaji,

D. P. Almond, P. J. Ford and D. A. Ladds, Phys. Rev. B 43 (1991) 5526.

E. V. Bezuglyi, N. G. Burma, I. G. Kolobov, V. D. Fil, I. M. Vitebskii,

A. N . Knigavko, N. M. Lavrinenko, S. N. Barilo, D. I. Zhigunov and

L. E. Soshnikov, Low Temp. Phys. 21 (1995) 65.

V. D. Fil, E. V. Bezugly, N. G. Burma, I. G. Kolobov, I. M. Vitebsky,

A. N. Knigavko, N. M. Lavrinenko, S. N. Barilo, D. I. Shigunov and

L. E. Soshnikov, Physica C 235-240 (1994) 121 5.

S. Jendal, T. Strach, T. Ruf, M. Cardona, V. Nekvansil, C. Chen,

B. M. Wanklyn, D. I. Zhigunov, S. N. Barilo and S. V. Shirayaev, J. Phys.

(:hem. Solids 59 (1 998) 1985.

C. Murayama, N. MGri, S. Yomo, H. Takagi, S. Uchida and Y. Tokura,

Nature 339 (1 989) 293.

B. Bucher, R. Rusiecki and P. Wachter, Condensed Matter 62 (1 989) 882.

B. Bucher, J. Karpinslu and P. Wachter, Physica C 157 (1989) 478.

A. Al-Kheffaji, J. Freestone, D. P. Almond, G. A. Saunders and Q. Wang,

Phys. Condens. Matfer 1 (1989) 5993.

Chang Fanggao, A. Al-Kheffaji, P. J. Ford, D. P. Almond, B. Chapman,

J. Freestone, D. A. Ladds, L1. Manosa, M. Cankurtaran and G. A. Saunders,

Stcpercond. Sci. Technol. 4 (1991) S 199.

M. Saint-Paul, J. L. Tholence, S. Pinol, X. Obradors, R. J. Melville and

S. B. Palmer, Solid State Commun. 76 ( 1 990) 1257.

W. A. Groen and H. W. ~ a n d b e r ~ e n , SoliciSfate Commun. 68 (1988) 527.

J. S. Olsen, S. Steenstrup, L.Genvard and B.Sundqvist, Physica Scripta

44(1991)211.

H.J.Liu, Q.Wang, G.A.Saunders, D.P. Almond, B.Chapman and

K. Kitahama, Phys. Rev. B 51 (1995) 9167.

J. S. Schilling, J. Diederichs, S. Klotz and R. Seiburger, in: T. Francavilla,

R. A. Hein and D. Leibenberg (eds.), "Magnetic Susceptibility of

Superconductors and Other Spin Systems," Plenum Press, New York,

1991.

R. Kubiak, K. Westerholt, G. Pelka, H. Bach and Y. Khan, Physica C

166 (1990) 523.

T. Yoneda, Y. Mori, Y. Akahama, M. Kobayashi and H. Kawamura, Jpn. J.

Appl. I'hys. 29 (1990) L1396.

S. Klotz, J. S. Schilling and P. Miiller, in: H. D. Hochheimer and

E. D. Etters (eds.), "Frontiers of f igh Pressure Research," Plenum Press,

New York, 199 1

H. Ledbetter, S. Kim and K. Togano, Physica C 185-189 (1 991 ) 935.

Ye-Ning Wang, in: Z. Z. Gan, S. S. Xie and 2. H. Zhao (eds.), "Proc.

Beijiny Int. Conf.: High Temperature Superconductivity," 1992, p. 23.

Seokwon Yoon, Zhen Yao, Hongjie Dai and C. M. Lieber, Science

270 (1995) 270.

X.-F. Chen, G. X . Tessema, M. J. Skove, M. V. Nevitt and J. E. Payne,

Physica C 235-240 (1994) 1523.

S. V. Lubenets, V. D. Natsik, M. N. Sorin, L. S. Fomenko,

N. M. Chaikovskaya, H.-J. Kaufmann and K. Fischer, Superconcluctivi&

Physics, C'lemislry, Technology 3 (1990) 5258.

B. G. Krishna, R. R. Reddy, P. V:Reddy and S. V. Suryanarayana, Mod.

Phys. Lett. 8 8 (1 994) 1097.

M. C. Shekhar, B. G. Krishna, R. R. Reddy, P. V. Reddy and

S. V. Suryanarayana, Supercond. Sci. Technol. 9 (1996) 29.

V. V. Aleksandrov, T. S. Velichkina, V. I. Voronkova, A. A. Gippius,

S. V. Red'ko, I. A. Yakovlev and V. K. Yanovskii, Solid State Commun.

76 (1990) 685.

G. A. Saunders, Chang Fanggao, Li Jiaqiang, Q. Wang, M. Cankurtaran,

E. F. Lambson, P. J. Ford and D. P. Almond, Phys. Rev. B 49 (1 994) 9862.

Jin Wu, Yening Wang, Huimin Shen, Jinsong Zhu, Yifeng Yan and

Zhongxian Zhao, Phys. Lett. A 148 (1990) 127.

M. Boekholt, J. V. Harzer, B. Hillebrands and G. Giintherodt, Physica (: 179(1991) 101.

Chang Fanggao, M. Cankurtaran, G. A. Saunders, A. Al-Kheffaji,

D. P. Almond and P. J. Ford, Supercond. Sci. Technol. 4 (1991) 13.

Jin Wu, Yening Wang, Pingsheng Guo, Huimin Shen, Yifeng Yan and

Zhongxian Zhao, Phys. Rev. B 47 (1 993) 2806.

Chang Fanggao, P. J. Ford, G. A. Saunders, Li Jiaqiang, D. P. Almond,

B. Chapman, M. Cankurtaran, R. B. Poeppel and K. C. Goretta, Supercond.

Sci. l'echnol. 6 (1993) 484.

S. Bhagavantam, "Crystal Symmetry and Physical Properties," Academic

Press. 1966.

K. Brugger, Phys. Rev. A 133 (1 964) 1161.