NISTIR 3980

OXIDES AND OXIDE

SUPERCONDUCTORS:

ELASTIC AND RELATED PROPERTIES

Ming Lei

Hassel Ledbetter

1 1

United States Department of Commerce

I I National Institute of Standards and Technology

NISTIR 3980

OXIDES AND OXIDE

SUPERCONDUCTORS:

ELASTIC AND RELATED PROPERTIES

Ming Lei^

Hassel Ledbetter

Materials Reliability Division

Materials Science and Engineering Laboratory

National Institute of Standards and Technology

Boulder, Colorado 80303-3328

August 1991

Visiting Scientist, on leave from the Institute of

Metal Research, Academia Sinica, Shenyang, China

U.S. DEPARTMENT OF COMMERCE, Robert A. Mosbacher, Secretary

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, John W. Lyons, Director
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Contents

1 Introduction 2

1.1 Elastic Constants 2

1.2 Superconductivity and Elastic Constants 4

1.3 Purposes of the Study 5

2 Previous Elastic-constant Studies 7

2.1 Perovskites 7

2.1.1 BaTiOa 7

2.1.2 SrTiOa 9

2.2 YBCO Superconductor 10

2.3 Other High-Tc Oxide Superconductors 13

2.4 Summary 14

3 Materials and Measurement Methods 15

3.1 Materials 15

3.2 Measurement Methods 17

3.2.1 Mass Density 17

3.2.2 Acoustic-wave Velocity 17

3.2.3 Low-temperature Measurements 20

4 Elastic-constant Measurement Results and Analysis 22

4.1 Perovskites 22

4.1.1 Polycrystal BaTiOs 22

4.1.2 Monocrystal SrTiOa 25

4.2 Polycrystal YBCO Superconductor 25

4.2.1 Ambient-temperature Measurements 25

4.2.2 Low-temperature Behavior 28

iii
5 Debye Temperature 33

6 Bulk Modulus 37

6.1 Calculation from Madelung Energy 37

6.1.1 Born Ionic Model 37

6.1.2 Madelung Energy 38

6.1.3 Bulk Modulus 39

6.2 Comparisons with Other Oxides 39

7 Monocrystal Elastic Constants of Orthorhombic YBCO 44

8 Electron—phonon Parameters 48

9 Valence Problem 52

10 Calculated Pressure Derivative and Uniaxial-stress Derivatives of Tc 56

11 Summary and Conclusions 62

Acknowledgments 65

References 66

Appendix A 92

iv
List of Tables

4.1 Measured elastic constants at T = 295 K 24

4.2 The fitted parameters of the moduli in the Varshni function. . 28

5.1 Sound velocities, bulk moduli, Debye temperature, and electron

phonon parameters 35

6.1 Bond distance and bulk modulus for YBCO 41

7.1 Measured and estimated monocrystal elastic constants for YBCO

superconductor. Units are GPa for Cij and TPa“^ for 5,j. . . 45

9.1 Crystallographic data for YBCO superconductor at room temperature 53

9.2 Site potentials and Madelung energies in YBCO for various

ion-charge configurations 55

10.1 Transition temperature Tc and AVa for metal-oxide superconductors 59

10.2 Estimated pressure (stress) and strain derivatives of Tc for

YBCO superconductor 60

10.3 For YBCO, estimated plane-stress and plane-strain derivatives

of Tc 61

A.l Measured elastic constants of polycrystal YBCO 92
List of Figures

3.1 Temperature variation of electrical resistance and magnetic

susceptibility of YBCO specimen 16

3.2 Functional diagram of ultrasonic pulse-echo sound-velocitymeasurement system 18

3.3 Composite oscilloscope display of transverse-wave pulse-echo

pattern at 4 K for YBCO specimen. Expanded first and sec- ond echoes are shown at bottom. Transit time was measured

between the first minimum in these adjacent echoes. Carrier

frequency equals 4 MHz 19

3.4 Low-temperature probe 21

4.1 Temperature variation of elastic constants of polycrystal barium titanate 23

4.2 Temperature variation of elastic constants of monocrystal strontium titanate 26

4.3 Temperature variation of elastic constants of polycrystal YBCO

superconductor 29

4.4 Schematic bulk-modulus-temperature curve in an Einsteinoscillator model. Bq denotes the harmonic-model value of the

bulk modulus 31

5.1 Temperature variation of Debye temperature for BaTiOa, SrTiOa,

and YBCO superconductor. The absolute values at room

temperature appear in Table 5.1 36

6.1 Crystal structure of YiBa2 Cu307 showing one unit cell 40

6.2 Dependence of bulk modulus on cation-anion distance for some

oxides 42

VI
8.1 Variation of Tc/0d with electron—phonon coupling parameter, A. Points for seven oxide superconductors based on Kresin

model. For comparison, the figure shows also curves from BCS

(weak-coupling), McMillan (moderate-coupling), and AlienDynes (strong-coupling) models 51

10.1 Tc-versus-AVA correlation 57

Vll

Abstract

Using both measurements and modeling, we studied the elastic and related

properties of some oxides and oxide superconductors. The polycrystal elastic constants were measured using a megahertz-frequency pulse-echo method

between 295 and 4 K and corrected to the void-free state by using a model

for a composite material containing spherical particles. The elastic moduli

of the high-Tc superconductor YiBa2 Cu307 (YBCO) were compared with

that of oxides, especially the perovskites BaTiOs and SrTiOa, which are

crystal-structure building blocks for the YBCO superconductor. The bulk

moduli were also calculated using a Born ionic model with two energy terms:

electrostatic (Madelung) and ion-core-repulsion. The calculated bulk modulus of YBCO, 98 GPa, agrees well with measurement, 101 GPa. Ba^ed on

monocrystal measurements combined with analysis-theory, elastic stiffnesses

Cij for orthorhombic YBCO were estimated. The bulk modulus obtained

from the estimated Cij by the Voigt—Reuss—Hill averaging method agrees

with the monocrystal measurement. From the measured polycrystal elastic

constants, the Debye characteristic temperatures, 0£>, were calculated. For

YBCO, = 437 K. The electron—phonon parameters. A, were estimated

from Tc and 0J) using Kresin’s model, which is valid for all values of A. For

YBCO, A = 2.24. Calculating the Madelung energy allowed us to study two

further features: the valence of copper and the electron hole distribution.

The results show that the hole prefers the Cu02 plane at the oxygen sites.

All the results are consistent with the assumption that all copper ions have

valences near -f2. Using a relationship between Tc and AVA, tbe difference

in Madelung site potentials for a hole at the apical and planar oxygens, the

pressure derivative and stress and stain derivatives of Tc were calculated.

The results show that Tc increases with decreasing a-axis, increasing fe-axis,

and decreasing c-axis.

Key words: Born model; bulk modulus; Debye temperature; elastic constants; electron-phonon parameter; Kresin model; oxides; pressure derivative; superconductors; YiBa2 Cu307.

1
Chapter 1

Introduction

1.1 Elastic Constants

A material’s elcistic properties are important because they relate to various fundamental solid-state phenomena [1]. These phenomena include interatomic potentials, equations of state, phonon spectra, and so on. Elastic

properties also link thermodynamically with specific heat, thermal expansivity, Debye temperature, and Griineisen parameter. In engineering design,

elastic constants enter many practical applications: load-deflection, thermoelastic stress, internal strain (residual stress), sound velocities, and fracture

toughness.

Elastic constants appear in the generalized Hooke’s law:

(Tij = Cijkl^kh (^•^)

Here, cr,j and £ki denote the second-rank stress tensor and strain tensor, re- spectively; Cijki indicates components of a fourth-rank elastic-constant tensor; and the usual summation convention is applied to repeated indices. The

elastic constants Cijki can be calculated from a solid’s internal energy U [2]:

, _1_ f

d^U \

K \dnijdr,uJ

(1.2)

Here, rjij denotes components of the Lagrangean elcistic-strain matrix, and

Va denotes atomic volume. By invoking the Voigt contraction convention.

2
the fourth-rank-tensor elastic constants can be arranged in a symmetrical

six-by-six matrix:

/ C11 C\2

C22

\

C\3 Ci4 C\5 Cie ^ C23 C24 C25 C26

C33 C34 C35 C36

C44 C45 C46

C55 C56

Cee /

(1.3)

The generalized Hooke’s law can also be expressed in the following way:

€ij — Sijkl(Tf;i. ( 1 . 4 )

Here, Sijki denotes components of a fourth-rank elastic-compliance tensor.

Similarly, the elastic-compliance tensor Sijki can also be arranged in a symmetrical six-by-six matrix. Generally, there is a simple relationship between

elastic stiffnesses and elastic compliances Sij:

[5.il = [Co-]-' . (1.5)

In the most general case (triclinic symmetry), there are twenty-one independent elastic constants. The number of independent elastic constants

is reduced by the crystal-class symmetry operations. For example, when a

material possesses orthorhombic symmetry there are only nine independent

elastic constants; a cubic crystal possesses three. For an isotropic material,

there are only two independent elastic constants: Cji and engineering

usage, there are four practical elcistic constants:

Shear modulus = G = ^(^11 “ ^12)5 (1*^)

Young modulus = E =

Bulk modulus = B =

(C?1 - C?2)(C?1 + 2C.°2)

C

O I /^O ’ 11 ^12

3(3G -E) 3

„ . . E C?2

Poisson ratio =v= — ~\ = go , go

(1.7)

( 1 . 8 )

(1.9)

3
These four elastic constants have particular physical meanings. For exarnple,

the bulk modulus describes a solid’s resistance to volume change; Young and

shear moduli describe a solid’s resistance to extensional strain and shear

strain, respectively; and the Poisson ratio represents a dimensionless ratio of

elastic constants that, for uniaxial stress, relates longitudinal and transverse

strains.

1.2 Superconductivity and Elastic Constants

In 1911, Kamerlingh-Onnes [3] found that the electrical resistance of mercury

vanished when the temperature dropped to 4 K. The phenomenon that materials lose their resistances when temperature is below a critical value Tc is

called superconductivity. When the temperature exceeds Tc, materials have

resistances typifying the normal state. The transition between normal and

superconducting states is a second-order phase transition, which is characteristic of order—disorder transitions. In a magnetic field, a superconductor can

stay in the normal state even though the temperature is below Tc. The minimum magnetic field that destroys superconductivity is called the critical field

He. Similarly, the minimum current density that destroys superconductivity

is called the critical-current density Jc.

For practical use of superconductivity, scientists are trying to raise these

three critical parameters, especially the critical transition temperature Tc, as

far as possible. Meanwhile, theorists try to link Tc to fundamental physical

properties. In 1957, Bardeen, Cooper, and Schrieffer [4] proposed a microscopic theory of superconductivity bcised on the interaction between electrons

and phonons. They gave a simple relationship that relates Tc to the Debye

temperature 0£>:

Tc = 1.140£>exp(-l/A). (1-10)

Here, A denotes the electron—phonon coupling parameter, which also depends

on 0£).

On the other hand, the Debye temperature 0£> can be calculated from

the following relationship [5]:

3N \

( 1 . 11 )

4
Here, h and k denote the Planck and Boltzmann constants, respectively; N

denotes the number of independent three-dimensional oscillators in a volume

V] and Vm denotes mean sound velocity calculated by the relationship

V

-3

m = 1 (.,-3 + 2.r*) ( 1 . 12 )

Here, V£ and Vt denote longitudinal and transverse velocities, respectively.

The electron—phonon parameter A is defined by [6]:

A

a^F{uj)

duj.

UJ (1.13)

Here, a^F{u:) represents the Elicishberg electron—phonon coupling function,

where F(lj) indicates phonon density-of-states. By assuming a Debye phonon

distribution, F{lo) ~ and taking to be frequency independent, we find

that

27t’^V^ ’ m

(1.14)

where

(1.15)

is the Debye cut-off frequency. Substituting Eqs. (1.15)

Eq. (1.14), we obtain

y

47T

J

and (1.11) into

(1.16)

Equations. (1.10), (1.11), and (1.16) show how the superconducting transition

temperature Tc depends on the elastic constants explicitly through both 0£>

and A.

1.3 Purposes of the Study

The present research focused on the elastic constants of high-Tc metal-oxide

superconductors, mainly YiBa2 Cu307 (henceforth written YBCO). This

study had nine objectives:

1. Determine the ambient-temperature intrinsic elastic constants of

YBCO.

5
2. Determine the temperature dependence of the YBCO elastic constants

in the 295—4 K temperature range.

3. Determine the temperature-dependent elastic constants of BaTiOa and

SrTiOa.

4. Compare YBCO with other oxides, especially the perovskites BaTiOa

and SrTiOa, which are crystal-structure building blocks for the YBCO metaloxide superconductor.

5. Calculate the bulk modulus of YBCO using a Born ionic model.

6. Estimate the nine independent elastic constants for orthorhombic

YBCO using measurements on monocrystals combined with some analysistheory.

7. Estimate some related physical parameters such as the Debye characteristic temperature Qd and the electron— phonon interaction parameter

A.

8. Consider the valence problem, for both the cations (Y, Ba, Cu) and

the anion (0). Especially, focus on Cu, which has two site symmetries, thus

two possible valences.

9. Estimate the stress dependence of the normal—superconducting transition temperature, dTc/dcij.

6
Chapter 2

Previous Elastic-constant

Studies

In this chapter, studies of elastic and related properties on high-Tc oxide

superconductors, especially YBCO, will be reviewed briefly. Since the crystal

structure of YBCO consists approximately of three stacked perovskite unit

cells, the review will also include the studies on BaTiOa and SrTiOs, two

representative perovskites.

2.1 Perovskites

2.1.1 BaTiOs

Since barium titanate is a displacive-type ferroelectric, most researches

on elcLstic constants of barium-titanate were combined with studies on dielectric and piezoelectric properties. In 1951, using an ultrasonic pulse method.

Bond, Mason, and McSkimin [7] first measured the elastic and dielectric constants of monocrystal barium titanate. For elastic constants and compliances,

they reported only 5n, 25i2 + 5'66, and C33. In 1956, Bechmann [8] measured

the complete elastic constants for transversely isotropic barium-titanate ce- ramics (an electric field was applied to the isotropic ceramics). A complete

set of elastic, piezoelectric, and dielectric constants of monocrystal barium

titanate was given by Berlincourt and Jaffe [9]. They reported the elastic

constants of both tetragonal (at 25° C) and cubic (at 150° C) crystal struc7
tures. Using relationships derived by Voigt, Berlincourt and JafFe calculated

the elastic constants of polycrystals from the monocrystal measurements.

They found good agreement with Bechmann’s [8] results.

The temperature dependence of elastic properties of BaTiOa was studied

by many researchers [9-17]. However, most of the studies focused on high

temperature, around 120°C (the Curie point). Only a few authors extended

the studies to low temperature, and all of the studies were within the temperature range of —50 to 190® C. Berlincourt and JafFe [9] and Huibregtse

and coworkers [10] observed a large temperature dependence of the elcistic

compliances of BaTiOs near the Curie point. Using a composite-bar method,

Fushimi and Ikeda [11] found a larger temperature dependence than [9,10]

near the Curie point. They concluded the difference may arise from electrical bias. The temperature dependences of the elastic moduli Cu and

E were measured by Strukov and Garland [12] and by Mader and coworkers [13], respectively. Singularities of the elastic properties of BaTiOa caused

by structural changes near the tetragonal— cubic phase transition were re- ported by Perro and Fritsberg [14]. Also, the higher-order elastic constants

were studied [15, 16]. Against expectation. Beige and Schmidt [15] found

that third-order elastic compliances depend more on temperature than do

the second-order. This suggests strong anharmonic effects.

Unlike the measurements, only a few model calculations of elastic properties of BaTiOs have been done [18-22]. Using a method of power expansion

of free energy, Devonshire [18] derived a complete set of elastic, dielectric,

and piezoelectric constants from the measurements of Bond and coworkers [7]. Devonshire’s elcistic constants show the type of anisotropy found

later by measurements [9], but to a larger degree. The reason might be

that his model fails to consider geometric changes at the Curie point [9].

Turik [19] proposed a general calculational method and derived equations

for determining the physical constants of a monocrystal with a 90° laminar

structure using known values of the constants of a monodomain tetragonal

crystal. Yokoyama, Pak, and Kinase [20] calculated the elastic compliances

of BaTiOa by using a point-dipole model, which considers the dipole-dipole

interactions and interatomic forces of the type (m =6, 9). Their model

calculation agrees with observation in the ferroelectric pha^e, but departs

from measurements on the parcielectric phase [10]. Beige [21] developed a

thermodynamic model that predicts the temperature behavior of secondorder and third-order elastic, dielectric, and electromechanical coefficients

8
for a first-order phase transition. For parcielectric BaTiOa, Beige concluded

that the polar defects produce a linear coupling between the order parameter

and the mechanical stress.

2.1.2 SrTiOa

Although SrTiOa and BaTiOa possess the same crystal structure, they show

different dynamical behavior. BaTiOa’s Curie temperature (400 K) is asso- ciated with the softening of the lower-zone-center transverse optic mode [22].

SrTiOa undergoes an antidistortive transition at 110 K associated with the

lowering of a zone-boundary optical frequency [22]. SrTiOa has attracted

increasing attention from different branches of solid-state physics. A principal reason for the interest in this material is that its structural transition

is associated with condensation of a lattice mode, often referred to as a soft

mode [23]. Previous studies on elastic properties of SrTiOa include both

measurements [15,24-39] and calculations [34,37-40].

Among the mecisurements, Poindexter and Giardini [24] measured elastic compliances Sij for cubic SrTiOs specimens by using a resonance

method. Also for cubic SrTiOa, Bell and Rupprecht [25] and Wachtman and

coworkers [26] measured elastic stiffnesses Cij using a pulse-echo method and

both found results in good agreement with those of Poindexter and Giardini except Cii. The temperature dependence of the elastic properties of

SrTiOa was studied extensively [15,25,27,28,30,31,34-37]. All the studies

confirmed that a phase transition occurs near 110 K in SrTiOs. But the

phase transition at 36 K reported by Sorge and Hegenbarth [28] remains

to be verified. Using a pulse-echo method, Rehwald [35] measured various

elastic stiffnesses and compliances in the 130-4 K temperature range, but

failed to find any evidence that might indicate a phase transition near 40 K.

Hochli and Bruce [37] measured the ela.stic compliance Su in the temperature

region 60—125 K using a resonance method. Near the phase-transition temperature, they observed a stronger singularity than that suggested by earlier

measurements [31,34,36] that used pulse-echo techniques. They think that

their results indicate a dynamic defect-controlled central-peak phenomenon.

Higher-order elastic stiffnesses [29,30,32,33,38,39] and compliances [15] were

also studied.

Only a few calculations on SrTiOa were reported. Using a rigid-shell

model with Coulomb and short-range central-force interaction, Achar,

9
Barsch, and Cross [38,39] calculated the third-order elastic constants of both

SrTiOa and BaTiOa. The SrTiOa results show that the model gives a consistent description of third-order elastic constants.

2.2 YBCO Superconductor

In 1964, Schooley and coworkers [41] first reported superconductivity in

SrTiOa, an oxide with perovskite crystal structure, with a quite low transition temperature, Tc = 0.3 K. In 1975, Sleight and coworkers [42] found a

higher superconducting transition temperature at 13 K in BaPbi-^Bia^Oa. In

1986, Bednorz and Muller [43] reported a remarkable superconducting transition at 30 K in Lai.ssBao.isCuOa (LBCO). Almost one year later, Wu and

colleagues [44] reported superconductivity in YiBa2 Cu307 (YBCO), with

Tc = 90 K. Since then, there appeared many studies on the elastic constants,

and related properties, of YBCO. Several review papers on this subject exist [45-50].

The studies on the elastic and related properties of YBCO can be divided

as follows: room-temperature sound velocities and elastic moduli [51-53,70-

72, 76-91]; temperature dependence of sound velocities and elastic moduli

[53-79,92-138]; temperature dependence of internal friction and attenuation

[55-62,73,93-108,139-158]; Debye temperature [51,54,62,68-70,78,83,86,95,

120,159-164]; linear thermal expansivity [62,68,69,79,85,95,120,166]; specific

heat [73,95, 159, 162-165]; temperature dependence of ultrasonic frequency

[59, 76, 106, 139, 143, 145-149, 167-169]. Among these studies, only thirteen

papers reported the single-crystal properties of YBCO [93,109-115,169-173].

Block and coworkers [52] measured the room-temperature bulk modulus

using a high-pressure X-ray-diffraction method and obtained B = 196 ±17

GPa. Using a similar method, Dietrich and coworkers [80] found B = 170

GPa and Fietz and coworkers [81] obtained 180 GPa; but Jaya [82] reported

a very different result, B = 92.9 GPa. Glazkov and coworkers [89] applied

neutron diffraction to study the compressibility k of YBCO at high pressure.

For the orthorhombic phase, they obtained k « 5.7 x 10“^ GPa“\ that is,

the bulk modulus B = 175.4 GPa. Using an ultrasonic pulse-echo method,

Ledbetter and coworkers [51] found a much lower value of B^ 72 GPa, cor- rected to the void-free state. Also by measuring ultrasonic wave velocities,

Al-Kheffaji and coworkers [83] and Cankurtaran and coworkers [84] reported

10
that B = 68.5 GPa for the nonporous YBCO. Some authors reported an even

lower value for the bulk modulus, for example, 46.3 GPa (at 205 K) by Almond and coworkers [53], 55 GPa by Yoshimoto and coworkers [77], 26.9 GPa

(at 220 K) by Kim and coworkers [78], 56.3 GPa by Salomons and coworkers [79], and 50.2 GPa by Bridge and Round [88]. Using a bending method

Alford and coworkers [85] measured the static Young modulus and estimated

for fully dense YBCO that E = 180 GPa. The static Young modulus of

polycrystalline YBCO was also measured by Chang and coworkers [90] using

Hertzian loading. They found at room temperature E = 226 GPa. Other re- ported elastic constants of polycrystal YBCO are listed in Appendix A. Some

argue that material defects explain the different results among the methods

applied. The material defects include voids, twins, microcracks, nonhomogeneities, texture, and residual stress. Recently, Ledbetter and Lei [174]

discussed this problem and concluded that the ultrasonic-velocity bulk modulus is more realistic because it is consistent with many physical properties:

atomic volume, cohesive energy, thermal expansivity, Debye temperature,

and Griineisen parameter. For the YBCO bulk modulus, they estimated

that B = 107 GPa.

The studies on temperature dependence of sound velocity and elastic

moduli show different results. Ledbetter and coworkers [54] measured the

shear modulus of polycrystalline YBCO between 275 and 4 K using a pulseecho method. They found that above Tc the shear modulus shows normal

behavior: it stiffens with decreasing slope during cooling, consistent with

the Varshni relationship [175]. Similar results have been reported by other

authors [56-61,63,65-67,70,71,74,77,78,94,95,98-100,107-109,113,121,122,

126,131,132,140]. Such a dependence of the elastic moduli has been observed

in single crystals [93,110,114,115].

During cooling below however, Ledbetter and coworkers [54] found

an abnormal behavior of the shear modulus: the slope increases after it

approaches zero. This behavior can be described by a “reentrant softening”

model [176]. The increased stiffness below the critical transition temperature

is the result of the softening being offset by the increased stiffness associated

with the developing superconducting phase. The calculated Debye temperature based on this model agrees with other experimental measurements. Also,

this model predicts that elastic constants in the superconducting state will

be lower than those in the normal state, consistent with the thermodynamics

of second-order phase transitions.

11
Strong thermal hysteresis in acoustic frequency and sound velocity was

observed by many researchers [53,59,73,92,95,106,117,135,147,149,177,178].

The cause of the hysteresis remains unclear. Some authors [53,73,92,117,147]

conjectured that this thermal hysteresis is caused by a phase transition or

by the coexistence of tetragonal and orthorhombic phases [59].

The changes of elastic properties at transition temperature were found

by some authors [63,76,97,143]. Remarkably, Migliori and coworkers [76] re- ported a large change of the resonant frequency at Tc, indicating a sharp drop

in the sound velocity with increasing temperature at the phase transition. In

contradiction, Laegreid and coworkers [50] did not observe any changes either

in the sound velocity or in the slope of the sound velocity at Tc. Also, elastic

softening in sound velocities above Tc were reported [75,95,102,177].

The reason for discord remains unclear. Possible reasons might be the

following: (1) Different frequencies. The frequency range applied in an ultrasonic method varies from 1 kHz to 30 MHz. (2) Mixtures of different

phases: orthorhombic I and orthorhombic II. These phases contain different

oxygen content and they show different properties [179]. (3) Impurities, such

as CuO. Anomalies of the temperature dependence of the sound velocity

and elastic moduli may arise from a magnetic phase transition in CuO. (4)

Hysteresis. Thermal history (and perhaps mechanical history) may affect

YBCO’s properties [92].

The anomalous stiffening below Tc observed in polycrystals was also re- ported in single crystals [93, 110, 111, 113, 114]. Hoen and coworkers [93]

suggested that the anomalous stiffening below Tc observed in polycrystalline

samples is not due to integranular effects, but is intrinsic to YiBa2Cu307.

Thermal hysteresis wa.s also observed in a single-crystal specimen [93].

As for the monocrystal elastic constants, only one set of complete Cij derived from the phonon frequencies of inelastic neutron scattering for a meantetragonal structure was reported [172]. Using sound-velocity measurements,

Saint-Paul and coworkers [111] estimated C33 and C44. Also from sound velocities, Golding and coworkers [113] calculated the pseudo-tetragonal elastic constants Cu and C33. Baumgart and coworkers [170, 171] determined

C'ii,C'33, and C44 by Brillouin spectroscopy.

For YBCO, some model calculations were reported. Using the interatomic

force constants, estimated from electronegativity values and bond lengths,

and the crystal structure information. Bridge and Round [180] calculated

the bulk modulus for single-crystal YBCO. They obtained B = 253 GPa.

12
This is much higher than the measured bulk modulus for the single-crystal

specimen [173], 115 GPa. Valkealahti and coworkers [181] studied structural properties of YBCO by static-lattice simulation methods and found

good agreement between the model calculations and the experimental results.

Also, by simple ionic potentials, they studied elastic constants. The elastic

and acoustic properties of a polydomain phase at the intrinsic ferroelectric

phase transition in the YBCO were studied by Vitebskii and coworkers [182].

Starting from the mean-field free energy, Millis and Rabe [183] derived ex- pressions for the singularities in the behavior of the lattice parameters and

sound velocities near Tc. Their theory predicts the existence of three types

of singularities dX Tc'. (1) a discontinuity in the temperature derivatives of

the lattice parameters; (2) a discontinuity in the elastic moduli and sound

velocities; (3) a discontinuity in the temperature derivatives of the elastic

moduli and sound velocities. For YBCO, they found inconsistencies between

the theory and the experimental results.

2.3 Other High-Tc Oxide Superconductors

La2 _rSrxCu04 (LSCO) is a compound quite similar to LBCO and has a

higher Tc of approximately 40 K. After the discoveries of LBCO (LSCO) and

YBCO, another two families of oxide superconductors with Tc above 100 K

were found: Bi-Sr-Ca-0 (BSCCO) and Tl-Ba-Ca-Cu-0 (TBCCO).

Although the compositions and the properties of the four types of materials are different, they have common features—Cu02 sheets of corner-linked

square-planar-coordinated copper, and oxygen nonstoichiometry or cation

disorder in layers that interleave the Cu02 sheets.

Studies similar to those for YBCO have been done for the other three

families: the temperature dependence of elastic moduli and sound velocities

for LSCO [56,100,104,105,107,115,116,118,133,161,184-196], BSCCO [98,

115, 160, 197-202], and TBCCO [203]; temperature dependence of internal

friction and attenuation for LSCO [56, 100, 104, 107, 133, 187-195], BSCCO

[98, 160, 169, 197, 198, 200,204], TBCCO [205]; and the room-temperature

elastic constants and sound velocities for LSCO [78,80,161,190,206-208] and

BSCCO [160,171,201,202].

Studies have shown different elastic constants for LSCO and BSCCO as

for YBCO. For example, for LSCO, a megahertz pulse-echo method finds

13
the bulk modulus to be 137 GPa [161] at room temperature; but a vibratingreed method yields 160 GPa [190]. Even the same method gives different

results sometimes. By high-pressure study, Takahcishi and coworkers [207]

reported 135 GPa for the bulk modulus, while Dietrich and coworkers [80]

found B = 160 GPa. For (Bi-Pb)2 Sr2 Ca2 Cu30io+x, the bulk modulus was

reported to be 17.4 [160] and 9.7 GPa [201] both by pulse-echo method.

The elastic moduli and sound velocities for LSCO show strong softening

when temperature is decreased [161,184,185,189-196], especially the shear

modulus softens up to 30% [161]. This lattice softening is usually ascribed

to a structural instability [184,190].

For BSCCO, the measured sound velocities and elastic moduli show stiffening when temperature decreases in both polycrystals [98,197,201,202] and

monocrystals [115,198-200].

Hysteretic behavior in sound velocities and elastic moduli was observed

also in LSCO [189,190], BSCCO [201,202], and TBCCO [203].

2.4 Summary

Many studies have been done on elastic and related properties of BaTiOa,

SrTiOa, and high-Tc metal-oxide superconductors, especially YBCO.

For the perovskites, the studies included room-temperature properties,

temperature dependences, and model calculations. Despite all the efforts, no

one measured the low-temperature elastic constants of BaTiOa, to 4 K; and

a phase transition near 40 K in SrTiOa remains controversial.

For the high-Tc oxide superconductors, the elastic properties display many

complexities. For example, the reported bulk modulus of YBCO varies in a

wide range: from 27 to 196 GPa. And the studies show different temperature

dependences of elastic constants, especially the behavior near Tc. The reasons

for discord need to be clarified.

14
Chapter 3

Materials and Measurement

Methods

3.1 Materials

The specimens used for this study include monocrystal SiTiOa, polycrystal

BaTiOa, and polycrystal YBCO superconductor.

The polycrystal BaTiOs and monocrystal SrTiOa specimens were obtained from commercial sources.

The YBCO specimen wa,s prepared at Arkansas University by Z. Sheng.

Well-mixed powders of Y2 O3 , BaCOa, and CuO were heated at 925° C in

air for 24 h. After being cooled slowly, the material was ground into fine

powder and pressed to a tablet-shape. Then the tablet was put into a tube

furnace with oxygen at 940° C for 12 h, and the temperature was kept at

650° C for another 2 h. Finally, the temperature Wcis reduced to room temperature. Thickness of the tablet-shaped specimen is 2.6 mm. Figure 3.1

shows the electrical resistance and magnetic susceptibility for the specimen,

which indicate that the superconducting transition occurs at 91 K. The oxygen content of the specimen was estimated from X-ray diffraction as 6.85,

that is, the molecular formula for the specimen is YiBa2 Cu306. 85 -

15
RESISTANCE,

SUSCEPTIBILITY

(arbitrary

uni

Figure 3.1: Temperature variation of electrical resistance and magnetic susceptibility of YBCO specimen.

16
3.2 Measurement Methods

3.2.1 Mass Density

The mass densities of BaTiOa and SrTiOa were determined by Archimedes’s

method using distilled water as a standard. Hence, density can be calculated

from the following relationship:

P =

Mo

Mo -

(3.1)

Here, Mo and Mt indicate specimen mass in the air and liquid, respectively;

and Pi denotes the density of liquid (distilled water), which can be found

in handbooks. The inaccuracy of this method is estimated as 2 parts in

10,000. Due to the chemical reactivity with oxygen and water, instead of

using Archimedes’s method, the effective mass density of the YBCO superconductor was determined by meaisuring macroscopic dimensions and mass,

with an inaccuracy of 4 parts in 10,000.

3.2.2 Acoustic-wave Velocity

To measure elastic constants, a megahertz-frequency pulse-echo method [209]

was used. The ultrasonic wave velocity v is determined by

2/ V = —

.

t (3.2)

Here, / denotes specimen length and t denotes round-trip time for an ultrasonic pulse to travel between two flat-and-parallel specimen faces. The

electric pulse is produced by a pulse generator and converted to an ultrasonic pulse by quartz and PZT piezoelectric transducers with frequencies

between 3 and 10 MHz. The transducer is bonded to the specimen surface

using phenyl salicylate and glycerine-alcohol. Figure 3.2 shows a diagram of

the electronics system, and Fig. 3.3 shows a typical pulse-echo pattern. The

travel time t is determined by measuring the time difference between consecutive echoes in a pulse-echo pattern using a dual-channel sweep-delayed

time-interval-microprocessor-equipped oscilloscope with a 0.001 ps resolution.

17
Figure 3.2: Functional diagram of ultraisonic pulse-echo sound-velocitymeasurement system.

18
Figure 3.3: Composite oscilloscope display of transverse- wave pulse-echo pattern at 4 K for YBCO specimen. Expanded first and second echoes are shown

at bottom. Transit time was measured between the first minimum in these

adjacent echoes. Carrier frequency equals 4 MHz.

19
The elastic constants C can be calculated from the general relationship

(3.3)

For polycrystal BaTiOa and YBCO specimens, which possess isotropic symmetry, we need only two measurements: longitudinal velocity vt and shear

velocity Vt. The longitudinal modulus Ci and shear modulus G can be calculated from Eq. (3.3) by the substitution of and respectively. Then

the bulk modulus B can be determined by

B = Ct- (3.4)

By using the interrelationships among isotropic-solid elastic constants, we

can also obtain other engineering elastic constants: the Young modulus

WG _ G[ZCi - 4G) ~ WTG ~ Ci-G

and the Poisson ratio _ E Ci-2G

''~2G ~2(Ct-G)'

(3.5)

(3.6)

For a monocrystal SrTiOa specimen, which possesses cubic symmetry,

there are three independent elastic constants, usually taken as C\\^C\2 ^ and

C44 . From the Voigt-Reuss-Hill [210] averaging method, the quasiisotropic

elastic constants can be obtained from monocrystal results.

The inaccuracy of elastic-constant-mecisurement is estimated to be 5 parts

in 10,000. This is based on measurements on standard monocrystal specimens

such as aluminum and copper.

3.2.3 Low-temperature Measurements

To obtain the temperature dependence of elastic constants, a lowtemperature probe was used. The probe was placed in the ullage of a helium

dewar and lowered stepwise to achieve cooling. Temperatures were monitored

with a chromel-constantan thermocouple contacting the specimen. Figure 3.4

shows the probe. The cooling rate was about 2 K/min.

20
4 cm

Electrical

Lead

Spring

Teflon

Brass

Spacer

Piezoelectric

Transducer

Specimen

Figure 3.4: Low-temperature probe.

21
Chapter 4

Elastic-constant Measurement

Results and Analysis

4.1 Perovskites

4.1.1 Polycrystal BaTiOs

The temperature dependences of elastic constants of the polycrystal BaTiOa

specimen in the range of 295-4 K are shown in Fig. 4.1. The elastic constants change smoothly within the temperature range without showing any

anomaly. The elastic moduli of BaTiOs show large changes and stiffen with

decreasing temperature, especially the shear modulus, which increases 28%.

The Poisson ratio changes in a different way—decreasing about 14%.

The room-temperature elastic constants of the polycrystal are lower than

those of a monocrystal reported by Huibregtse and coworkers [10]. Due to

the thinness of the monocrystal specimen for the present study, only one

elastic constant Cu was obtained, which is close to the results of Huibregtse

and coworkers. Therefore, the monocrystal measurement results at roomtemperature by Huibregtse and coworkers are preferred and tabulated in

Table 4.1 for further use.

22
(

962

)

0

/(

1)0

Figure 4.1: Temperature variation of elastic constants of polycrystal barium

titanate.

23
to

Oi

CM

tt

(A Ws

cd

(Acou—<u ns

a;ImaM

cd

a; —M(d H

CO

lu 1 1 1 1 rr CM 00 to o CO •o* to CO CM CM CM o o o o

QO CM

cd CO t^

CL 05 cb 00 o 05 QO o' CM ' O CO 05

?o 05 o 00

CQ CL cb CT5 tb o

CO tw

TT o 00 05 o

cd A CM CO to 05

P-i ib tb tb 00 o TT to '—

^ 

CO o O' to

cd o t^ 00 to o

CL CO cb tb 05 O CO CO bCM CO , , (A CO o t^ to

to 00 00 o c

CO CM CO co o C5 o o , CAa. CO CM CO CM

05 05 CO c

to 00 ^ to cu o o O o

eo cc o t^ 00 E 05 f—

u 05 to CO

bO tb tb tb cb « eo o Q O O o

cd

H pa o

CQ

CQ CO >- >-

<«-A <d

I/) u

«> vis

'o>o

t* -o

a;uu

h>MoUa

24
4.1.2 Monocrystal SrTiOa

The measured elastic stiffnesses C,j of monocrystal SrTiOa at ambient temperature agree well with those determined using the same method by Bell

and Rupprecht [25] and by Wachtman and coworkers [26]. The quasiisotropic

elastic constants calculated from the Cij using the Voigt— Reuss—Hill [210]

averaging method are listed in Table 4.1.

For the monocrystal SrTiOa, Fig. 4.2 shows the temperature dependences

of the elastic moduli and the Poisson ratio. In the temperature range 295-

108 K, the elastic constants show similar behavior with that of BaTiOa

G, B increasing and v decreasing, in a smaller magnitude than BaTiOa,

with decreasing temperature. A cubic-tetragonal structure phase transition

occurs at 108 K where the elastic moduli drop drastically. Below 108 K,

the Young and shear moduli decrease further and reach a minimum at 66 K.

Different from the behavior of Young and shear moduli, the bulk modulus

increases below 108 K and reaches its first maximum at 66 K and the second

maximum at 20 K. Unlike BaTiOa, the Poisson ratio of SrTiOa shows a

large incre2ise below 108 K and goes through two maxima at 66 and 20 K,

respectively. All the changes of elastic constants below 108 K indicate some

kind of phase transitions occur in this region. Since little or no hysteresis

was observed in SrTiOa specimen, these transitions may be approximately

second order.

4.2 Polycrystal YBCO Superconductor

4.2.1 Ambient-temperature Measurements

The sound velocities and elastic constants of YBCO measured at room temperature are also tabulated in Table 4.1.

Since the polycrystalline oxide superconductor contains voids, the measured elastic constants must be corrected to the void-free state to get the

intrinsic-material elastic constants. First, the porous ceramic superconductor was considered as a composite containing spherical particles—voids. Because the void density pp = 0, the void volume fraction c can be determined

from the X-ray mass density pm and macroscopic density p:

p = pm{l - c) + ppC = pm(l - c). (4.1)

25
C(T)/C(295)

1.3

1.2

1.1

1.0

0.9

0.8

0.7

SrTi 03

1

100 200 300

TEMPERATURE (K)

Figure 4.2: Temperature variation of elastic constants of monocrystal strontium titanate.

26
Here, the subscripts m and p denote matrix and particle.

Then, a model given by Ledbetter and Datta [211] for particle-reinforced

composites was used to correct the measurements to the void-free state. Assuming randomly distributed spherical voids and taking the voids to possess

zero elastic resistance to both dilatation and shear, we obtain expressions for

shear and bulk moduli for a material containing spherical pores:

B = B.

and

G = G

^Gm + ^cBm ’

(1 — c){9Bjn + 8G'm)

(4.2)

(4.3)

(9 -f 6c)Bm 4* (8 -f 12c)Gt

Solving Eqs. (4.2) and (4.3) for Gm and Bm^, we obtain the following results:

-(1 — c)G^ + (3 - 2c)B - (5 + ic)G

o

G'^-3(l+c)BG' = 0, (4.4)

and

Brn = ^GmB

4(1 — c)Gm — ^cB

Applying the usual quadratic-equation formula, Eq. (4.4) becomes

(4.5)

where

1 —A2 + (^2 — 4^1^43)2 ,

= (3 - 2c)B - (5 + 4c)G,

A3 = —3(1 T c)BG.

(4.6)

(4.7)

Equations (4.5) and (4.6) represent bulk and shear moduli in the void-free

state. Using Eqs. (3.5) and (3.6), we obtain the void-free Young modulus

and Poisson ratio.

The corrected results are listed in the fourth row of Table 4.1. The cor- rected bulk modulus agrees well with the monocrystal measurement, 114.8

GPa, based on high-pressure X-ray diffraction, by Aleksandrov and coworkers [173].

27
Table 4.2: The fitted parameters of the moduli in the Varshni function.

Co

(GPa)

s (GPa) (K)

Bulk modulus B 105.98 3.25 147.91

Young modulus E 157.94 4.68 115.41

Shear modulus G 63.09 1.88 111.16

4.2.2 Low-temperature Behavior

Figure 4.3 shows the 295-4-K temperature dependences of elastic moduli

and Poisson ratio of polycrystalline YBCO superconductor. Obviously, the

elastic constants of the YBCO show very different temperature dependences

from that of SrTiOs. They also show smaller changes than BaTiOa, the

shear modulus of the YBCO increasing only about 7%, although they show

a similar tendency with temperature. Against expectation, the elastic constants of this YBCO specimen showed practically no hysteresis. Similar to

some previous studies, within the measurement error, no significant changes

in the elastic stiffnesses were observed near Tc. But the Poisson ratio behaves otherwise. Near Tc, rather than decreasing smoothly during cooling,

the Poisson ratio shows an irregular change: near 100 K it begins to decrease

irregularly. This reflects some internal change in YBCO, because, among all

the elastic constants, Poisson ratio shows the strongest sensitivity to changes

in interatomic bonding.

The behavior of all the elastic stiffnesses of the polycrystalline YBCO

obeys the Varshni function [175]

Ct = Co

exp(0jE;/T) - 1 (4.8)

Here, Ct and Co denote an elastic constant as a function of T and at zero

temperature, respectively; s is a constant related to zero-point energy; and

Qe denotes the Einstein temperature. Table 4.2 shows the fitting results for

these parameters.

The following analysis shows that the parameter s can be calculated from

some fundamental physical properties such as the Einstein temperature and

28
C(T)/C(295)

1.08

0.98

1

« « 0 100 200 300

TEMPERATURE (K)

Figure 4.3: Temperature variation of elastic constants of polycrystal YBCO

superconductor.

29
the Griineisen parameter. The bulk modulus will be considered as an example. For the bulk modulus, in the high-temperature limit, Eq. (4.8) can be

rearranged to the form

(4.9)

Figure 4.4 shows the the Varshni function and Eq. (4.9) for the bulk modulus.

At zero temperature, Eq. (4.9) predicts that

Bq — Bq

Here, Bq denotes the harmonic-model value of the bulk modulus.

On the other hand, zero-point energy Uzp can be written as [212]

(4.10)

9

8

Uzp — ^^Ox). (4.11)

Here, k indicates the Boltzmann constant. Then, we find the bulk modulus

caused by zero- point energy:

Bzp

Since

Bo-Bo = k(^

7 =

v=v^

=h

8

d^eD

dv^ V=Va

din Of)

dlnV ’

Od

where 7 is Griineisen parameter,

d0£)

~dV~ ' V

Substituting Eq. (4.14) into Eq. (4.12), we obtain

Qkyl'y + 1)0D

(4.12)

(4.13)

(4.14)

Bo — Bq — 8K

(4.15)

Comparing Eq. (4.15) with Eq. (4.10), yields an expression for s in terms of

Einstein temperature, Griineisen parameter, and atomic volume:

s =

-f l)0f;

(4.16)

30
BULK

MODULUS

Figure 4.4: Schematic bulk-modulus-temperature curve in an Einsteinos dilator model. Bq denotes the harmonic-model value of the bulk modulus.

31
In this equation, the relationship 0£> = 40£;/3 between Debye temperature

and Einstein temperature was used.

For YBCO, the Debye temperature 0£> is 440 K [213]. Hence, the Einstein

temperature 0^; = O.750£> = 330 K. Also for YBCO, Ledbetter [214] found

that the Griineisen parameter is around 1.35. Using these numbers for Qe

and 7 together with K = 13.51 in Eq. (4.16), we find s = 3.21 GPa,

which agrees well with the fitted value 3.25. The fitted Einstein temperature

is lower than the value calculated from the measured Debye temperature.

The reason for this is not known. Nevertheless, the Varshni function fits the

measurements well. And the fitted result of s is in a good agreement with

the simple Einstein-model calculation.

32
Chapter 5

Debye Temperature

The Debye characteristic temperature is a fundamental parameter that cor- relates with many physical properties of solids, such as specific heat, elastic constants, and melting temperature. Moreover, for superconductivity,

as mentioned in the introduction, the Debye temperature relates to the

electron—phonon parameter and to the critical transition temperature. Besides specific heat and elastic constants, there are various experimental methods to determine Debye temperature. For example, it can be determined

using X-ray diffraction, Mossbauer effect, and electrical resistivity measurement. A complete review on this subject was given by Herbstein [215]. Here,

only the elastic-constant method will be considered.

In Debye’s theory, the Debye temperature is defined through the cutoff frequency vd of the Debye spectrum for the elastic vibrations in an elastic

continuum:

0£) = (5.1)

Instead of considering a single frequency as proposed by Einstein, Debye

assumed that the frequency distribution function takes the form

N{v)dv =

47t ^

-f 2v^ V jy < I'D

0 U > I'D

(5.2)

where N{i') is the number of frequencies between v and u dv. With the

constraint that the total number of normal vibrations must be 3A^, where N

33
is the number of atoms in the crystal,

Hence,

where Vm is defined by Eq. (1.12). Substituting Eq. (5.4) into Eq. (5.1), we

find Eq. (1.11).

For YBCO, using Eq. (1.11) and the mean sound velocity and the atomic

volume given in Table 5.1, we find that Qd = 423 K, which agrees well with

the specific-heat Debye temperature, 440 K [213]. The Debye temperature

at T = 0 K, 0j), can be obtained from the mean sound velocity at the

same temperature by extrapolation of ela.stic constants using their temperature dependences. For YBCO, = 437 K, which is almost identical with

the specific-heat value. Theoretically, at T = 0 K, the Debye temperature

calculated from elastic constants is exactly the same as that from specific

heat [216]. Table 5.1 shows also the calculated Debye temperatures from

elcLstic constants for BaTiOa, SrTiOa, LSCO, and BSCCO.

Figure 5.1 shows the temperature dependence of 0£> calculated from measured elastic-constant temperature dependences. These three materials show

strongly different behavior. For BaTiOa, 0£> increases smoothly about 12%.

But, for SrTiOa 0£> decreases about 10%, with a sharp decrease near 108 K

caused by the paraelectric—improper-ferroelastic transition. As for YBCO,

0£) increases almost linearly about 3% and shows no discontinuity around

Tc.

34
Table

5.1:

Sound

velocities,

bulk

moduli,

Debye

temperatures,

and

electron—

phonon

parameters. 0 0 •n* 40 rr 05 cs 1 cs tx 0 40 cs C<5 00 d d TT d td u

"eu QQ

CuU

164 183 138 118 94 00

05 96 98

oQ© £ 1 627 195“ 368 240“ 437 319 270“

lAOx.

n cs nC) 0:1 05 1 1 es 1 © 40 n nc

00 10 tx

n 00 es

05 cs 00 es 05 t-; 05

:s

B

d xXd 00 xX cs tx‘ es

05 tx xX tn 40 10 40 1 1 1 n1 es1 CS1 CS1 CS1

tx

M 0 xX Tt<

a

n 05 05 tx 00 lO cs 40 CN XX d d «r ed d XX XH T"1 ^x

'U' 05 05 tx oq Ox cd d 1 t^ 1 d d

Tf 1 2.

(O

^x tx 0 _ - _ (Aa. TT cs 40 05 C^ Sp

05 e^

m c

Lf3 1 CO 1 CO cs 1 By d d d d d

_ - ^ 1 CN 00 C'l xX

m kO 1 V 00 05 0 es

XX0na

n0 V0

GO 0 lA

Ues a 0

io.25

B0U5

CN su

GOd0 0es

lx

e^ aU

n0

P5 0

OQo

h.d

.O

CT

C/5

00

<5* Ues

lx

ei a0es a

c/3es

Ox

ft u

cd CQ

cs H

H<c 03

Hkx CD

Ox

fS 00

(4 x3

c/3es CO

03

•H (Bi-

(A

lxUMoou

"Cs

lx

4;

(AB0Q©

XX

nsV1 u

u:uVa

c/^

"d

4)lxs

41B

"osc

xx

(Asou

.0

35

Calculated

for

O7.0.
0o(T)/0d(295)

TEMPERATURE (K)

Figure 5.1: Temperature variation of Debye temperature for BaTiOa, SrTiOa,

and YBCO superconductor. The absolute ©£> values at room temperature

appear in Table 5.1.

36
Chapter 6

Bulk Modulus

6.1 Calculation from Madelung Energy

6.1.1 Born Ionic Model

In general, there are two kinds of interactions between particles in a crystal:

attraction and repulsion. The first is usually long range, while the second

is usually short range. Atomic bonds due to these two interactions can be

divided into primary and secondary in terms of interaction strength. Primary

bonds include ionic, covalent, and metallic. Van der Waals and hydrogen

bonds are considered secondary, being much weaker. For an ionic crystal,

according to the Born model, the atomic interaction potential contains

only two energy terms and can be written cls (see Brown [218]) = ( 6 . 1

)

Here, Z, is the charge number of the ith ion; h' is a constant and n the Born

exponent; e indicates electronic charge and r{j denotes the distance between

ions. Then, the lattice energy, f/, can be determined by considering the

contributions from each ion:

Me^ b

+ —

•

( 6. 2)

Here, indicates a sum not including j = i; r denotes the nearest-neighbor

distance, b the repulsive constant, M the Madelung constant, which depends

37
only on the crystal structure. In Eq. (6.2), the first term represents Ipngrange point-charge Coulomb interactions and is often called the Madelung

energy. The second term is the short-range core-core repulsion interaction,

often called the Born-Mayer repulsive energy. At equilibrium.

dr

Me^ nb

ro ^2 = 0 , r=ro

SO

h =

Me2 ,n—

1

n

Then, Eq. (6.2) becomes

(6.3)

(6.4)

U =

r

(6.5)

Therefore, the bulk modulus is determined by

\ny')v.v.

Here, K denotes atomic volume, and

Em = —

9Va

Me^

ro

is the Madelung energy.

( 6 . 6 )

(6.7)

6.1.2 Madelung Energy

To calculate bulk modulus using Eq. (6.6), the Madelung energy (or the

Madelung constant) has to be calculated. Because of long-range electrostatic

interactions, the series of terms in the evaluation of the Madelung constant

converges slowly. Several methods (direct and indirect) for evaluating the

Madelung constant exist. In 1921, Ewald [219] introduced a general method

that involves the reciprocal lattice and achieves rapid convergence. Although

the Madelung constants for most simple crystal structures are known, they

are unknown for general ionic charges and for complicated crystal structures.

In this study, an Ewald method described by van Gool and Piken [220] was

38
used to calculate the Madelung constants for perovskitelike metal-oxide su- perconductors, especially YBCO.

To verify the computer program, the Madelung constants were calculated

for all the crystal structures listed by Tosi [221], the perovskite crystal structures reported by Naimon [222], and the oxide-superconductor crystal structures considered by Iguchi and Yonezawa [223]. For all cases, the previous

results were reproduced.

For this study, the unit-cell dimensions, ionic positions, and labeling for

YBCO established by David and coworkers [224] were adopted. Figure 6.1

shows the crystal structure for orthorhombic YiBa2 Cu307. For valences, we chose that -f3 for Y; H-2 for Ba, Cul, and Cu2; —2 for 01

and 04; and —1.75 for 02 and 03. These choices are discussed in Chapter

IX.

6.1.3 Bulk Modulus

Before the bulk modulus can be calculated from Eq. (6.6), we also have to

estimate the Born exponent for YBCO. For oxides, the Born exponent n

was reported in the range between 3.79 and 4.95 [225]. For BaTiOa and

SrTiOa, from the the Madelung energies and the measured bulk moduli, n

were estimated to be 4.40 and 4.30, respectively. Hence, the average, 4.35,

was taken to be the Born exponent for YBCO. With the atomic volume K =

13.51 and the Madelung energy Em = —283.98 eV/molecule, we find the

bulk modulus B = 97.9 GPa. This result agrees well with the measurements

of both polycrystal, 100.9 GPa (see Table 4.1), and monocrystal, 114.8 GPa

[173].

6.2 Comp8irisons with Other Oxides

As described in Chapter II, the reported bulk modulus for polycrystal YBCO

varies in a wide range: from 27 to 196 GPa. The result given by the present

study is much lower than those obtained from high-pressure studies. It is

necessary to clarify what is the true bulk modulus of YBCO.

Since BaTiOa and SrTiOa are the crystal-structure building blocks for

YBCO, it is useful to compare the bulk modulus for YBCO with those

for the perovskites. Table 4.1 shows that the bulk modulus for YBCO is

39
I

I

40
lower than those for BaTiOa and SrTiOa. This is not surprising because

the YBCO structure is formed approximately by stacking three unit cells

of perovskite and removing two oxygen atoms. Such a structure essentially

must be elastically softer than the complete structure, as typified by BaTiOa

and SrTiOs- Furthermore, for the real superconductor YiBa2 Cu307_5 , 8 is

between 0 and 0.5; thus, an additional oxygen site is occupied incompletely.

In YBCO, although some Cu-0 bonds are shorter, hence stronger, than in

CuO, a structure’s overall macroscopic stiffness depends more strongly on

the weaker links: Y-0 bonds, Ba-0 bonds, and especially the missing Cu-0

bonds.

Studying many monoxides and perovskites, Ledbetter and Lei [174] found

a correlation between bulk modulus and cation-anion bond distance shown in

Fig. 6.2. In YBCO, there are 27 metal-oxygen bonds (8 Y-0, 10 Ba-0, 9 Cu0). Table 6.1 shows these bond distances given by David and coworkers [224].

Table 6.1: Bond distance and bulk modulus for YBCO.

Bond Distance (A) N B (GPa) (= 3630/r'*)

Y-03 2.387 4 111.81

Y-02 2.402 4 109.05

Ba-04 2.741 4 64.31

Ba-Ol 2.877 2 52.98

Ba-03 2.949 2 48.00

Ba-02 2.986 2 45.66

Cul-04 1.843 2 314.63

Cul-01 1.942 2 255.22

Cu2-02 1.929 2 262.17

Cu2-03 1.958 2 246.98

Cu2-04 2.306 1 128.37

Using the bond lengths in Table 6.1 together with the curve in Fig. 6.2,

we find that for YBCO the bulk modulus equals 138 GPa. However, this

calculation Wcls done with an assumption of a rocksalt-structure coordination

number: six. Correcting the strong Cu-0 bonds for reduced coordination

41
BULK

MODULUS

(GPa)

CATION-ANION DISTANCE (A)

Figure 6.2: Dependence of bulk modulus on cation-anion distance for some

oxides.

42
number (four and five) leads to B = 116 GPa, close to the measured bulk

modulus B — 101 GPa.

43
Chapter 7

Monocrystal Elastic Constants

of Orthorhombic YBCO

In the previous chapter, the bulk modulus of YBCO was discussed. Here, we

consider the monocrystal elastic constants Cij.

Since the YBCO superconductor possesses orthorhombic symmetry

(space group Pmmm, no. 47), it exhibits nine independent Cij. Although

many studies have been done on both polycrystal and monocrystal YBCO,

only one set of complete Cij was reported, as mentioned in Chapter II. Table 7.1 shows the results reported by Reichardt and coworkers [172]. Although useful, the results reported by Reichardt and coworkers present some

problems. First, their Cij were obtained for a mean-tetragonal structure,

thus only six out of nine independent Cij. Second, the bulk modulus obtained from their Cij by Voigt—Reuss—Hill [210] averaging disagrees with the

monocrystalline-measurement, B = 115 GPa, reported by Aleksandrov and

coworkers [173]. Third, errors in elastic constants determined from phonondispersion-curve slopes duj/dk are usually larger than those measured ultrasonically.

Table 7.1 shows also some incomplete Cij reported by other authors [111,

113,170,171].

Here, for orthorhombic YBCO, an estimate of Cij based mainly on measurements on monocrystals combined with some analysis— theory was made.

Polycrystal measurement results were excluded. Especially, all high-pressure

measurement results were avoided since they present some problems [174],

not yet understood, related perhaps to pressure-induced phase transitions.

44
V

eS00

esMoucOuMV

OhP

in oomM

CO

Ch

(d

COpOuu• CO

'o3

CO>»

IHuOco

'TS

a;6 :?

*43 Co

CO« s

-dC

cd

«H

;3CO

cd

a; I(d pL, H

T3c

<d

'S c2 H O

Tf* TT T»i

qS

O 1-H CN cs M

ca Tf

cs

05

m rH lA

eo rH

rH

n o

93 Cj orH

fO o

89 o 'P^ o

100 37

<0 lA t- o 00 Oi o t- o lA m O m o n lA to o oo m O o 00

cO lA CO CO

1-^ f^

cs

O Tfc CO

cs

VCM

rH o CO n CO cs o es es CN n rH f 1 O

b-

?r

b- • rH .

-«-• V

"3 e 13 13 >o

*;3 •**V

*>V

wM

TJ

:a

-«-> «> -•-> ’13 2cd o 9

ctf

»w00

009 ti 1^

**9 CJM9o

CO ,i4V s99 2"o

JS

CJ 'S

-*j9

'S

4^

cn4>M

to < ffl o Pi CA (X

i<o00

*«

c«VM9Vs

45
The ultrasonically measured C\\ and C33 by Golding and coworkers [113]

(see Table 7.1) were adopted. Since these two elastic constants were measured

at 80 K, they have to be adjusted to ambient temperature. Applying the

polycrystal ratio C^(80)/C^(295) = 1.05, we find that at room temperature

C\i = 223.4 GPa and C33 = 138.4 GPa, respectively.

From a ratio C22 ICW = 1.091 calculated by Bcietzold [226] from an empirical two-body-potential method, C22 wcls determined.

To determine G12, G13, and C23, a simple elastic problem was considered.

For a hydrostatic pressure P, Hooke’s law gives —P = C\\S\ + C\ 2^2 H" ^'1363, —P = C\2S\ + C22^2 + C23£3 i —P = Ci36i + C23^2 “1" 03363.

Solving these equations, we obtain

C33 + (1 — G — ^)C23

^ _ (PCii — b^C22 {b — d)C33 — 6(1 -f a — b)C23

"" a(l + 6-a) ’ C23 — A I (a -f- o6 — a^)[9Bv — {C\\ + C22 C^sa)]

-f4a^Gii — 26(a + 6 — 1)(722 2(1 a — 6)C33| ,

where

A = i . 2[2(a T 6 + ah) — (1 + + 6^)]

Here, a = £ 1/63 and 6 = £2/^3 are strain ratios. Using the three principal

linear compressibilities

3 = * = 1,3 (7.6)

J =l

found by Aleksandrov and coworkers [173] (see Table 7.1), we determined

that a = 0.58 and 6 = 0.50. In Eq. (7.4),

By = — [Cii -|- C22 + O33 -f 2 (C12 + Gi3 -f C23)]

y

Ci2 —

"F (1 — <i)bC22 —

a(l + 5

(7.1)

(7.2)

(7.3)

(7.4)

(7.5)

46
represents the Voigt-bound bulk modulus. We chose By = 115.9 GPa

to make the Hill-average bulk modulus from the C,jS equal the value of

monocrystal measurement by Aleksandrov and coworkers.

To determine the three shear moduli C44 ,Cs5 , and Cee, again, two Cij

ratios, CaaICqq = 0.633 and = 0.485, derived by Baetzold [226]

were used. As above, more confidence was taken in Baetzold’s Cij ratios of

diagonal terms than in his absolute CijS. The final problem then is to estimate Cee- The criterion was chosen that the Debye temperature calculated

from the Cij [5] equals the value determined from specific heat. Using the

temperature dependence of the Debye temperature given in Chapter V and

the value at 0 K (specific heat value) = 440 K, we can obtain the value

calculated from Cij at room temperature = 426 K. Hence Cq& = 96.8

GPa.

The estimated elastic constants for orthorhombic YBCO are shown in

Table 7.1. Except C12 , the estimated Cij differ from those measured by

Reichardt and coworkers less than 10% on average. The large discrepancy

between the estimated and measured C12 is unclear. Because C12 is not a

simple elastic constant (no wave speed in the crystal depends only on C12 ),

it is difficult to calculate from basic principles and difficult to reason about

on physical grounds.

47
Chapter 8

Electron-phonon Parameters

Electron—phonon interactions in solids play an important role in the theory of superconductivity. The interaction between electron and phonon is

described by a parameter A, called the electron—phonon coupling parameter. In conventional superconductors, electrons indirectly attract each other

through exchanging phonons and form Cooper pairs, which are principally

responsible for superconductivity. Therefore, the superconducting transition temperature Tc is sensitive to the electron— phonon interactions and the

phonon frequency. Since the phonon describes the vibrations of the lattice,

the phonon frequency strongly depends on the ion maiss, M. Hence Tc is a

function of ion mass. In fact, the isotope effect provides a simple relationship between Tc and ion mass: Tc ^ Af~“, where a is a parameter. The

isotope effect indicates that Tc approaches zero when M —> oo. When the

ion mass approaches infinity, ions can not move, hence, no lattice vibrations

occur. Therefore, an isotope effect shows clearly that an electron— phonon

interaction is the origin of superconductivity.

By measuring the isotope effect, Zettl and coworkers [115] studied the

electron—phonon interactions in LSCO and YBCO high-temperature oxide

superconductors. Within the error limits, they found shifts in Tc when re- placing ^^0 with and that a = 0.140 ±0.008 and 0.023 ±0.005 for LSCO

and YBCO, respectively. The measured values of a for LSCO and YBCO are

smaller than 0.5, a value expected from the conventional phonon—mediated

paring mechanism. As described by Pickett [227], the isotope effect is not

as simple as is usually assumed because “it involves not only the change in

phonon frequencies but also the readjustment of the amount of retardation

48
of the electron-phonon interaction arising from the change in frequencies”.

In fact, several elemental superconductors show values of a strongly different from 0.5. For example, Zr and Ru have small values of a, near zero. a-U even shows a negative value. Therefore the small values of a for LSCO

and YBCO can not exclude electron-phonon intreaction as a possible paring

mechanism for oxide superconductors.

If it is assumed that electron-phonon interaction is responsible for su- perconductivity in high-Tc oxide superconductors, then A can be estimated

from Tc and 0d using the BCS relationship. However, the BCS expression

was obtained for a weak-coupling approximation: A <C 1. The well-known

McMillan [228] expression is valid up to A = 1. The Allen— Dynes [229]

expression applys only for A > 10. Therefore, instead of either the BCS re- lationship or the McMillan or the Allen— Dynes relationships, the expression

by Kresin [230] is preferred.

Directly from the Eliashberg equation, Kresin derived a simple analytical

expression describing Tc for a full range of A:

Te = O.250D[exp(2/A) - 1]“'/^ (8.1)

Unlike the BCS, McMillan, and Allen— Dynes relationships, Kresin’s relationship predicts that elastic softening always increcises Tc.

Using the information of Tc and 0£) in Table 5.1, the

electron-phonon interaction parameter A can be estimated from Eq. (8.1).

The results are shown in Table 5.1.

Table 5.1 shows that the calculated lambdas vary over a wide range, from

0.20 for SrTiOa to 7.82 for TBCCO. The estimated values for A in Table 5.1

provide a test as to whether an electron—phonon coupling mechanism can

explain superconductivity in these oxides. If the lambdas are unreasonably

large, then electron-phonon coupling is unlikely.

For LSCO, Rammer [231] calculated A w 2 using a strong-coupling theory.

From specific-heat measurements, Schlesinger and coworkers [232] estimated

A « 1-2. Using a method based on an analysis of heat capacity data, Kresin

and Wolf [233] estimated that A « 1. 8-2.0. Using the Alien-Dynes relation,

Horie and coworkers [58] found that A = 2.4.

For YBCO, Rammer [231] similarly obtained A « 5. Horie and coworkers [58] got an even higher value, A = 6.5. Zeyher and Zwicknagl [234]

calculated the superconductivity-induced phonon self-energy in the strongcoupling limit and found that A = 2.9.

49
For BSCCO and TBCCO, no reported estimates of A are available.

Independent calculations and estimates support the derived values of

lambda from Kresin’s expression, at least up to A « 5. This leaves the

highest value, A = 7.82 for TBCCO, without theoretical or experimental

support.

Figure 8.1 shows how Tc changes with A. Besides the Kresin relationship,

it shows lines representing the BCS, McMillan, and Allen— Dynes equations.

As shown in the figure, almost for the full range of A, from 2 to 10, BCS

theory predicts too high a Tc, and McMillan too low a Tc. It is not a surprise

because the values of A in this range are beyond the limits of validity for

both the BCS and the McMillan expressions. Figure 8.1 also shows that the

Allen— Dynes curve lies near and parallel to the Kresin line. In fact, for large

A (strong-coupling limit), the Kresin relationship reduces to the form:

Tc = O.1770cA'/^ (8.2)

which is very close to the Allen—Dynes expression:

Tc = O.1820£,A'/^ (8.3)

The behavior of the Kresin curve for strong coupling supports the

Allen— Dynes contention that for large A, A^/^0£>, rather than A, governs

high Tc

In the weak-coupling limit, the Kresin relationship fails to come to the

BCS expression. Instead, it takes the form

Tc = O.250£) exp(— 1/A). (8.4)

50
1.0

0.8 - <?/

/

/

/

ELECTRON-PHONON PARAMETER, X

Figure 8.1: Variation of Tc/0£) with electron— phonon coupling parameter,

A. Points for seven oxide superconductors based on Kresin model. For comparison, the figure shows also curves from BCS (weak-coupling), McMiUan

(moderate-coupling), and AUen—Dynes (strong-coupHng) models.

51
Chapter 9

Valence Problem

Superconducting properties in YBCO are very sensitive to the defect structure of oxygen ions, which determines the copper—oxygen ionic-charge configurations and the content of electron-hole carriers. Therefore, for understanding the mechanism of superconductivity in YBCO, it is important to

know the ionic-charge distributions.

Calculations of lattice properties for high-Tc metal-oxide superconductors

based on an ionic model were reported. By calculating the Madelung site

potentials, Iguchi and Yonezawa [ 223] determined the most stable Cu'*'^ position and showed the remarkable stabilization of oxygen ions in YiBa2Cu307,

compared with YiBa2Cu30s. Wright and coworkers [235] focused on forces

in YiBa2Cu307 and found that mechanical stability imposes constraints on

copper-charge distribution. Kress and coworkers [ 236] predicted phonondispersion and phonon-density-of-states curves and concluded that interatomic interactions in YiBa2Cu307 resemble those in perovskites and other

oxides. Further, they concluded that the ionic polarizability suggests strong

electron—phonon interactions. Cohen and coworkers [ 237] considered lattice dynamics in La2Cu04 and YiBa2Cu307 and reached similar conclusions:

important ionic contributions and large electron— phonon matrix-element en- hancements. Also, they failed to find evidence for Cu'*'^. Using a Madelung

site-potential calculation, Kondo and coworkers [ 238] considered the holecarrier distribution and found that the holes enter the Cu02 plane.

For present purposes, the Madelung site potentials were calculated to

determine the possible valence configuration for YiBa2Cu307. The Madelung

52
( 9 . 1

)

site potential for the ith ion is defined by

^. = E rj - ri

The same computer program mentioned in Chapter VI was used for the

calculation.

Again, to do the calculation, the unit-cell dimensions, ion positions, and

labeling established by David and coworkers [224] were adopted. Table 9.1

shows this information.

Table 9.1: Crystallographic data for YBCO superconductor at room temperature.

Ions

Wyckoff

symbol X y Z

Y Ih 1/2 1/2 1/2

Ba 2t 1/2 1/2 0.1844

Cul la 0 0 0

Cu2 2q 0 0 0.3554

01 le 0 1/2 0

02 2s 1/2 0 0.3788

03 2r 0 1/2 0.3771

04 2q

a = 3.8187 A

0

6 = 3.8833 A

0

c = 11.6687 A

0.1579

Source: David and coworkers [224].

For valences, the seven possible copper—oxygen ionic-charge configurations suggested by Muller [239] were used. To deduce these valences, Muller

made two assumptions: (1) unit-cell charge neutrality, and (2) oxygen ions

possess only —1 or —2 charges. From the measurement results of 772 = 1 and

772 = 0, where 771 and 772 denote the asymmetry parameters at the Cul and

Cu2 sites, Muller concluded that 02 and 03 must have the same valences.

The nominal charges -f3 and +2 were assigned to yttrium and barium, re- spectively.

53
The ion-charge configurations and the site potentials are shown in Table 9.2. For each configuration, Table 9.2 shows also the associated total

ionic energy, Madelung constant, and the calculated bulk modulus using the

method given in Chapter VI.

As shown in Table 9.2, the Madelung energy depends strongly on ioniccharge configuration and varies over a wide range: from —153 to —336

eV/molecule. The site potentials also change remarkably with ionic charges.

In the actual superconductor, YiBa2 Cu307_5 , S varies between 0 and 0.5.

Thus, one of the four oxygen sites is occupied incompletely. This site may be

electrostatically unstable, that is, possess the highest potential energy. Beech

and coworkers [240] identified this partly occupied site as 01. According to

Table 9.2, only ionic-charge configurations I and III have a high potential

energy at the 01 site. In these two configurations, the site potential for

Cu2 exhibits a big difference: —4 eV in I and —50 eV in III, that is, Cu2

is much more stable in III than in I. Therefore, the two possibilities can be

distinguished by the difference of site potential for Cu2 in the configurations

I and III. The possibilities of configurations I and III can be also distinguished by

considering the bulk moduli: 53 and 113 GPa in I and III, respectively.

In this study, the measured-polycrystal bulk modulus is 101 GPa. Using

X-ray-diffraction, Aleksandrov and coworkers [173] found for monocrystal

B = 11b GPa. Thus, the observed bulk modulus supports strongly charge

configuration III. Since no one reported evidence for a valence of -f*3 at the Cu2 site, a

perturbation of configuration III is proposed. The results are shown in Table 9.2 as configuration A. Obviously, configuration A can be also considered

as a perturbation of configurations V and VII. In configuration A, the va- lences for Cul and Cu2 are equalized as -|-2, the usual oxidation state. Also,

one hole was introduced into the Cu02 plane, on the oxygen sites. Followed

Muller’s idea [239], we assumed that the hole was equally distributed at the

two oxygen sites 02 and 03. We think that superconductivity in YBCO

arises from hole pairs in the Cu02 planes. In configuration A, compared

with configuration III, even though the site potential of 01 was remarkably

decreased, 01 remains the most unstable site among the oxygen sites. The

bulk modulus was reduced only slightly: from 113 to 98 GPa.

54
Table

9.2:

Site

potentials

and

Madelung

energies

in

YBCO

for

various

ion-charge

configurations.

m

tLO o es t- A<

O

m 00 o r-^ o q CO q 00 00 O) CO t—

00 < CN CN CN1 1 1

CN

d

t-1 •V1 LO

CA

A"1

CO

es1

es

A<1 1

COCO

CO

00

CN1

CN1

lO

CD

0> LA O) 9> 00 00 CO CO lO CN

Oi

iA CO CN CM es es CN

00dlf5 •dLAdtq

CA

q t-- cCO

CO

CO CN 89.

2C > 1 1 1 o> e«5 00 LA LA CA ro COCO1

CO 1 1 1 1 1 1 1 1 1

00 t- CA

O A< A* o lO

^9\ 00 LA o rr CA q 00 q lO lO

CO es es es 00 es eA 00

d A* CN CA CN

9* > 1 1 1 1 e»5 7

es1 'f1 es1001 1 7

CNt

CA

es1

CN1 CO o00

LA to CA e- LA CO A< CO

OD 00 o> es to LA q CO lO q

0) > cs CM es es es d « d to* e-* t-* d CO s C^ A 1 1 1 1 o>1 to1 fA1 LA1 1 CA1 CO1 1

00es

CN1

0)

0>

w 1 *»

IduS

bO « t- to es LA es o CO

—

65.25

C 00 at es

qqqq

q CO

wCN oU > CN es es 00 00 00 d d A< d 00

N>4 1 1 1 1 to1

es1 1 es1 es1 7 1 CN1

001 1 OLO

cr> 00 to o> tn CA to

O on es r- es «A e»; AJ q q q CO vu

00 0> NH «»? es es es es d d eA to* LA1

00* fA t-* CO00

CN 1 1 1 1 m1 1 CA1 1 to1 to1 es1

es

CA1

CO1

LA to A* LA A* es t- o> O)

LO

w#4 es es es

e^ t-;

q

qqq LO

q

q 00 es d d e-" d es LA A> d A< A* Km 1 1 1 1 001 «1

f-H1 A<1 CA1 A<1 A*1 1

LA

es1

es1 00

es es LA o> O) LA oo o> to to CN

es LA

q « qq COq

o>

q LA

T-H es es 1 1 1 1 CO

00 LA CA es CA to 0() es

d

Q 1 V1 CA1 1 1 es1 es1 1

LA1 1 A< lO

>>

a->e a.

bO tc Id

u

*e<

LaV9 vc M9 (0 [o V %; O 9 U

's

tj

U

u "3

Vu

Co

*5

.0

Ji0

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“o9

bO9

_9 0£ e B c '13 N es

Id

‘-a-. <d •—>

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M 9 es eo ct 9 9 es CA A< o

?> o s

Id 9 U CJ o o o o Co 03 u u o o o o H H n

55
Chapter 10

Calculated Pressure Derivative

and Uniaxial-stress Derivatives

of Tc

The critical-temperature pressure derivative dTc/dP plays a central role in

the thermodynamics of superconductors [241].

In nontransition-metal conventional superconductors, dTJdP is usually

negative because pressure increases the elastic stiffness, the Debye temperature, and the average phonon frequency. In transition-metal conventional

superconductors, dTc/dP may be negative or positive because of a competing

effect: the increase in Tc from an increase in the ionic potential [242].

For YiBa2 Cu307_5 , reported dTJdP values range from —0.8 to 4.3

K/GPa [243,244]. The dTc/dcrij values disagree even more [110,183,245-247].

Therefore, an estimate of dTJdP based on theory might reduce the uncertainty in dTcjdP.

As described in Chapter IX, the ionic model has been successfully used to

describe the charge distributions in YBCO metal-oxide superconductor. Besides the charge distributions, Ohta and coworkers [248] used an ionic model

to study the dependence of Tc on the Madelung site potential. Defining AV^

as the difference in Madelung site potentials for a hole between the apex and

in-plane oxygens (in David’s notation, 04 and 02,03, respectively), they

found a remarkable correlation between Tc and AV^, shown in Fig. 10.1, for

all known metal-oxide superconductors. The compounds shown in Fig. 10.1

can be identified in Table 10.1, which shows the names, Tc, and calculated

56
Figure 10.1: Tc- versus- A

V

4 correlation.

57
AVa of the compounds. From the curve, they concluded that AV4 governs

Tc in hole-carrier superconductors. They also explained in terms of the Tc~

AVa curve why the pressure coefficient dTc/dP should be proportional to the

gradient dTc/dAVA of the Tc~AVa curve. In this chapter, the Tc~AVa corre- lation will be used to calculate the pressure derivative coefficient dTc/dP and

the three principal uniaxial-stress derivatives dTc/daa^ dTc/dab, and dTc/dcTc

for YBCO.

To use Ohta’s correlation, an analytic expression for the Tc-AVa relation

is needed. Using a least-squares fit, we obtain an empirical expression for Tc

in terms of AVaTc = 10.57 + 30.25(1.43 + (10.1)

where the units are in K for Tc and eV for AV^.

The site potential for a hole at the position of zth ion Vi (the electrostatic

energy to bring a hole from infinity to the z-site) is defined by

( 10 . 2 )

Here, f*j denotes the position of jth ion, and Zjt the associated charge. Again,

it is assumed that the total charge of the lattice is neutral: Zj = 0. The

ionic charges were assigned to each individual ion as +3 for Y, +2 for Ba,

+3 for Cul, +2 for Cu2, and —2 for 01—04. An average potential of 02

and 03 sites was taken to be the in-plane oxygen site potential. The values

for 02 and 03 sites differ only by about 0.5%.

To evaluate the site-potential change under certain pressure (or

stress), we need to know how the corresponding unit-cell dimensions change

(assuming the ions inside the unit cell will change homogeneously), hence the

information about the strain. This leads to a need for the elastic constants.

Since no measured monocrystal elastic constants as a full set (nine for or- thorhombic phase) for YBCO are available, the estimated values for Cij were

used (see Table 7.1). The unit-cell dimensions and ionic coordinates of David

and coworkers [224] were used for the Madelung-site-potential calculations.

Table 10.2 shows the results for a pressure, and for compressive uniaxial

stresses, of 0.5 GPa. Values calculated for lower pressure or stress differ up to

a few percent. Values for higher pressure or stress differ similarly, but may

58
Table 10.1: Transition temperature Tc and AV^ for metal-oxide superconductors.

Compounds Tc(K) AV:4 (eV)

La2SrCu206.2 0 -2.639

Lai.9Cai.iCu206 0 -1.316

Lai.85Sro.i5Cu04(I) 38 -0.750

Lai.85Sro.i5Cu04(II) 38 -1.267

Lai.85Bao.i5Cu04 30 -0.861

Ndi.66Sro.205Ceo.i35Cu04 28 -0.224

SmLao.75Sro.25CuO3.95 37 -0.463

(Bao.67Euo.33)2(Euo.67Ceo.33)2Cu308+z 48 0.630

Yo.8Cao.2Ba2Cu306.ii(I) 50 -0.141

Yo.8Cao.2Ba2Cu306.ii(n) 50 -1.691

Yo.8Cao.2Ba2Cu306.ii(III) 50 -0.770

YBa2Cu30e.5 60 0.815

YBa2 Cu307 93 3.533

ErBa2Cu306.53(I) 60 -0.698

ErBa2Cu30e.53 ( 11 ) 60 0.716

ErBa2 Cu307 92 3.679

Yo.9Cao.iBa2Cu408(I) 90 3.244

Yo.9Cao.iBa2Cu408(II) 92 3.362

(Cao.sL£lo.5)(Bai.25Lzio.75)Cu306+^ 90 3.896

Pb2Sr2Yo.5Cao.5Cu308(I) 83 -0.242

Pb2Sr2Yo.5Cao.5Cu308(n) 83 1.560

Pbo.5Tlo.5Sr2CaCu207 no 11.033

Pbo. 5 Tlo. 5 Sr2 Ca2 Cu309 125 12.008

Tl(Bao.6Lao.4)2Cu05_^ 52 1.739

TlBa2 CaCu207_^ 103 11.812

TlBa2Ca2Cu309_^ 120 12.584

Tl2Ba2CuOe 87 8.942

Tl2Ba2CaCu208 no 11.112

Tl2Ba2Ca2Cu30io 125 11.346

Bi2Sr2Cu06+« 80 9.088

Bi2 Sr2 Ca0 . 9Y0 . 1 Cu2 O8.24 93 10.901

Bi2Sr2Ca2Cu30io+tf no 8.300

Bi2Sr2(Gdo.82Ceo.l8)2CU20io.24 40 5.056

59
Table 10.2: Estimated pressure (stress) and strain derivatives of Tc for YBCO

superconductor.

a b c Hydrostatic

dTJdaij (K/GPa) 0.093 -0.063 0.102 0.093

dTc/dsij (K) 23.3 -6.9 12.2 —

Note that throughout this chapter, against usual convention, a compressive stress is

considered positive. This facilitates easy comparison between pressure and stress

effect.

lack meaning because high pressure may change the relative ion positions

within the unit cell.

Compared with measurements [243,244], which range from negative values to 4.3 K/GPa, the result for dTc/dP falls among the lower positive values.

This can be interpreted as follows. This calculation Wcls based on a highoxygen-content material: YiBa2Cu306.85- Several authors [242,243] showed

that dTc/dp tends to decrease with increasing Tc and probably approaches

zero when Tc becomes about 100 K or when (b — a)/a reaches 0.018 [244].

All this emphasizes the role of oxygen content and oxygen atom positions

in affecting both Tc and dTc/dP. We conjecture that the effect of pressure

is to place oxygen atoms in positions more favorable for high Tc. But, in

high-oxygen-content materials (near O7) pressure produces little effect and

may decrease TcThe anisotropy in dTddaij is interesting: dTc/dcTc ~ dTc/dcja^ but

dTc/duk negative. Contrary to expectation, compression along the c-axis,

which pushes the Cu02 planes together, fails to increase Tc significantly

more than compression along the a-axis, which pulls them apart through a

Poisson effect. The negative dTcjdat, value agrees with results derived from

neutron-diffraction studies by Millis and Rabe [183]. All three predicted

dTcfdaij are consistent with increasing b unit-cell dimension resulting higher

Tc. The increase of Tc with increasing b is well established in many studies,

for example Cava and coworkers [249].

60
Table 10.3: For YBCO, estimated plane-stress and plane-strain derivatives

of T,.

a-b a-c

dTc/dffij (K/GPa) 0.04 0.17

dTJdsij (K) 17.24 35.00

Table 10.2 shows also the strain derivatives dTc/deij for the three principal

axes. The uniaxial strain applied to each principal axis was 3%. These

derivatives avoid knowing the elastic constants Cij, and they represent more

fundamental properties. The results show the largest effect on Tc associated

with compressing the a-axis. They confirm that a compressive strain along

the c-axis produces nothing remarkable and that a strain along the 6-axis

lowers Tc.

Because both a-axis and c-axis compressions increase Tc, the effect of

plane stress and plane strain in the a-c plane should be considered. Table 10.3 shows the results together with those for the a-b plane. As expected,

compressing the a-c plane gives a large effect, about twice that of hydrostatic

pressure.

61
Chapter 11

Summary and Conclusions

The elastic and related properties of some oxides and high-Tc oxide superconductors were studied using both measurements and modeling. The physical

parameters considered in this study include longitudinal and shear sound

velocities vi and Vt\ maiss density p; elastic moduli B^E^G and Poisson ra- tio 1/ of polycrystals, especially the bulk modulus B; elastic stiffnesses Cij

of monocrystals; atomic volume 14; Einstein temperature 0£;; Debye temperature 0£>; Griineisen parameter 7; electron-phonon interaction parameter A; Madelung energy Em and site potential critical superconducting

transition temperature Tc, ionic charge Z,; and pressure derivative, dTJdP^

stress derivative, dTc/daij^ and strain derivative, dTc/dsij. For three oxides

BaTiOa, SrTiOa, YBCO—elastic constants were measured between 295 and

4 K.

The present study led to ten conclusions:

1 . Using a megahertz-frequency pulse-echo method, the intrinsic elastic constants of monocrystalline SrTiOa and polycrystalline YBCO at room

temperature were determined. The measurements for SrTiOa agree well with

those reported by Bell and coworkers and Wachtman and coworkers. The re- sults for YBCO were corrected to the void-free state using a model for a

composite material containing spherical particles. The measured bulk modulus B = 101 GPa agrees well with the monocrystal measurement, 115 GPa,

from a high-pressure X-ray-diffraction study by Aleksandrov and coworkers.

Compared with BaTiOa and SrTiOa, its perovskite-crystal-structure building blocks, YBCO is elastically softer. For example, the bulk modulus equals

101 versus 167 and 180 GPa.

62
2. The elastic constants of YBCO show different temperature dependences from that of the perovskites, especially SrTiOa. Unlike LSCO and

the A15 superconductors, YBCO fails to show elastic-stiffness softening during cooling. Such softening represents a zone-center soft lattice-vibration

mode, which many consider related strongly to superconductivity.

3. For the polycrystalline YBCO specimen used in this study, the elasticmodulus-temperature behavior is nearly regular. It fits the Varshni function:

exp(0£;/T)-r

Here, Co denotes an elastic constant at T = 0 K, 0£; the Einstein temperature, and s a parameter that relates to zero-point vibrations and can be

calculated from 0£;, K, and 7. Contrary to many other reports, this specimen showed practically no Ct hysteresis. The Poisson ratio provides an

exception to the regular (Einstein-model) temperature-dependent behavior.

Near Tc, rather than decreasing smoothly during cooling, the Poisson ratio

shows an anomalous drop, which reflects some internal change in YBCO.

4. Within measurement error, about 5 in 10"*, and contrary to many other

reports, the elastic stiffnesses showed no change near Tc during cooling. This

agrees with the small thermodynamically predicted change: about 1 in 10^.

5. The bulk modulus of YBCO was calculated using a Born ionic model,

which contains only two terms: Madelung energy and ion-core repulsion

energy. The calculation supports the measurement result. This indicates that

covalent forces are weak compared with ionic interactions and that metallic

bonding makes no large contribution to the bulk modulus.

6. From the measured polycrystal elastic constants of YBCO, the Debye

characteristic temperature was determined. The result, 0^ = 437 K, agrees

well with the specific-heat value, 440 K. The temperature dependences of the

Debye temperatures were determined also. The results showed that BaTiOa,

SrTiOa, and YBCO show different temperature dependences. Thus, these

oxides show no common Cj behavior.

7. For orthorhombic YBCO, nine independent elastic stiffnesses Cij

were estimated based on monocrystal measurements combined with analysistheory. The results are shown in Table 7.1. The Debye temperature and the

bulk modulus calculated from the estimated Cij agree with the corresponding values determined from specific-heat measurement and a high-pressure

X-ray- diffraction study.

63
8. For several oxide superconductors, the electron-phonon parameters

A were calculated from Tc and Gjj using Kresin’s model, which is valid for

all values of A. For YBCO, A = 2.24. This realistic A value allows for a

possible BCS superconductivity mechanism. The calculations indicate that

a stronger electron-phonon interaction yields a higher superconducting transition temperature.

9. By Ewald’s method, the ion-site potentials and Madelung energy were

calculated to determine the possible valence configuration for YiBa2 Cu307. Among seven suggested possibilities, only one ion-charge configuration allows for vacancies at the 01 site and also gives a realistic bulk modulus. A

perturbation of this configuration by equalizing the Cul-Cu2 charges as -f 2

and introducing a hole into the Cu02 plane at the oxygen sites also gives

recLSonable agreement with observation.

10. By calculating the Madelung site potentials and using the estimated

Cijy we determined the difference in Madelung site potentials for a hole at

the apical and planar oxygens AVA for hydrostatic pressure, compressive

stresses, and compressive strains. Applying Ohta’s relationship for Tc and

AVA, the pressure derivative dTc/dP^ uniaxial-stress and plane-stress derivatives dTc/daij^ and uniaxial-strain and plane-strain derivatives dTddsij were

calculated. The results show that Tc increases with decreasing both a and c axes and with increasing 6-axis. This is consistent with the current view that

Tc increases as the c-axis decreases, as the Cu02 planes move closer together.

The results show also that compressing the a-c plane increases Tc the most,

about twice that of hydrostatic pressure. These calculations imply several

possibilities for increasing Tc.

64
Acknowledgments

This publication represents the thesis submitted by Ming Lei to the University of Colorado (Boulder) to satisfy part of the requirements for the Ph.D.

degree. Dr. Ledbetter directed the thesis research.

The authors thank S. Kim at NIST for his assistance with measurements

and Dr. Z. Sheng at University of Arkansas for supplying the YBCO specimen. Critical readings were provided by Professors S. Datta, A. Hermann,

J. Scott, and P. Zoller at the University of Colorado.

65
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YBa2Cu307_5 in thermal cycling,” to be published.

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in Y—Ba—Cu—0,” Sov. Phys. Solid State 30,510 (1988).

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91
Appendix A

Measured Elastic Constants of

Polycrystal YBCO

Table A.l: Measured elastic constants of polycrystal YBCO.

First author Temp.

(K)

B

(GPa)

E

(GPa)

6

(GPa)

V

Alford [85] 300 214.00

Al-Kheffaji [83] 295 42.40 78.00 32.70 0.194

Al-Kheffaji [83] 295 56.40 116.00 50.10 0.157

Al-Kheffaji [83]“ 295 68.50 135.00 57.40 0.149

Almond [53] 205 46.30 88.50 37.40 0.182

Block [52] 295 196.00 235.00

Bridge [88]“ 300 50.20 99.80 42.70 0.169

Chang [90]“ 300 226.00 87.00 0.300

Dietrich [80] 300 170.00

Fietz [81] 300 180.00

Glazkov [89] 300 175.40

Glazkov [89] 300 112.40

Kim [78] 220 26.90 0.194

Jaya [82] 295 92.86

Ledbetter [5l]“ 295 72.20 90.80 35.20 0.290

Ledbetter [51]“ 295 56.10 101.80 42.50 0.198

Lee [70]“ 295 145.10 59.00

Round [86] 300 38.00 64.00 26.00 0.220

Salama [91] 300 117.00 48.00

Salomons [79] 300 56.30

Severin [87] 300 112.00 0.250

Staines [7l]“ 295 118.00

Tallon [72] 295 115.00 104.00 39.00 0.350

Wolf [67] 220 38.80 0.190

Yoshimoto [77] 295 55.08 84.01 33.72 0.246

“Corrected to void-free state.

92
BL-1 14A U.S. DEPARTMENT OF COMMERCE

(5-90) NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

BIBLIOGRAPHIC DATA SHEET

1. PUBUCATION OR REPORT NUMBER

NISTIR 3980

2. PERFORMINQ ORGANIZATION REPORT NUMBER

a. PUBUCATION DATE

August 1991

4. TITLE AND SUBTITLE

OXIDES AND OXIDE SUPERCONDUCTORS:

ELASTIC AND RELATED PROPERTIES

5. AUTHOR(S)

Ming Lei and Hassel Ledbetter

S. PERFORMINQ OROANBATION (IF JOINT OR OTHER THAN NIST, SEE INSTRUCTIONS)

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECNNOLOOY

BOULDER, COLORADO SOSOS^SS

7. CONTRACT/QRANT NUMBER

t. TYPE OF REPORT AND PERIOD COVERED

9. SPONSORJNQOROANIZATION NAME AND COMPLfTE ADDRESS (STREET, CITY. STATE, ZIR)

10. SUPPLEMENTARY NOTES

Using both measurements and modeling, the elastic and related properties of some oxides and

oxide superconductors were studied. The polycrystai elastic constants were measured using

a MHz -frequency pulse-echo method between 295 and 4 K and corrected to the void- free state

by using a model for a composite material containing spherical particles. The elastic

moduli of the high-T^, superconductor YiBa2 Cu307 (YBCO) were compared with that of oxides,

especially the perovskites BaTiOs and SrTiOg , which are crystal -structure building blocks

for the YBCO superconductor. The bulk moduli were also calculated using a Born ionic model

with two energy terms: electrostatic (Madelung) and ion-core-repulsion. The calculated

bulk modulus of YBCO, 98 GPa, agrees well with measurement, 101 GPa. Based on monocrystal

measurements combined with analysis—theory , elastic stiffnesses C^j for orthorhombic YBCO

were estimated. The bulk modulus obtained from the estimated C^j by the Voigt—Reus s—Hill

averaging method agrees with the monocrystal measurement. From the measured

polycrystalline elastic gonstants, the Debye characteristic temperatures, Q-q, were

calculated. For YBCO, ®D - 437 K. The electron—phonon parameters. A, were estimated from

T^ and ©d using Kresin's model, which is valid for all values of A. For YBCO, A - 2.24.

By calculating the Madelung energy, two further features were studied: the valence of

copper and the electron hole distribution. The results show that the hole prefers the CUO 2 plane at the oxygen sites. Ail the results are consistent with the assumption that all

copper ions have valences near +2. Using a relationship between Tq and AV^, the difference

in Madelung site potentials for a hole at the apical and planar oxygens, the pressure

derivative and stress and strain derivatives of T^ were calculated. The results show that

Tp increases with decreasing a-axis, increasing b-axis, and decreasing c-axis.

1Z KEY WORDS (6 TO 12 ENTRIES; ALPHABETICAL ORDER; CAPITALiZE ONLY PROPER NAMES; AND SEPARATE KEY WORDS BY SEMICOLONS)

Born model; bulk modulus; Debye temperature; elastic constants; electron-phonon

parameter; Kresin model; oxides; pressure derivative; superconductors; YiBa2 Cu 30 y.

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UNUMITED 100

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