The role of electron-phonon interaction in oxide superconductors

PHYSICAl/

Physica B 186-188 (1993) 816-821 North-Holland

The role of electron-phonon interaction in oxide superconductors Kazuko Motizuki a and Masafumi Shirai b

aDepartment of Physics, Faculty of Science, Shinshu University, Matsumoto, Japan bDepartment of Material Physics, Faculty of Engineering Science, Osaka University, Toyonaka, Japan The electron-phonon (EP) interaction and lattice dynamics are investigated microscopically for oxide superconductors, BaxK ~ xBiO3 (BKB) and Laz xSr~CuO4 (LSC), on the basis of their realistic tight-binding bands fitted to the first-principle bands of each compound. By using the EP interaction and renormalized phonons we have calculated the spectral functions of BKB and LSC. In the framework of the phonon-mediated pairing mechanism, superconducting properties such as the transition temperature, the gap function and the tunneling spectra are calculated by solving the Eliashberg equation. The results for BKB are in good agreement with observations. For LSC, the calculated results are compared with recent observations and discussed in connection with those for BKB.

I. Introduction

Oxide superconductors BaxK 1 ~,BiO3 (BKB) having a cubic perovskite-type structure show the highest transition temperature T c of about 30 K for x = 0.60.7 [1,2] among Cu free oxide superconductors, and Tc indicates a remarkable dependence on the potassium concentration [3,4]. BKB contains no transitionmetal elements and lacks two-dimensional structural features which are widely believed to be the essential factor in producing a high Tc in cuprate oxide systems. Furthermore, no magnetic ordering has been observed in BKB [5] and the magnetic susceptibility in the normal state shows a Pauli paramagnetic behavior [6,7]. Therefore, it is expected that the electronphonon (EP) interaction is responsible for the origin of superconductivity in BKB. Previously, we have studied theoretically the superconductivity of BKB in the framework of the phononmediated pairing mechanism [8]. First, the EP interaction of BKB has been investigated microscopically by using the realistic electronic band structure of BaBiO 3 [9], and the lattice dynamics is studied by taking account of the effect of the EP interaction. Superconducting properties of BKB, such as T c, its isotope effect, the superconducting energy gap, and tunneling spectra, have been studied in detail on the

Correspondence to: K. Motizuki, Department of Physics, Faculty of Science, Shinshu University, Asahi 3-1-1, Matsumoto 390, Japan.

basis of the strong coupling theory developed by Eliashberg [10]. For a cuprate oxide system La 2 xSrxCuO4(LSC), the first theoretical study on the EP interaction has been carried out by Weber [11]. After that, many theoretical and experimental efforts have been made extensively to clarify the mechanism of the superconductivity in LSC and other cuprate oxide systems. As the result, it has been found that some experimental facts contradict Weber's predictions. Thus it is necessary to reexamine theoretically a role of the EP interaction in the lattice dynamics and the superconductivity of LSC. For this purpose, we have recently extended our studies of EP interaction, lattice dynamics and superconductivity to the cuprate oxide system LSC. In the derivation of the EP interaction we have adopted the tight-binding (TB) band fitted to the accurate electronic band structure [12]. To improve the lattice dynamical calculation we have used experimental data on phonons in LSC available so far [13-15]. In this paper, our theoretical results for BKB and LSC are reviewed and compared with observations.

2. BaxKl_xBiO 3 (BKB)

2.1. Electron-phonon ( EP) interaction The electronic band structures of BaBiO 3 and Bao.sK0.sBiO 3 have been originally calculated by Mattheiss and Hamann [9] using the self-consistent

0921-4526/93/$06.00 © 1993- Elsevier Science Publishers B.V. All rights reserved

K. Motizuki, M. Shirai / Electron-phonon interaction in oxides

linearized augmented-plane-wave ( L A P W ) method. Recent band calculations for BKB performed by several authors give similar results to those of Mattheiss and Hamann. According to their result, the Fermi level crosses a conduction band which consists of Bi 6s and O 2p orbitals and has (sp~) antibonding character. It has been confirmed that the conduction band of BaBiO 3 is little affected by substitution of K for Ba. Therefore, a so-called rigid-band approximation is applicable for doping of K atoms. At x = 1 the conduction band is half-filled and at x = 0 it is empty. The conduction band of BaBiO 3 is well reproduced by the TB model with three kinds of Slater-Koster transfer integrals between nearest neighboring O and Bi atoms; t(spcr)= 2.2 eV between Bi 6s and O 2p orbitals, t(pp~) = 2.7 eV and t(pp~r) = - 0 . 9 eV between Bi 6p and O 2p orbitals. The three orbital energies are taken to be E ( 6 s ) = - 4 . 1 e V , E ( 6 p ) = 3 . 5 e V and E(2p) = - 1.9 eV. In the TB approximation the intra-band EP coupling coefficient for the conduction band (c) is given by V ' ( k , k - q) = Z.~ ~

1

e . , ~ . ( q ) g : ( c k , ck - q)

(1)

where 3' specifies the phonon mode, ev,~,(q) denotes the phonon polarization vector, and M~ is the mass of the /xth atom. The coupling coefficient g~ between two Bloch states ck and ck - q caused by displacement of t h e / ~ t h atom in the unit cell along the a direction ( a = x, y, z) is expressed in terms of transformation

817

coefficients, which diagonalize the Hamiltonian matrix of the undistorted states, and a quantity expressed by using derivatives of transfer integrals, t'(sptr), t'(pp~r) and t'(pp~r). By using the electronic band structure for the distorted phase of BaBiO 3 and the observed direct and indirect gaps, the following values are estimated: t ' ( s p ~ ) = - 4 . 0 5 e V A 1, t , ( p p ~ ) = _ 4 . 1 7 e V A - ~ and t'(pp~r) = 3.78 eV/~ ~. Figure 1 shows the EP coupling coefficient calculated as a function of the wavevector k for the fixed value of q at the R point. It is found that the EP coupling has remarkable dependences on the wavevectors and the phonon modes. Apparently the EP coupling for the A~g phonon mode (so-called breathing mode) is stronger than those for the other modes almost throughout the Brillouin zone. The stretchingtype deformation of BiO 6 octahedra (Eg mode) has the next strongest EP coupling. The coupling coefficients for the other phonon modes are negligible compared with those for the A;g and Eg modes. The coupling caused by t'(pp~r) and t'(pp~r) is confirmed to be much weaker than that caused by t'(sp~). Thus, it is concluded that t'(spcr) mainly contributes to the EP coupling. This result is a direct consequence of the nature of the conduction band in which both O 2ptr and Bi 6s components are dominant except near the F and R points.

2.2. Lattice dynamics The generalized electronic susceptibility tensor x ( q ) is given by

Bax Kl_XBi 0 3

X .,~t3 ~ ( q ) = - 2 ~ g~(ck, ck - q)g~(ck, ck - q)* k

z

2.0

--

O

oz ~ _2_ _; . s 0.¢ ~

Alg

. . . . Eg(1) ..........Eg (2)

X

1.0

Z

o

0.5

lad ..J

"'

0

X

1,4

6° Y~ A1g

R

F

WAVE VECTOR

k

~ : EgO)

Pi Eg(2)

Fig. 1. The electron-phonon coupling coefficient V~(k, k q) as a function of wavevector k for the fixed value of q at the R point (T/a, it~a, It~a).

E o o f(,_.)-f(e,) E o -

o Ek-q

(2)

The Fourier transform of X~,~(q) corresponds to the effective long-range interatomic force caused by the EP interaction. To obtain the phonon frequencies we have diagonalized the dynamical matrix D(q) which can be written as the sum of x(q) and D°(q). Here, D°(q) denotes contributions other than x(q) and is usually expressed in terms of the Fourier transform of interatomic short-range forces. For short-range forces we have considered seven force constants and determined them so as to reproduce seven phonon frequencies (shown by the triangles in fig. 2(a)) observed for BaPboTsBi0.1503 by inelastic neutron scattering measurements [16]. These phonon frequencies are almost unaffected by the EP interaction. In our study we have neglected the difference between the shortrange force constants associated with Ba atoms and

818

K. Motizuki, M. Shirai

(a) !

1

8ao.TKo3BiO3 (111) (b)

(11o)

(100)

Electron-phonon interaction in oxides

O(slJretching / breathing)

60i~

Iv3`(k, k')l 2

3" x ~ ( E ° - E F ) f ( E 2, - e~)6(~o - o~,, ,).

(3) ',

I

t t t

,, ,, -- ....

rr 40 l.u z klJ z

zo

, -.-==:=

0 20 1-

/

,,

// \

%/ ,,/

OF

-- -.

.,~

XF

IF

WAVE VECTOR

.5 .5 .5 .5 L DENSITY OF STATES

Fig. 2. (a) The phonon dispersion curves and (b) the partial phonon density of states calculated for BaoTKo.3BiO3 (x = 0.7) including the effect of the electron-phonon interaction. The full curves represent the transverse modes and the broken curves the longitudinal modes. The triangles indicate the experimental data [16] utilized in determining the shortrange force constants. those associated with K atoms. This approximation is not so bad, because vibrations of Ba or K atoms are hardly coupled with the conduction band states and therefore the vibrations of O atoms, which play an important role in superconductivity, are little affected by the short-range force constants associated with Ba or K atoms. The dispersion curves and the phonon density of states (DOS) calculated for x = 0.7 are shown in figs. 2(a) and (b), respectively. We have found remarkable frequency renormalization around the M and R points for the longitudinal mode of O stretching and/or breathing vibrations, whose frequencies lie originally near 60 meV. This result originates from the nesting effect of the Fermi surface as well as the remarkable wavevector and mode dependences of the EP coupling. The overall feature of the phonon DOS in fig. 2(b) is in agreement with the result of a molecular dynamics simulation and of neutron scattering measurements [17]. The renormalization increases as x increases. For x > 0.9 the frequency of the O breathing phonon at the R point vanishes and hence the lattice becomes unstable against formation of the distorted structure described by that phonon. This kind of distortion has been experimentally confirmed in BaBiO 3 [4,18].

The results for x =0.5 and 0.7 are shown by full curves in fig. 3 together with the total phonon DOS (broken curves), aEF(to) has a frequency dependence entirely different from that of phonon DOS and takes large values in the frequency range where the longitudinal O stretching and/or breathing mode phonons lie. This implies the importance of these phonons for the superconductivity. As x increases, some main peaks in a2F(to) shift to the lower frequency side, reflecting the phonon frequency renormalization, and the magnitude of a2F(~o) increases remarkably in the whole frequency range up to 60 meV. This considerable change in a2F(to) is expected to bring a remarkable dependence of Tc on x. The dimensionless coupling constant A is evaluated by integrating c~2F(to)/o~. Due to the renormalization of the O stretching and/or breathing mode phonons, A increases rapidly with increasing x and exceeds unity for x =0.7, which indicates that BKB belongs to strong-coupling superconductors• The transition temperature Tc has been evaluated by solving the linearized Eliashberg equation with the effective screened Coulomb repulsion constant/~* = 0, 0.05, 0.10 and 0.15. The calculated Tc increases rapidly with increasing x as long as lattice instability does not occur, and reaches 31.3K for x = 0 . 7 ( A = 1.09, ~ * = 0 . 1 ) . Our results for Tc agree well with the observed Tc in BKB [3,4]. The characteristic exponent a defined by T ~ M j (M o is the atomic mass of O) is evaluated as a = 0.39, 0.40, 0.43 for x = 0.5, 0.6, 0.7

Bax Kl-xBi03 "3 15

x=0.5

~~

10 ~

~,

gos~ Z D

Oj

.

20

u_ r ~ 15 t

20 2.3. Superconductivity We have calculated the spectral function a2F(to) defined by

ENERGY

40

60

/~

x=0.7

40

60

It) (meV)

Fig. 3. The spectral function aZF(to) (full curves) calculated for BaxKt xBiO3 (x = 0.5 and 0.7)• The phonon density of states (broken curves) is also shown.

K. Motizuki, M. Shirai / Electron-phonon interaction in oxides (/x* =0.10), respectively. These values of a are in agreement with those observed by Kondoh et al. [7] and by Hinks et al. [19], but are rather larger than those observed by Batlogg et al. [6]. It should be noted that a differs from the BCS value (a = 0.5) for compound superconductors [8,20]. To evaluate the energy-dependent gap function A(e), we have solved the Eliashberg equation for T = 0 K by using the spectral function shown in fig. 3. 3(e) has sharp and prominent structures reflecting the peaks in aZF(w). The superconducting energy gap defined by A0 = A(A0) is found to be 4.8meV for x = 0 . 7 and 1.0meV for x = 0 . 5 , where / t * = 0 . 1 is used. The result for x = 0.7 is in good agreement with A0 = 4.3 meV determined by optical measurements for Bao 6Ko 4BiO 3 (T c = 29 K) [211. We have calculated the differential conductance dI/ dV and its derivative, d2I/dV 2. The results for x = 0.5 and 0.7 (full curves in fig. 4) clearly deviate from the BCS results (broken curves). The fine structures in d 2 I / d V 2 are drastically reduced for x = 0.5 compared with those for x = 0.7. This is due to a decrease of the magnitude of A(e) as x decreases from 0.7 to 0.5. Tunneling spectroscopy measurements [22-24] have revealed d2I/dV z curves similar to those shown in fig. 4. In particular the dips in d2I/dV 2 observed below 60meV for Ba0.625K0.a75BiO 3 (T c = 29K) [22] show good correspondence to the negative peaks in fig. 4. Furthermore, a drastic reduction of prominent structure in d2I/dV 2 spectra has been observed for BaosK0.sBiO 3 (T~ = 13 K) [23] as predicted by our theory.

819

3. La2_~Sr~CuO 4 (LSC)

3.1. Electron-phonon ( E P ) interaction The electronic band structure of L a 2 C u O 4 has been calculated by several authors. The conduction band consists of Cu 3d(x 2 - y2) orbitals and 2per orbitals of O(1) atoms (located in the CuO 2 plane) and is halffilled. The Fermi level lies just above a sharp peak of the electronic DOS and the Fermi surface shows a good nesting feature for the wavevector of the X point ( ~ / a , ~ r / a , O). By substituting Sr for La, holes are doped in the conduction band. De Weert et al. have well reproduced the conduction band of L a 2 f u O 4 by a TB model [12]. In the present paper, we use the same TB parameters with those determined by De Weert et al. The EP interaction in LSC is calculated in the same manner as in the case of BKB. The main contribution to the EP interaction arises from a derivative of transfer integral t'(dptr). Thus, we consider only t'(dpcr) and use a value of 2.6 eV ~ - 1 for t'(dpcr) which was evaluated by Weber [11]. In fig. 5 the EP coupling coefficient VV(k, k - q) is shown as a function of wavevector k for the fixed value of q at the X point. Apparently the EP coupling shows a remarkable dependence on wavevectors and phonon modes, and is especially strong for the O(1) breathing-type deformation. Displacements of Cu atoms in the CuO2 plane also contribute appreciably to the EP coupling.

La2.x Sr x C u 0 a Bax Kl-xBi 03

O.Ol t

0.6 > 0(1 ) (breathing)

~D Z

-0.01 I0102 ~ /

.

.

.

.

X : 0 "5

&A

0.4

A

(D

0.010

o 0

20

40

6'0

~

o -0.01 -0.02~ 0

A

Z O Z O T (3.

t, ~x n

C u ( i n plane)

0.2

O

x=o.7 20 40 60 BIAS VOLTAGE (my)

Fig. 4. The second derivative spectra d211dV 2 calculated for BaxK1 xBiO3 (x=0.5 and 0.7). The parameter /z* =0.1 is adopted. The broken curves represent the result obtained within the BCS weak-coupling theory.

o

o

o

•oo EOo o oo o oo x

o

~ o

o

oOOo oOOo °n~n° o

z

WAVE VECTOR

k

Fig. 5. The electron-phonon coupling coefficient VV(k, k q) as a function of wavevector k for the fixed value of q at the X point Or~a, ax/a, 0).

820

K. Motizuki, M. Shirai

Electron-phonon interaction in oxides

3.2. Lattice dynamics

due to the breathing-type phonon but a tilt-mode phonon.

We have studied lattice dynamics of LSC by taking account of long-range inter-atomic forces caused by the EP interaction. For short-range forces we have considered 28 kinds of forces and determined these forces so as to reproduce the experimental data by Raman scattering [13], infrared absorption [13,14] and inelastic neutron scattering [15]. The dispersion curves and the phonon DOS calculated for Lal 9Sr0.1CuO 4 (x = 0.1) are shown in figs. 6 (a) and (b), respectively. The characteristic feature of the phonon DOS agrees well with that of a generalized phonon DOS for La~.85Sr0.~sCuO 4 obtained by inelastic neutron scattering [25]. We have found that the EP interaction causes a remarkable frequency renormalization for the highest A~-branch around (~x/a, 0, 0) and the highest ]~lbranch around the X point (~x/a,~/a,O). The wavevector dependence of renormalization for the former phonon is mainly caused by a wavevector dependence of the EP interaction, and that for the latter is a direct consequence of the nesting feature of the Fermi surface. Recently, the. renormalization of the former phonon has been confirmed by inelastic neutron scattering measurements [26]. It is noted here that in our result no structural instability takes place for any Sr concentration. This is in contrast with the Weber's result, where the structural instability associated with the breathing-mode phonon takes place for a wide composition range (0~< x <~ 0.1). In fact, experimental results indicate that the structural instability observed in LSC is not Lal 9Sr0 iCuOa (a)

(]0o)

(001)

(11o)

(b) 0(2)

O(1) Cu La(Sr) Total ! ..... ]

3.3. Superconductivity We have calculated the spectral function a2F(to). The results for x = 0.1, 0.2 and 0.3 are shown by full curves in fig. 7, together with the total phonon DOS (broken curves), c~2F(to) reveals a characteristic structure in a wide frequency range below 85 meV. Main contributions to a2F(to) arise from the breathing-type vibration of O(1) atoms, which extends almost over whole frequency range below 85 meV due to the significant frequency renormalization around the X point. Below 40 meV, however, contributions from the in-plane vibrations of Cu atoms cannot be disregarded, where the vibrations of Cu atoms hybridize well with those of O(1) atoms. Recently, ct2F(~o) has been derived from the optical spectra for Lal 86Sro 14CUO4 [27] and shows a good correspondence with that calculated by us. By using a2F(to), A is evaluated to be 1.06, 0.50 and 0.35 for x = 0.1, 0.2 and 0.3, respectively. By solving the linearized Eliashberg equation with p.*= 0.1, T for x = 0.1, 0.2 and 0.3 are obtained as 17.2, 5.2 and 1.7 K, respectively. The calculated maximum value of T c is lower compared with that observed in LSC (T c = 35-40 K). We have solved the Eliashberg equation at T = 0 K and calculated d l / d V and d21/dV z. Results for x = 0.1 are shown in fig. 8 by full curves. Both curves deviate clearly from the BCS results shown by broken curves, and reveal prominent structures which are characteristic behavior for strong coupling superconductors. Recently, d l / d V and d21/dV 2 similar to those shown in La2_xSrxCuO4 10 1

l '° 60

~

,,5

i

~o

4o

~-

o5

2o,~ Or

~

ZA r

WAVE VECTOR

i x

,. i!!:?. ~

o

lO 10 DENSITY OF STATES

!

'

Fig. 6. (a) The phonon dispersion curves and (b) the phonon density of states (DOS) calculated for LalgSr01CuO 4 (x = 0.1) including the effect of the electron-phonon interaction. The broken curves represent the partial DOS for bending modes of O atoms.

~

!

x =0.1

'

o ] :

x=0.2

o x-03

0

20

40 ENERGY

60 W (meV)

80

Fig. 7. The spectral function a2F(w) (full curves) calculated for La 2 xSrxCuO4 (x = 0.|, 0.2 and 0.3). The phonon density of states (broken curves) is also shown.

K. Motizuki, M. Shirai / Electron-phonon interaction in oxides

[4] [5] [6] [7] [8]

La 19Sro.1Cu04

(a) 103r

> "°LI

(b) 00l 0

20

40

60

80

[9] [10]

-o.o~

[11] [12]

°-o.o I / , -oo3•

~o z~

6o ~-

BIAS VOLTAGE (rnV)

Fig. 8. (a) The tunneling conductance d l / d V and (b) its derivative d21/dV 2 calculated for Lal.9Sr0.1CuO 4 (x =0.1). The parameter ix* = 0.1 is adopted. The broken curves represent the result obtained within the BCS weak-coupling theory.

fig. 8 have been observed for Lal.8~Sr0.~sCuO 4 by tunneling spectroscopic measurements [28].

References [1] L.F. Mattheiss, E.M. Gyorgy and D.W. Johnson, Jr., Phys. Rev. B 37 (1988) 3745. [2] R.J. Cava et al., Nature 332 (1988) 814. [3] D.G. Hinks et al., Nature 333 (1988) 836.

[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

[27] [28]

821

S. Pei et al., Phys. Rev. B 41 (1990) 4126. Y.J. Uemura et al., Nature 335 (1988) 151. B. Batlogg et al., Phys. Rev. Lett. 61 (1988) 1670. S. Kondoh et al., Physica C 157 (1989) 469. M. Shirai, N. Suzuki and K. Motizuki, J. Phys.: Condens. Matter 1 (1990) 3553. L.F. Mattheiss and D.R. Hamann, Phys. Rev. B 28 (1983) 4227; Phys. Rev. Lett. 60 (1988) 2681. G.M. Eliashberg, Sov. Phys. JETP 11 (1960) 696; Sov. Phys. JETP 12 (1961) 1000. W. Weber, Phys. Rev. Lett. 58 (1987) 1371. M.J. de Weert, D.A. Papaconstantopoulos and W.E. Pickett, Phys. Rev. B 39 (1989) 4235. S. Sugai, Phys. Rev. B 39 (1989) 4306. V.A. Maroni et al., Phys. Rev. B 39 (1989) 4127. H. Rietschel et al., Physica C 162-164 (1989) 1705. W. Reichardt and W. Weber, Jpn. J. Appl. Phys. Suppl. 26-3 (1987) 1121. C.-K. Loong et al., Phys. Rev. Lett. 62 (1989) 2628. D.E. Cox and A.W. Sleight, Acta Cryst. B 35 (1979) 1. D.G. Hinks et al., Nature 335 (1988) 419. T.W. Barbee III et al., J. Phys. C 21 (1988) 5977. Z. Schlesinger et al., Phys. Rev. B 40 (1989) 6862. J.F. Zasadzinski et al., Physica C 158 (1989) 519. J.F. Zasadzinski et al., Physica C 162-164 (1989) 1053. F. Morales et al., Physica C 169 (1990) 294. B. Renker et al., Z. Phys. B 67 (1987) 15. W. Reichardt et al., in: Electron-Phonon Interaction in Oxide Superconductors, ed. R. Baquero (World Scientific, Singapore, 1991) p. 1. J. Tanaka et al., Physica C 176 (1991) 170. T. Ekino and J. Akimitsu, JJAP Ser. 7 (1992) 260.