Phonon dispersion curves of Bi2Sr2CaCu2O10−x and Bi2Sr2Ca2Cu3O12−y

e'~"'~ Solid State Communications, Vol. 68, No. 7, pp. 655-658, 1988 ~%~_~Printed in Great Britain

0038-1098/88 $3.00 + .00 Pergamon Press plc

PHONON DISPERSION CURVES OF Bi2Sr2CaCu2010_x AND Bi2Sr2Ca2Cu]O12_y S. Mase and T. Yasuda Department of Physics, Kyushu University, Fukuoka 812, Japan (Received 8 August 1988 by T. Tsuzuki) By making use of an algebraic method in classifying the normal modes, we calculated phonon dispersion curves and density of states curves of the high T c superconductors, Bi-Sr-CaCu-Olo x and Bi2Sr2Ca2Cu3012 - . Here, Z 2 Z the crystal structures are approximated by tetragonal ones wit~ one formula units, instead of an orthorhombic one with 5 formula units proposed by Kajitani et al. Some of the force constants between atoms are estimated from the results in our previous paper. Similarly to that of YBa2Cu307, the density of states curve roughly splits into two parts in both Bi2Sr 2 CaCu2010 x and Bi2Sr2Ca2Cu3Ol2_y , the higher energy part being mainly contributed from Cu-O, 0-0, Ca-O and Sr-O bonds. The role of phonons in high T c superconductivity is briefly discussed in relation with YBa2Cu307.

I. Introduction

the detail of the Fermi surface with, probably, a small dimension. Figure I shows the unit cell of Bi2Sr2CaCu 2 O10 together with Bi2Sr2Oa2Cu3012 approximated to have a tetragonal symmetry. Three translation vectors are of the body centered tetragonal type. The first Brillouin zone is shown in Fig. 2. The symmetry operations are the same as those in the tetragonal (Lal_xSrx)2Cu045 and consist of the following 16 elements:

The oxide superconductors Bi2Sr2CaCu2010-x and Bi2Sr2Ca2Cu3Ol2-y, I in particular the latter, show a high superconducting transition temperature, and some studies have been made to clarify the crystal structures, 2-4 electromagnetic properties and others. The superconductivity of this kind of substances also might come from some non-phonon mechanisms as in the case of YBa2Cu307. However, the mechanism of the high T e superconductivity in the latter is still not yet settled at present. Therefore, the information of the phonon aspect may be very useful, because even if the non-phonon mechanism is eventually found to play the most important role in superconductivity we must make clear why the electron-phonon Y ~ e r a c t i o n is not so effective in the high T c superconductivity in oxides, in spite of having some advantages in symmetry properties associated with Cu-O bond.5, 6 Here we present calculated results of the phonon dispersion curves and the density of states curves in these oxide superconductors. As for the higher T c compound, Bi2Sr2Ca2Cu3012 _y, the crystal structure is not yet settled, so that here we assume a possible structure.

R : E, C2(z), C4+(z) , CA-(z), C2(x), C2(Y),

C2(xy), c2(~y), I • R's.

(I)

Let us denote the number of atoms to be invariant in position by an operation R as NR. Then, we have the following results for Bi2Sr 2 CaCu2010 :

E Ce(z) C4+(z)C4-(z) C2(x) C2(y) N R 17

17

11

11

C2(xY) C2(~y) NR

3

I

3

I

I

IC2(z) IC4+(z) IC4-(z)

I

I

3

3

IC2(x) IC2(y)IC2(xy)Ic2(~y)

2. Symmetry Consideration

N R 17 According to Kajitani et al., 2 the unit cell of this substance contains 5 formula units, and the crystal structure is of an orthorhombic type. However, this crystal structure may be approximated to be a tetragonal type containing one formula unit. Possible slight deviation of positions of oxygen atoms from some metal-ion planes is taken into account, but small values of x and y are neglected in calculation. As far as the phonon aspects are concerned, any essential point may not be lost by this approximation, though in the band structure calculation this approximation might give rise to a crucial effect on

17

11

11

Following the prescription refs. 5 and 6, 51 normal modes into several classes belonging irreducible representations of of ~. Here we write down only

(2)

described in are classified to different group or algebra the results:

D51(F) = 6FI+2F4+8F5+8F2'+IF3'+9F 5' D51(A ) = 17AI+8A2+I7A3+9A4, D51(Z) = 15zI+gZ2+I6Z3+IIZ 4, D51(A ) = 14AI+3A4+ITA 5. 655

(3)

PHONON DISPERSION CURVES

656

¢_ 2

Vol. 68, No. 7

Ca Fig. 2.

I

First Brillouin zone of a body centered tetragonal structure.

F2'(A2u), F5'(E u)

Y

: infrared active,

FI(AIg), F4(B2g), F5(Eg)

: Raman active.

(5) In the case of Bi2Sr2Ca2Cu3012 the mathematics is quite the same as in the former case, except that the role of the Ca layer in Bi2Sr 2 CaCu2010 is replaced by the thrse layers Ca; Cu, O, O; Ca, and that the number N R in Bi2Sr 2 CaCu2010 is replaced by the followings: +

C2(z) c4 (z) c4-(z) C2(x) C2(y) N R 21

21

13

13

3

3 +

C2(xY) C2(~y) NR

3

I

3

3

IC2(z) IC 4 (z) IC4-(z) 3

3

3

IC2(x) IC2(Y) IC2(xY) IC2(~y) N R 21

21

13

13

(6)

Thus, we obtain the following decompositions of 63 normal modes in+to irreducible representations of group of k: D63(F) = 7FI+2F4+9F5+10F2'+2F3'+I2F5' , D63(A) = 21AI+9A2+21A3+12A4, D63(~) = 19EI+IIX2+1913+I4~4, Fig. I.

Crystal structure of Bi2Sr2CaCu2010_. (a) and Bi2Sr2Ca2Cu3Ol2_y (tentative$ (b).

The character t~bles of irreducible representations for each k and the+compatibility relations between different k were given in ref. 5. For a benefit of readers we refer to only that between F and A: FI A1

F2 A4

F3 £4

F4 F5 gl A2+A3

F I' F 2' F 3' F 4' F 5' A2 A3 A3 A2 AI+A4

(4) The activeness of infrared absorption and Raman scattering are also easily determined for 51 normal modes (the symbol in the parenthesis is due to usual notation in molecular spectroscopy):

D63(A) = 17AI+4A4+21A 5.

(7)

First, we must classify all atoms into sets of non-separable atoms by any symmetry operation. We called these sets families.5, 6 The classification of atoms into families can be made by operating the idempotents ef(~)'s and ef'(~)'s (f = the consecutive number of i~reducible representations) belonging to each k, which are described in ref. 5, on the atomic displacement u(k,l) of k-th atom in the direction of i. The results of the operations efu(k,l) or ef'u(k,l)'s are shown in Table I. Here, the atom number j in Caj, Srj, Bij, Cuj and Oj is numbered as follows: Ca ion (Cal) is taken as the inversion center and other ions are numbered in the order of the upper ÷ lower ÷ upper ÷ -- layers. It is noted, however, the first four 0 atoms in CuO 2 layers are numbered as 1, 3 (upper); 2, 4 (lower). No. in the table means the number of independent sets of linear combinations of all families, i.e. No.

PHONON DISPERSION CURVES

Vol. 68, No. 7

Table I.

I Cal 100

Classifications of atoms into families (Roman numerals) for each irreducible representation: Bi2Sr2CaCu2010 . For E-line the family of VI is written in the lower line.

A4

001 O10

II III Cul Cu2 Srl Sr2 100 100 100 100 00100T 00100T 010 OTO 010 OTo 100 TOO 100 TOO 001 001 001 001 010 010 010 010

E1

I Cal 110

II Cul Cu2 110 110

III Srl Sr2 110 110

IV Bil Bi2 110 110

00100T

00100T

00100T

1TO TIO

1TO TIO

1TO TIO

AI A2 A3

r2

11o TTo

r3

E4

AI A4

A5:

11o TTo

IV Bil Bi2 100 100 00100T 010 OTO 100 TO0 001 001 010 010

V 01 02 100 100 00100T 010 OTO 100 TOO 001 001 010 010 V, 01 02 100 100 010 010 00100T

IX 09 010 100 100 00100T 010 OTO 100 TO0 001 001 010 010

00100T

100 TOO OTO 010 010 OTO TOO 100 001 001 001 001

1TO TIO

1TO TIO

11o TTo

lOO Too OlO oTo

11o TTo

11o TTo

1To Tic

8

010 OTo 100 TOO 001 001 001 001 001 001

001 001

001 001

8

1To 1TO

1TO 1To

9

1TO

1To 1TO

1TO 1TO

1To 1TO

I II III IV V Vl Vll Ca1 Cul Cu2 Srl Sr2 Bil Bi2 001 001 001 001 001 001 001 IV V VI VII Srl Sr2 Bil Bi2 100 100 100 100 010 010 010 010

100 100 OTo oTO 010 010 Too Too 001 ooT OOT 001

IX O~ 010 110 110

No. 9 8 8 8 9 9

00100T

001 001

NO.

9 6

110 TTO

8

1

1To 1TO

001 OOT 2

VIII IX X Xl XII X m XlV 01 03 02 04 05 06 07 08 09 010 No. 001 001 001 001 001 001 001 001 001 001 l& 00100T 00100T 001 00~ 3 VNI IX X Xl XII X]K XIV XV XVI X V K 01 02 0~ O 4 05 06 07 08 09 010 No. 100 100 100 100 100 100 100 100 100 100 17 010 O10 010 010 010 010 010 010 010 010 17

is equal to the number of normal modes having the specified polarization in each irreducible representation. The symmetry properties in the case of Bi 2 Sr2Ca2Cu3012 are essentially the same as those in Bi2Sr2CaCu2010 , as found from eqs. (3) and (7). 3. Dispersion and Density-of-States

Vlll 07 08 100 100 00100T 010 OTO 100 TO0 001 001 010 010 VIII 07 08 110 110

001 001

III Cu2 100 010

Vll 05 06 100 100 00100T 010 OTO 100 TOO 001 001 010 010 Vll 05 06 110 110

001 001

II Cul 100 010

VI 03 04 100 100 00100T 010 OTO 100 TOO 001 001 010 010 Vl O~ 04 010 010 100 100 00100T

001

I Cal 100 010

657

Curves

For the force constants K's between atoms we use the following ones shown in Table II. The values are estimated or deduced from our previous paper on YBa2Cu307.7 In the absence of data from inelastic neutron diffraction, EXAFS and other measurements which make possible to determine the values of K's, the present values are rather tentative. The deficiency of oxygen atoms, i.e. the fact x ~ O , y ~ O , may be partly be represented by replacing the <'s values related with O-atom. Here, however, most of important ones is of the type M-0, so that the present result is regarded as the case of a random distribution of the deficiencies of Oatoms. In the calculations of the dispersion curves the symmetry consideration mentioned above was fully taken into account. Only the results in A-line for Bi2Sr2CaCu2010 are shown

Table II. Force constants <'s between atoms. Atomic distances are a little varied from those in ref. 2, because of minimization of different kinds of bonds. Atom-Atom Cu-O Bi-O Ca-O Sr-0 0 - 0 Ca-Cu Sr-Cu

d (i) I .9 2.7-2.8 I .5 2.5-2.7 2.4 2.2 2.7-2.9 2.7-2.9 3.2 3.1

K (kd~ne/cm) 100 40 120 AO 60 60 &O £0 30 30

in Fig. 3. As for the density of states curve D(E), however, it was derived by solving the original 17 x 3 or 21 x 3 dimensional equation for every limited point of ~ inside the first Brillouin zone. Namely, the first Brillouin zone was divided into 30 x 30 x 10 cells and calculated 51 or 63 eigenvalues in each cell. Then, we obtained the D(E) at each phonon frequency ~ (E = ~ ) by counting the number of

PHONON

658

. ~#x

I ~.

~#y

DISPERSION

"KIIZ

N.IIZ

A3

A!

CURVES

Vol.

68,

No.

/

•

A2

o

o o

~

o,

o

o

~6

.'

o

o

o i

o

d i

3

hi .

°r Fig. 3.

x

r

xr

x r

xr

.

.

.

XF

X

E

/

Phonon dispersion curves along A-line for Bi2Sr2OaCu2010 for ~ // x-axis. is the polarization vector.

I

I

/

20

0

I

60

E cells whose eigenvalues belong to the frequency interval from ~ to ~ + A~. The result is shown in Fig. 4(a) for Bi2Sr2CaCu2010 and in (b) for Bi2Sr2Ca2Cu3012. In the sense of phonon mechanism of superconductivity the important parts in the D(E) curve are the peaks closely related with the Cu-0 and 0-0 bonds. The phonon energy levels dominantly contributed from these bonds are examined by mainly inspecting the amplitudes of the eigenvectors. The specification with arrows in Fig. 4 is thus obtained, together with the contributions from other bonds. For both Bi~Sr~CaCu~O.~ and Bi~Sr~Ca~Ou~O.~ L Z ~ iU--X ~ Z L D I --y an essential feature of the shape of the D(E~ curve, i.e. D(E) being roughly split into two parts, and the general pattern of the contributions from short distance bonds are nearly the same as those in YBa2Cu307. In Bi2Sr2Ca 2 Cu3Ol2_y , however, additional three layers contribute to additional enhancement of peaks as compared with the corresponding one in Bi2Sr 2 CaCu2010_ x. See, for example, the Cu-0 peak. The split into two parts in D(E) is characteristic to the layered perovskite structures. Thus, from only the view point of symmetry properties of wave functions related with the Cu-O bond the present Bi-Sr-Ca-Ou-O system must also have an advantage of a strong electronphonon interaction. If the origin of high Tc superconductivity comes from some non-phonon mechanism as proposed by Anderson, 8 and also as the quite weak isotope effect in YBa2Cu3079 suggests, then the magnitude of the electron-phonon interaction matrix element itself must be fairly small in

I

40

,

8O

(meV)

i

,

i

i

i

3 i °~

c

,n ."

O ~ ' ~

I

dl

~ (3~

O

I

'

/

O "

,4

A

iii t7 v

/ 0

20

40 E

Fig. 4.

60

80

(meV)

Phonon density of states curves for Bi2Sr2CaCu2010 (a) and for Bi2Sr2Ca 2 Cu3012 ( b ) .

spite of the above-mentioned advantage. Thus, as far as the phonon aspects are concerned, the Bi-Sr-Ca-Cu-0 system gives nothing more useful clue about the origin of the high T c superconductivity than YBa2Cu307. AcknowledgementThis work is supported by a Grant-in-Aid for Scientific Research on Priority Areas "Mechanism of Superconductivity" from the Ministry of Education, Science and Culture.

References I. H. Maeda, Y. Tanaka, M. Fukutomi and T. Asano, Jpn. J. Appl. Phys. 27, L209 (1988). 2. T. Kajitani, K. Kusaba, M. Kikuchi, N. Kobayashi, Y. Syono, T. B. Williams and M. Hirabayashi, Jpn. J. Appl. Phys. 27, L587 (1988). 3. D. Shindo, K. Hiraga, M. Hirabayashi, M. Kikuchi and Y° Syono, Jpn. J. Appl. Phys. 27, LI018 (1988). 4. S. Ikeda, H. Ichinose, T. Kimura, T. Matsumoto, H. Maeda, Y. Ishida and K. 0gawa, Jpn. J. Appl. Phys. 27, L999 (1988). 5. S° Mase, Y. Horie, T. Yasuda, M. Kusaba and T. Fukami, J. Phys. Soc. Jpn. 57, 607 (1988).

6. S. Mase, T. Yasuda, Y. Horie, M. Kusaba and T. Fukami, J. Phys. Soc. Jpn. 57, 1024 (1988). 7. T. Yasuda and S. Mase, JJAP Series I, Superconducting Materials, ed. 8. Nakajima and H. Fukuyama, P.265. Publication Office, Japanese Journal of Applied Physics, Tokyo (1988). 8. P. W. Anderson, Science 235, 1196 (1987); P. W. Anderson, G. Baskaran, Z. Zou and T. Hsu, Phys. Rev. Lett. 58, 2790 (1987). 9. K. J. Leary, H. C. zur Loye, S. W. Keller, T. A. Faltens, W. K. Ham, J. N. Michaels and A. M. Stacy, Phys. Rev. Lett. 59, 1236 (1987).

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