CORRELATIONS AMONG C/B, TC, AND MADELUNG POTENTIALS IN THE SYSTEM OF RBA2CU3OX SUPERCONDUCTORS

(1)

PHYSICAL REVIEW B VOLUME 40, NUMBER 13 1NOVEMBER 1989

Correlations

among

clb,

T„and

Madelung

potentials

in

the

system

of RSa2Cu30

supereonductors

Shiow-Fon Tsay and Shou-Yih Wang

Department

_of

Physics, National Tsing Hua University, Hsinchu, Taiwan 30042,Republic

of

China Lance Horng and

T.

J.

Watson Yang

Electrophysics Department, National Chiao Tung-Universt'ty, Hsinchu, Taiwan 30042,Republic

of

China (Received 6 March 1989;revised manuscript received 30May 1989)

We have observed a correlation between the lattice-constant ratio c/b and T, in R-13a-Cu-O superconductors (where R is arare-earth element). Madelung potentials and the bond length of Cu(1)

—

O(3) are calculated and correlated with the distribution of T, over the various

RBa2Cu307 superconductors. Further consideration of YBa2Cu30„with variable

x

ofsuitable

ox-ygen content leads to an explanation ofthe correlation ofdecreasing c/b ratio with increasing T,

.

A Coulomb-type ionic mechanism is therefore suggested sothat shortening of the

c

axis and

in-creased chain length provides

(1)

achannel ofcharge-transfer excitation from chain to Cu-O

lay-ers and (2)a long-range Coulomb interaction between the two Cu-O layers, both ofwhich result in stronger coupling between the two Cu-O layers and higher T,

.

Since Chu and Wu ' and their collaborators Grst

discovered in 1987the superconductor YBa2Cu307 with high transition temperature

T„many

studies

of

the

Y-Ba-Cu-0

quaternary system have been performed inorder toattempt tounderstand the underlying mechanism

of

su-perconductivity. Research on YBa2Cu30„concerning the correlation between

T,

and crystal structure have pro-ceeded along two routes: one by varying the oxygen con-tent

(6

&

x

&

7),

and the other by replacing

Y

with other rare-earth elements, viz.

RBa2Cu30„.

For example, the

T,

dependence

of

the lattice constants

of

the unit cell

(a,

b, and

c),

3_upon _{the distortion} _—,'

_c

—

a,

4 upon the unit-cell volume, 3 the orthorhombic splitting, (b

—

a)/a

etc.

, has shown that small structural changes and the critical temperature

T, of

the superconducting phase

of

YBa2-Cu30» are related. We report now the c/b ratio for vari-ous cases

of

RBa2Cu30 and YBa2Cu30

(6

&

x

&

7).

To

explore why the ratio c/b isso strongly correlated with

T„we

have calculated the Madelung potential

of

RBa2Cu307.

It

is found that the variation

of

Madelung potentials among superconductors with different

R

is simi-lar to that

of

the bond length

of

Cu(1)

-0(3).

The calcu-lation and description are given in detail. Finally, we dis-cuss why the Coulomb interaction plays so important a role and draw conclusions from this.

We follow the notation

of Ref.

5 for various ion posi-tions in the structure

of YBa2Cu30„and

use the values

of

their lattice parameters.

Cu(1)

and

0(1)

are in the chain direction (baxis); with

0(2)

the vacant position

(a

axis);

0(3)

in the

c

axis (apex

of

the pyramid

of

the Cu site); and

Cu(2)-0(4)-0(5)

as the copper oxide layer. Wenote that samples

of

the same composition can beprepared, via different processes, to exhibit different

T,

values. Thus

we make comparisons and analyses

of

data from the labo-ratory that provides the most complete data

of

the RBa2Cu 307 system. We 6nd that for the samples

RBa2Cu30„

from each

of

the laboratories investigat-ed, ' _the _ratio _{c/b always tends} _{to decrease} _as

_T,

in-—_3.₀₁20 3.0080 CL st' -3.0040 95.

0-93.

0-91.

0-t,fTlIC!) I I I I I '\ —_3,₀₀₀₀

89.

0— $c(zeal I I I I I I I I Y Sm Eu Gd Dy Ho Er Tm Yb

FIG.

l.

The c/b ratio and T,in the RBa2Cu307 system, R

be-ing the rare-earth element.

creases. As shown in Fig. 1, the variations in the c/b ratio are

of

the order

of

magnitude

of

0.001.

Consequently, the validity

of

the correlation between c/b and

T,

depends mainly onthe accuracy

of

measurement

of

the lattice con-stants but also on the quality

of

the sample. As displayed in-Table

I,

which was obtained from the data from seven

papers by Takita and co-workers, with quite accurate measurements, we analyzed these different RBa2Cu3G7 samples on the relation between c/b and

T,

.

The result,

shown in Fig. 1, correlates the

T,

at midpoint and atzero resistivity with the c/b ratio, via the series

of

samples

of

substitutional replacement

of

the rare-earth

(R)

element. The relation isclear that

T,

increases with decreasing c/b

(2)

CORRELATIONS AMONG c/b,

T„AND

MADELUNG.

. .

9409 TABLE

I.

The lattice parameters of RBa2Cu30„and their

T„T,

width, AT„and the bond length of

Cu(1)

—

O(3)(see Ref.

5).

Y Sm Eu Gd Dy Ho Er Tm Yb a (A) 3.

8179(1)

3.8440(1) 3.8384(2) 3.8350(2) 3.8246(1) 3.8192(1) 3.8128(1) 3.8087

(1)

3.8018(2) b (A) 3.8828

(1)

3.9018(2) 3.8973(3)

3.

8947(2) 3.8865

(1)

3.

8850(1) 3.8781

(1)

3.8746(2)

3.

8710(2) c (A) 11.6809(3) 11.7248(4) 11.7069(8) 11.6992(6) 11.6857(3)

11.

6770(2)

11.

6644(3)

11.

6655(6) 11.6576(5) T,(mid) (K.

)

91.

5 93.8 94.5 94.7 93.0 93.4 93.2 90.5 90.0 2.5 3.0 1.3 1.5 1.2 1.0 1.4 3.0 3.0 dcu())-o(3) (&) 1.849 1.872 1.851 1.885 1.794 1.918 1.836 1.820 1.838 C b 3.0120 30080 R

~30„

3,020 YBa2Cu3 x 3.0040 3.010 3. 0000-I I I I I I 90.0 92.0 94.0 3000 {b) 25 50 75

FIG. 2. (a) The c/b ratio dependence of T, in the RBa2-Cu30 system. (b) The c/b ratio dependence of T, in

YBa2Cu30, with variable oxygen contents (6

(

x&

7).

and viceversa. The relation between c/b and

T,

is accord-ingly depicted in Fig.

2(a),

where the

T,

s at each mid-point

T,

(mid) and their widths AT, areshown.

A similar relation is also found in the

YBa2Cu30,

su-perconductors with variable oxygen content. Basedon ex-perimental 6ndings, 'o higher oxygen content gives higher

T„a

gradual decrease in the lattice constant c, and a slight increase inthe lattice constant b. Thenet result isa decreasing c/b with increasing

T„as

displayed in Fig.

2(b).

The widths /),

T,

are also indicated similarly, as in Fig.

2(a).

Taking into consideration both the results

of

Figs.

2(a)

and

2(b),

we 6nd in both cases an increasing

T,

with decreasing c/b. In the comparison we further found the correlation is more pronounced when the lattice con-stants are known more accurately. We also tried to relate the c/a ratio to

T„but

did not 6nd such a good correla-tion as c/b with

T,.

In order tounderstand the close rela-tion between c/b and

T„we

proceeded to calculate the Madelung potential for the system

of

the RBa2Cu307 su-per conductors.

The valences

of Y

+ and

Ba

+ in the

YBa2Cu30„sys-tem

(x

6-7)

have been generally accepted.

It

is also

known that

YBa2Cu30„

is an ionic insulator when

x

6, but is metallic when

x

&

6.

5.

However, its charge-carrier concentration

(-10

' cm

)

in the metallic phase is so

low that one can assume a high ionicity in

YBa2Cu30„

and deal with an ionic model. In such a crystal, the long-range Coulombic interactions (Madelung potential ener-gy) contribute dominantly tothe total potential

of

the ions

(over 80%

of

the total potential, although they vary with

the ionicity).

"'2

The mechanically repulsive interactions between nearest ions also contribute to some extent, but are found to be small compared with Coulombic interac-tions. We neglect contributions from the repulsive in-teractions and also van der Waal's attractive interaction

in this work. We remark further that our Madelung-potential calculation, treating only the relative effect aris-ing from substitutional replacement

of

the

R

element in RBa2Cu307 rather than treating a general Madelung problem, is expected

to

yield more accurate and meaning-fulpredictions.

Next, we calculated the Madelung potential per YBa2-Cu307 "molecule"

(i.e.

, formula unit) according to the well-known Ewald's method. '3 The desired total potential

of

the reference ion iinthe 6eld

of

all the other iona inthe crystal is )ir(i)

4x

QS(G)G

exp Q2 4 ' 1/2

+g

"F«»",

),

(1)

x

J rJ

where 6, isthe volume

of

the unit cell,

G

isthe reciprocal lattice vector and ri is a control parameter which allows

both sums

of

Eq.

(1)

to converge rapidly. The functions

F(x)

and

S(G)

are de6ned, respectively, as

2 g2

F(x)

_J

e

s ds,

S(G)

-

gq/exp(iG

rj),

J

in which

S(G)

is the structure factor

of

the crystal. Based on the condition

of

electric neutrality and some ex-perimental results, ' we assume that there are one Cu + at the

Cu(1)

site, two Cu + at the two equivalent

Cu(2)

sites, and vacancies atthe

O(2)

sites. InTable

II

we show

the Madelung energies for various ions in the RBa2Cu307 molecule

(R

Y,

Sm, Eu, Gd, Dy, Ho,

Er,

Tm, and

Yb).

The total Madelung potential energy which isthe sum

of

all the iona inthe molecule is shown in

Fig.

3(a).

Now we

compare the total Madelung potentials with the sum

of

the Madelung potentials

of

Cu(1)

and

O(3)

as shown in Fig.

3(b).

The variation

of

the total Madelung potential

(3)

TSAY, WANG, HORNG, AND YANG

TABLEIL Madclung potential ofeach Ion and the total Madelung potential ofaRBaqCu307 unit cell (units in eV). In obtaining the total Madelung potential, the values of Ba,Cu(2),

O(3),

O(4),and O(S)are each doubled due tothe symmetry ofthe unit cellof

RBa2Cu307.

Cu(l)

CU(2) R Ba

O(l)

O(3)

o(4)

o(s)

Total Y

-88.

701

-58.

139

-100.

142

-34.

390

-S4.

161

-46.

987

—

_37.584

-37.

392

-335.

996

-89.

162

-57.

657

—

_94.₅₄₄

—

_35.₃₈₆

—

53.188

-45.

696

-37.

935

-38.

016

-333.

138 Eu

-90.

136

—

_57.428

—

_94.₁₈₉

-35.

524

—

52.888

-45.

641

-38.

290

—

38.344

-333.

837

—

87.777

—

_58.210

—

_94.₈₇₃

—

_35.414

—

_53.970

—

_45.₅₄₅

—

37.819

—

37.701

-333.

003

—

91.

889

-57.

293

—

101.315

—

33.967

—

_52.984

-47.

928

-37.

838

-37.

488

-337.

593 Ho

—

_85.322

-58.

862

-99.

522

—

_34.761

—

55.289

—

45.824

-37.

295

-37.

138

—

333.968 Er

—

_90.₆₆₉

—

_57.615

—

_97.913

—

_35.₀₄₅

—

_53.₁₆₅

—

_46.₅₁₂

—

38.410

—

_38.254

—

_336.712 Tm

—

_91.

₂₉₅

—

_57.₅₂₇

-97.

819

—

_35.₀₅₈

—

_53.001

—

_46.₆₇₇

—

_38.550

—

_38.₂₇₅

—

_337.₁₄₇ Yb

—

_92.₁₀₀

—

_57.₂₅₈

—

94.251

—

_36.₀₄₈

—

52.367

—

_45.649

—

_39.359

—

_39.015

—

_336.₆₉₁ --333.0 I L

(a)

I 1 II I 1 I I 1 I 1 I I _I {eV) t -&3&.0— I ) 1 I 'L I' I I L) I I I A I -135.0— I d' 1 I I '1 I ll , I LI

(b)

f l50 -~ Ig o Ig 4y I -&33.0— —&37.

0-1.900~ -&3S.

0---334.0 --335.0 --336.0 ~ —-337.0

--338.

0 (eV} Y Sm EQ Gd Dy Ho Er Tfn Yb

FIG.

3.

(a) The total Madelung potential (eV) of RBa2-Cu307. (b)The sum ofthe Madelung potentials (eV)of

Cu(l)

and O(3)ofRBaqCu3O7. The inset represents thebond length of

Cu(l)-O(3)

inthe RBaqCu3O~. The abscissa isthe same asin

the main part ofthe Sgure.

with various

R

elements has very clearly the same tenden-cy as the sum

of

the Madelung potentials

of

Cu(l)

and

O(3).

This implies that the sum

of

the potential

of

Cu(1)

and

O(3)

mainly reflects the variation

of

the total Madelung energy. The dominance suggests

a

correlation between the Madelung potential and the bond length

of

Cu(1)-O(3)

because

of

the nature

of

Coulombic interac-tion. In fact, DyBa2Cu307 has the smallest total Madelung potential, the smallest potential sum

of

Cu(1)

and

O(3),

and also the smallest bond length

of

Cu(1)-O(3).

On the other hand, HoBaqCu30q has relativel

larger total Madelung potential, largest

Cu(l)

and

O(3

Madelung potential sum, and largest bond length

of

Cu(l)-O(3).

These are manifested in Figs.

3(a)

and

3(b),

and in the inset

of

the flgure, respectively. Thus, the total Madelung potential energy

of

RBa2Cu307 is quanti-tatively correlated tothe bond length

of

Cu(1)

—

O(3).

Now we turn back to

T,

of

Fig. 1 and compare

T,

with

the total Madelung potential

of

Fig.

3(a).

There exists the same tendency in the variations over the superconduc-tors with different

R

elements except Sm and Tm for

which the cause, while not known for certain, might be due to some individual feature

of

the ions themselves. Aside from this small discrepancy we have observed a consistent correlation among these quantities. A decrease

in c/b is correlated to an increase in

T„Madelung

total potential, Madelung potential sum

of

Cu(l)

and

O(3),

and the bond length

of

Cu(l)

-O(3).

Having obtained the correlation among those quantities from substitutional replacement

of

the

R

element in RBa2Cu307, we reconsider the situation

of YBa2Cu30„

with variable oxygen content. The experimental result' is that greater oxygen content will cause longer bond

length

of

Cu(1)-O(3),

and thus shorter bond length

of

Cu(2)-O(3).

This can be explained qualitatively from

our concept

of

the ionic model due to Coulomb interac-tion. As oxygen content isincreased, the occupation prob-ability

of

O(1)

sites would be increased. This would in-crease the length

of

the b axis. The bond length

of

Cu(l)-O(3)

would be increased due to the Coulomb

repulsive force on

O(3)

by

O(1)

and possibly the presence

of

O(2).

The reaction force from

O(3)

would further

re-pel

O(l)

and lengthen the b axis. As

O(3)

is pushed

to-ward

Cu(2)

the increase in the Coulomb attraction be-tween

O(3)

and

Cu(2)

would make

Cu(2)

come closer to

O(3)

such that the overall

c

axis isshrunk. Thus the ionic Coulomb interaction as displayed by using increasing

oxy-gen content tends to yield a decreasing c/b, increasing bond length

Cu(1)

-O(3),

increasing Madelung potential, and increasing

T,

.

During the process

of

decreasing in

c/b,

etc.

, there could be charge transfer excitation and

simultaneously long-range Coulomb interaction between the two

Cu-0

layers. This therefore suggests a Coulom-bic mechanism that, increasing chain length and decreas-ing the

c

axis provides:

(i)

a channel

of

charge transfer'

'

from the chain to the

Cu-0

layers and

(ii)

a long-range Coulomb interaction between the two

Cu-0

layers, both

of

which result in stronger coupling between

(4)

CORRELATIONS AMONG

clb, T„AND

MADELUNG.

. .

9411 the two

Cu-0

layers and higher

T,.

The part

of

the

super-conductivity mechanism occurring within the Cu-O lay-ers, which still exists there, is not addressed. To summa-rize, we Grst observed the correlation between the ratio c/b and

T,

.

Then we calculated the total Madelung po-tentials, the sum

of

Madelung potentials

of

Cu(1)

and

O(3),

and also the bond length

of

Cu(l)

—

O(3)

for the various RBaqCu 307 superconductor

s.

Comparison

of

these calculated results with the distribution

of

T,

over these superconductors shows a correlation between the bond length

of

Cu(1)-O(3)

and

T,

.

On the other hand, still using the ionic Coulomb interaction

(i.

e.

, the same physics

of

Madelung potentials) but considering the su-perconductors

YBaqCu30„with

variable

x,

we explain the experimental result

of

decreasing c/b ratio with increasing oxygen content and increasing

T,

.

To

conclude, acorrelation between c/b and

T,

has been observed and explained. Consequently, a Coulomb-type charge-transfer excitation mechanism is suggested that, as a result

of

decreasing

c

and increasing b, charge transfer from chain to copper-oxide layers could happen and simultaneously along-range Coulomb interaction between the two Cu-O layers could prevail, resulting in enhanced coupling between the two

Cu-0

layers and higher

T,

.

The authors are indebted to

Dr. W.

P.

Hsieh, Chiem-Jamg Wu, Ching-Jium Wu, and Chein-Ming Tsai for their helpful and stimulating discussions. This research was 6nancially supported by the National Science Coun-cil, Taiwan, Republic

of

China, under Grants No.

NSC78-0208-M007-65

and No.

NSC78-0208-M009-13.

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