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Isotope effect of oxide superconductors
Volume 143, number 9
PHYSICS LETTERSA
5 February 1990
I S O T O P E EFFECT OF OXIDE S U P E R C O N D U C T O R S Xi-yu SU Department of Physics, Qufu Normal University, Qufu 273165, Shandong, PR China
Jiang SHEN and Li-yuan Z H A N G Department of Physics, Peking University, Beijing 100871, PR China
Received 5 May 1989; revised manuscript received 3 November 1989; accepted for publication 5 December 1989 Communicated by D. Bloch
The free-carrier-negative-U-centerinteracting mechanism model has been used to investigate the isotope effect of the oxide superconductors. The properties of the systemscan be interpreted within this model. Some predictions are made.
It has been demonstrated by experiments [1-3] that a very weak but nonzero isotope effect exists for the high-To oxide superconductors, and the higher the transition temperature, the smaller the isotope effect is. This fact cannot be understood within the conventional electron-phonon mechanism or within the nonelectron-phonon mechanism model. Although phonon exchange [4] seems capable of producing a transition temperature above 30 K as required [5 ] for doped LaECuO4, it is more difficult to imagine that it can be responsible for superconductivity above 90 K as attained in Y - B a - C u - O , BiS r - C a - C u - O , T I - B a - C a - C u - O and related materials, while in the nonelectron-phonon mechanism the isotope effect will vanish. Thus, Cohen et al. [ 6 ] and Marsiglio et al. [ 7] have proposed the electronphonon-nonelectron-phonon (phenomenologically) and the electron-phonon-exciton-exchange coupled models, respectively, and explained some experimental results partly. In fact, as pointed out by Zhang and Yao [8,9], the free-carrier-negative-U-center interacting mechanism model is very possibly responsible for the highTc superconductivity of the oxide superconductors. They also explained some properties of the systems. Although so far there is no direct experimental evidence for the existence of the negative-U center, it is certain that there are experiments which support the free-carrier-negative-U-center mechanism. For in-
stance, from the copper L3 X-ray absorption spectra, Bianconi et al. [ 10,11 ] have given evidence for itinerant holes in the oxygen derived band (ligand hole) in YBa2Cu307_~ and LaL85Sro.15CuO4. They have also indicated that a strong correlation between the local Cu 3d and itinerant ligand holes and large hybridization between the O 2p and Cu 3d orbitals exist. Thus, similar to the argument by Hirsch [ 12 ], it is possible to form the negative-U center on the Cu site in the case with the on-site correlation U and a nearest-neighbor correlation V (repulsion between Cu and O). Relating to this, we call attention to the paper by Ihara et al. [ 13 ], in which they have shown from the photoemission study of the chemical bond nature of YBa2Cu307_,~that trivalent or monovalent Cu ions are always coexistent with divalent Cu ions, in agreement with Pauling's point [ 14 ]. In our opinion, at least, the possibility of the near unscreening charge variations (fluctuations) on a small scale must be considered in the high-To oxide superconductors [9]. Another point, it is also possible that a bipolaron is constructed from two d-holes of neighboring Cu ions, which has been suggested by Kamimura using the superexchange mechanism [ 15 ]. Finally, it is worthwhile to note the work by Cooper et al. [ 16 ], in which the existence of a continuum of electronic states inside the gap of superconducting YBa2Cu307_,~ has been shown by a Raman scattering experiment. Early in February 1987 [ 8 ], we pre-
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PHYSICS LETTERS A
dieted by the free-carrier-negative-U mechanism that a small density of states due to the mixing interaction between the free carrier and the localized carrier exists within the region of the energy gap in the highTc oxide superconductors. In this paper, we use the free-carrier-negative-Ucenter interacting mechanism model to investigate the isotope effect of the oxide superconductors. It has been shown that the experimental results can be interpreted within this model. Furthermore, we have made some predictions. Within the free-carrier-negative-U-center interacting mechanism model, the earlier works by Ting et al. [17] and Zhang [ 18 ] have revealed that the mixing interaction between the free carrier and the negative-U center is to increase the transition temperature, since the attraction between the free carriers is strengthened by the attractive interaction within the negative-U center. Even in the zero limit of the electron-phonon interaction (•ph)of the free carriers there is still a high value of Tc. This means that the negative-U center provides an extra attractive interaction to the free carriers. The extra attraction may be electron-phonon or nonelectronphonon interaction [ 8 ]. Here, we only consider the latter case. For simplicity, we suppose the extra attraction can be written as - V ¢ (V c > 0 ) and is independent of the phonon frequency (to). Thus, the total attraction between the free carriers is V,(to) = - Vph (to) -- V~,
x
fd¢' [v,.(¢-¢')+ Vc]J(¢')
1- 2 f ( ¢ ' ) 24'
'
1- 2 f ( ~ ' )
(3)
24'
A is independent of ~. The equation for J is A(¢) = N ( 0 ) j d~' Vph(~--~' )A(~')
x
1-2f(¢' ) +A. 24'
(4)
When [~1 > COD,the integral is not large, because the factors V p h ( ~ - - ~ ' ) a n d I / 4 ' cannot be simultaneously large. Thus in this region we may roughly set A(~) =A. On the other hand, the integral is important when I~1 is small. Let B be the average value of A in the region toD> I~1. Then we have rOD
B~-N(O)Vph f d~' B I-2f(~') +A 24'
--
COD
~-N(O) VphBIn(1.14toD/T¢) +A,
(5)
where Vph is some average of Vph(to) over the interval - toD < to < COD"Finally, eq. (3) defining A may be rewritten as
A"~N(O)V~
-- £oc
-- ~OD
O)D
~-N(O) Vc[BIn(1.14toD/Tc) +A
In (to¢/toD) ],
(6) toc is a high-frequency cutoff for Vc. Noting that eqs. (5) and (6) are compatible, we arrive at the condition 1 =ln(1.14too/T~)
( ×
N(O)Vc N(0)Vph+ 1 - N ( 0 ) V ~ l n ( t o ¢ / t o o ) J '
(7)
or
(2)
where o ~ = ~ - ¢ ' , ~ is the energy measured from the Fermi level, f(~)=f(~/T=) is the Fermi function, and N ( 0 ) is the density of states on the Fermi level. To solve eq. (2) we first separate the Vc term and call its contribution A, 490
(" A = N ( 0 ) J d ~ ' VcA(~')
(1)
as it has been demonstrated by experiments [ 19-21 ] that in the oxide superconductors the superconducting carrier is stillthe Cooper pair. Thereby, similar to rcf. [22 ], the equation for the transition tcrnperature becomes under tlacweak-coupling condition (using units k a = h = 1 ) A(~)=N(O)
5 February 1990
Tc = 1.14top exp
2ph+
'
(8)
where 2ph = N ( 0 ) Vph, N ( 0 ) Vc Uc= I - N ( 0 ) V c ln(toc/toD) '
(9) (10)
Volume 143, number 9
PHYSICS LETTERS A
0,5 0.4.
~o.~ ~o.2 gl
g~ o
.~ o , t 0 o
I
I
)
lo
20
30
!
!
40
5O
Uc/X.ph Fig. 1. Isotope effect a as a function of Uc/Xph. Uc is the effective attraction provided by the negative-U center. Obviously, Uc enhances T¢ greatly (here, we only consider the case Uc > 0). As usual, the isotope effect ( a ) can be derived from eqs. (8) and (10) as o~=
dlnM
2
1-
2ph-~Uc
'
(11)
where M is the oxygen mass (COD~CM1/2). First of all, we discuss the isotope effect in two limits. ( 1 ) In the pure electron-phonon mechanism, i.e. 2 p h ¢ 0 and U~=0, we have ol= ½. This is the wellknown result o f the conventional BCS theory. (2) In the pure negative-U center mechanism (nonelect r o n - p h o n o n ) , i.e. U~¢0 and 2ph=0, we have c~=0. This is just the result o f the usual nonelectronphonon mechanism. These results make us believe in the correctness o f our model. In general, ot is a function of Ud2vh, the relative strength o f the attraction Uc compared to the elect r o n - p h o n o n coupling, as shown in fig. 1. It is sim-
5 February 1990
ilar to the result by Marsiglio et al. [ 7 ]. From the figure we see that a remains near zero for a large range below the pure negative-U center mechanism. What is most important is that using the value of Uc/ 2 ph = 2 1 estimated from the EELS experiment [ 23 ] for YBa2Cu307_6 system we have a = 0 . 0 4 4 , which is in good agreement with the experiments [24,25]. This shows again the reasonableness o f our model. Relating to this, we derive the values o f the parameters from the experimental data of Tc and ot for different systems. In this process we let COD=300 K be fixed, since the phonon cutoff frequency cannot be very different for these systems [26 ]. The results are shown in table 1. Now, let us consider the isotope effect as a function o f the transition temperature. For simplicity, setting 2ph = 0.052, the average value of that in table 1, we have the solid line shown in fig. 2. What is interesting is that the experimental results (shown by dots) are distributed near the theoretical line in a reasonable way. It reminds us that all the oxide superconductors may be treated as a kind of class of superconducting system. In this system, the higher the To, the smaller the c~ is. This fact can be understood easily within our model. We see from table 1 that the effective attraction Uc of the negative- U center plays an important role in the oxide superconductors. The higher the transition temperature, the stronger the strength of the effective attraction U~ and the relative strength of it compared to the electronphonon coupling are. The former leads to an increase in the transition temperature, and the latter leads to a decrease in the isotope effect, respectively. Besides another prediction can be made by taking table 1 into account, and it is easier to see from fig. 3, T~ as a function o f Uc/2ph, that when Tc is low, it increases rapidly as Uc/2ph increases; when Tc is high, its increasing speed is small. So we can conclude that
Table 1 The values of some parameters obtained from experiments. System
Tc (K)
a
Uc//~ ph
•ph
Ue
BaPbBiO LaSrCuO YBaCuO BiSrCaCuO T1BaCaCuO
11 36 92 110 125
0.22 [ 1] 0.16 [27] 0.043 [24,25 ] 0.026 [3] 0.02 [3]
3.0 4.7 22 37 48.5
0.073 0.079 0.033 0.023 0.02
0.218 0.371 0.727 0.857 0.970 491
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t e m , the e f f e c t i v e a t t r a c t i o n Uc p r o v i d e d by the nega t i v e - U c e n t e r e n h a n c e s the t r a n s i t i o n t e m p e r a t u r e greatly, the i s o t o p e effect decreases as the r e l a t i v e strength o f the e f f e c t i v e a t t r a c t i o n Uc c o m p a r e d to the e l e c t r o n - p h o n o n c o u p l i n g increases, a n d the transition temperature of some oxide superconductor m a y reach 150 K.
I
B~bBiO 0,4
Z~SrCuO
YgaCuO
I
-p
~o.~
5 February 1990
BiSrCaCuO
o
References
~O.t -1
I
I
I
I
40
0
ao Te (K)
I e
I
120
160
Fig. 2. Isotope effect a as a function of Tc (2 ph= 0.052 ). The dots are the experimental data for different systems.
160
T1BaC a ~iS~-CaCuO
t20
C u O ~
-'---- YBaC~O v
o80
4O
~aSrC~o LBI~bBIO 0
I
I
I
I
I
10
20
"~0
40
50
Uc/xpk Fig. 3. Tc as a function of fc/~-ph.
the t r a n s i t i o n t e m p e r a t u r e o f the o x i d e s u p e r c o n d u c t o r has a s a t u r a t i o n v a l u e (Tcs). H e r e we get T~s~ 150 K. T h a t is to say, the t r a n s i t i o n t e m p e r a ture o f s o m e o x i d e s u p e r c o n d u c t o r m a y reach 150 K. A s i m i l a r result has b e e n o b t a i n e d in a n o t h e r w a y by Eab a n d T a n g [ 2 8 ] . In c o n c l u s i o n , the e x p e r i m e n t a l results o f the isot o p e effect o f the o x i d e s u p e r c o n d u c t o r s h a v e b e e n i n t e r p r e t e d w i t h i n o u r m o d e l . We h a v e s h o w n that all the o x i d e s u p e r c o n d u c t o r s m a y be t r e a t e d as a k i n d o f class o f s u p e r c o n d u c t i n g system. In this sys492
[ 1] B. Batlogg, R.J. Cava and M. Stavola, Phys. Rev. Lett. 60 (1988) 754. [ 2 ] M. Tachiki and S. Takahashi, Charge fluctuation mechanism of high Tc superconductivity and isotope effect in oxide superconductors, preprint. [ 3 ] H. Katayama-Yoshida, T. Hirooka, A. Oyamada, Y. Okabe, T. Takahashi, T. Sasaki, A. Uchiai and T. Suzuki, Oxygen isotope effect in the superconducting Bi-Sr-Ca-Cu-O system, preprint. [4] W. Weber, Phys. Rev. Lett. 58 (1987) 1371, 2154(E). [5] J.G. Bednorz and K.A. Miiller, Z. Phys. B 64 (1986) 189. [6] R.W. Cohen, M.H. Cohen and M.L. Cohen, Isotope effect as a probe of pairing interactions on oxide superconductors, preprint. [ 7 ] F. Marsiglio and J.P. Carbone, Phys. Rev. B 36 ( 1987 ) 3937. [8] Zhang Li-yuan, Solid State Commun. 62 (1987) 491. [9] Zhang Li-yuan and Yao Chen-yuan, Physica C 153-155 (1988) 1207. [10]A. Bianconi, J.Budnick, A.M. Flank, A. Fontaine, P. Lagarde, A. Marcelli, H. Tolentino, B. Chamberland, C. Michel, B. Raveau and G. Demazeau, Phys. Len. A 127 (1988) 285. [ 11 ] A. Bianconi, A. Congiu Castellano, M. De Santis, P. Delogu, A. Gargano and R. Giorgi, Solid State Commun. 63 (1987)
1135. [ 12 ] J.E. Hirsch, Talk presented at Workshop on Mechanism of high Tc superconductivity, Theoretical Physics Institute, University of Minnesota (25-27 October 1987 ). [ 13 ] H. Ihara, M. Jo, N. Terada, M. Hirabayashi, H. Oyanagi, K. Murata, Y. Kimura, R. Sugise, I. Hayashida, S. Ohashi and M. Akimoto, Physica C 153-155 (1988) 131. [ 14] L. Pauling, Phys. Rev. Lett. 59 (1987) 225. [ 15 ] H. Kamimura, Japan. J. Appl. Phys. 26 (1987) L627. [16] S.L. Cooper, M.V. Klein, B.G. Pazol, J.P. Rice and D.M. Ginsberg, Phys. Rev. B 37 (1988) 5920. [ 17 ] C.S. Ting, D.N. Talwar and K.L. Ngai, Phys. Rev. Lett. 45 (1980) 1213. [ 18 ] Zhang Li-yuan, Acta Phys. Sin. 32 ( 1983 ) 1435. [19] C.E. Gough, M.C. Colelough, E.M. Forgan, R.G. Jordan, M. Keene, C.M. Muirhead, A.I.M. Rae, N. Thomas, J.S. Abell and S. Sutton, Nature 326 (1987) 855. [20] J.S. Tsai, Y. Kubo and J. Tabuchi, Phys. Rev. Lett. 58 (1987) 1979.
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[ 21 ] T. Yamashita, A. Kawasaki, T. Nishihara, Y. Hirotsu and M. Takata, Japan. J. Appl. Phys. 26 (1987) L635. [22] P.G. De Gennes, Superconductivity of metals and alloys (New York, 1966) p. 126. [23] M. Tachiki and S. Takahashi, Phys. Rev. B 38 (1988) 218. [24] 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 (1987) 1236.
5 February 1990
[25]T.A. Faltens, W.K. Ham, S.W. Keller, K.J. Leary, J.N. Michaels, A.M. Stacy and H.-C. Zur Loye, Phys. Rev. Lett. 59 (1987) 915. [26] S. Uchida, H. Takagi, T. Hasegawa, K. Kishio, S. Tajima, K. Kitazawa, K. Fueki and S. Tanaka, Electronic states in high Tc oxide superconductors, preprint. [27]B. Batlogg, G. Kourouklis, W. Weber, R.J. Cava, A. Jayaraman, A.E. White, K.T. Short, L.W. Rupp and E.A. Rietman, Phys. Rev. Lett. 59 (1987) 912. [28] C.-H. Eab and I-M. Tang, Phys. Lett. A 134 (1989) 253.
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5 February 1990
I S O T O P E EFFECT OF OXIDE S U P E R C O N D U C T O R S Xi-yu SU Department of Physics, Qufu Normal University, Qufu 273165, Shandong, PR China
Jiang SHEN and Li-yuan Z H A N G Department of Physics, Peking University, Beijing 100871, PR China
Received 5 May 1989; revised manuscript received 3 November 1989; accepted for publication 5 December 1989 Communicated by D. Bloch
The free-carrier-negative-U-centerinteracting mechanism model has been used to investigate the isotope effect of the oxide superconductors. The properties of the systemscan be interpreted within this model. Some predictions are made.
It has been demonstrated by experiments [1-3] that a very weak but nonzero isotope effect exists for the high-To oxide superconductors, and the higher the transition temperature, the smaller the isotope effect is. This fact cannot be understood within the conventional electron-phonon mechanism or within the nonelectron-phonon mechanism model. Although phonon exchange [4] seems capable of producing a transition temperature above 30 K as required [5 ] for doped LaECuO4, it is more difficult to imagine that it can be responsible for superconductivity above 90 K as attained in Y - B a - C u - O , BiS r - C a - C u - O , T I - B a - C a - C u - O and related materials, while in the nonelectron-phonon mechanism the isotope effect will vanish. Thus, Cohen et al. [ 6 ] and Marsiglio et al. [ 7] have proposed the electronphonon-nonelectron-phonon (phenomenologically) and the electron-phonon-exciton-exchange coupled models, respectively, and explained some experimental results partly. In fact, as pointed out by Zhang and Yao [8,9], the free-carrier-negative-U-center interacting mechanism model is very possibly responsible for the highTc superconductivity of the oxide superconductors. They also explained some properties of the systems. Although so far there is no direct experimental evidence for the existence of the negative-U center, it is certain that there are experiments which support the free-carrier-negative-U-center mechanism. For in-
stance, from the copper L3 X-ray absorption spectra, Bianconi et al. [ 10,11 ] have given evidence for itinerant holes in the oxygen derived band (ligand hole) in YBa2Cu307_~ and LaL85Sro.15CuO4. They have also indicated that a strong correlation between the local Cu 3d and itinerant ligand holes and large hybridization between the O 2p and Cu 3d orbitals exist. Thus, similar to the argument by Hirsch [ 12 ], it is possible to form the negative-U center on the Cu site in the case with the on-site correlation U and a nearest-neighbor correlation V (repulsion between Cu and O). Relating to this, we call attention to the paper by Ihara et al. [ 13 ], in which they have shown from the photoemission study of the chemical bond nature of YBa2Cu307_,~that trivalent or monovalent Cu ions are always coexistent with divalent Cu ions, in agreement with Pauling's point [ 14 ]. In our opinion, at least, the possibility of the near unscreening charge variations (fluctuations) on a small scale must be considered in the high-To oxide superconductors [9]. Another point, it is also possible that a bipolaron is constructed from two d-holes of neighboring Cu ions, which has been suggested by Kamimura using the superexchange mechanism [ 15 ]. Finally, it is worthwhile to note the work by Cooper et al. [ 16 ], in which the existence of a continuum of electronic states inside the gap of superconducting YBa2Cu307_,~ has been shown by a Raman scattering experiment. Early in February 1987 [ 8 ], we pre-
0375-9601/90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)
489
Volume 143, number 9
PHYSICS LETTERS A
dieted by the free-carrier-negative-U mechanism that a small density of states due to the mixing interaction between the free carrier and the localized carrier exists within the region of the energy gap in the highTc oxide superconductors. In this paper, we use the free-carrier-negative-Ucenter interacting mechanism model to investigate the isotope effect of the oxide superconductors. It has been shown that the experimental results can be interpreted within this model. Furthermore, we have made some predictions. Within the free-carrier-negative-U-center interacting mechanism model, the earlier works by Ting et al. [17] and Zhang [ 18 ] have revealed that the mixing interaction between the free carrier and the negative-U center is to increase the transition temperature, since the attraction between the free carriers is strengthened by the attractive interaction within the negative-U center. Even in the zero limit of the electron-phonon interaction (•ph)of the free carriers there is still a high value of Tc. This means that the negative-U center provides an extra attractive interaction to the free carriers. The extra attraction may be electron-phonon or nonelectronphonon interaction [ 8 ]. Here, we only consider the latter case. For simplicity, we suppose the extra attraction can be written as - V ¢ (V c > 0 ) and is independent of the phonon frequency (to). Thus, the total attraction between the free carriers is V,(to) = - Vph (to) -- V~,
x
fd¢' [v,.(¢-¢')+ Vc]J(¢')
1- 2 f ( ¢ ' ) 24'
'
1- 2 f ( ~ ' )
(3)
24'
A is independent of ~. The equation for J is A(¢) = N ( 0 ) j d~' Vph(~--~' )A(~')
x
1-2f(¢' ) +A. 24'
(4)
When [~1 > COD,the integral is not large, because the factors V p h ( ~ - - ~ ' ) a n d I / 4 ' cannot be simultaneously large. Thus in this region we may roughly set A(~) =A. On the other hand, the integral is important when I~1 is small. Let B be the average value of A in the region toD> I~1. Then we have rOD
B~-N(O)Vph f d~' B I-2f(~') +A 24'
--
COD
~-N(O) VphBIn(1.14toD/T¢) +A,
(5)
where Vph is some average of Vph(to) over the interval - toD < to < COD"Finally, eq. (3) defining A may be rewritten as
A"~N(O)V~
-- £oc
-- ~OD
O)D
~-N(O) Vc[BIn(1.14toD/Tc) +A
In (to¢/toD) ],
(6) toc is a high-frequency cutoff for Vc. Noting that eqs. (5) and (6) are compatible, we arrive at the condition 1 =ln(1.14too/T~)
( ×
N(O)Vc N(0)Vph+ 1 - N ( 0 ) V ~ l n ( t o ¢ / t o o ) J '
(7)
or
(2)
where o ~ = ~ - ¢ ' , ~ is the energy measured from the Fermi level, f(~)=f(~/T=) is the Fermi function, and N ( 0 ) is the density of states on the Fermi level. To solve eq. (2) we first separate the Vc term and call its contribution A, 490
(" A = N ( 0 ) J d ~ ' VcA(~')
(1)
as it has been demonstrated by experiments [ 19-21 ] that in the oxide superconductors the superconducting carrier is stillthe Cooper pair. Thereby, similar to rcf. [22 ], the equation for the transition tcrnperature becomes under tlacweak-coupling condition (using units k a = h = 1 ) A(~)=N(O)
5 February 1990
Tc = 1.14top exp
2ph+
'
(8)
where 2ph = N ( 0 ) Vph, N ( 0 ) Vc Uc= I - N ( 0 ) V c ln(toc/toD) '
(9) (10)
Volume 143, number 9
PHYSICS LETTERS A
0,5 0.4.
~o.~ ~o.2 gl
g~ o
.~ o , t 0 o
I
I
)
lo
20
30
!
!
40
5O
Uc/X.ph Fig. 1. Isotope effect a as a function of Uc/Xph. Uc is the effective attraction provided by the negative-U center. Obviously, Uc enhances T¢ greatly (here, we only consider the case Uc > 0). As usual, the isotope effect ( a ) can be derived from eqs. (8) and (10) as o~=
dlnM
2
1-
2ph-~Uc
'
(11)
where M is the oxygen mass (COD~CM1/2). First of all, we discuss the isotope effect in two limits. ( 1 ) In the pure electron-phonon mechanism, i.e. 2 p h ¢ 0 and U~=0, we have ol= ½. This is the wellknown result o f the conventional BCS theory. (2) In the pure negative-U center mechanism (nonelect r o n - p h o n o n ) , i.e. U~¢0 and 2ph=0, we have c~=0. This is just the result o f the usual nonelectronphonon mechanism. These results make us believe in the correctness o f our model. In general, ot is a function of Ud2vh, the relative strength o f the attraction Uc compared to the elect r o n - p h o n o n coupling, as shown in fig. 1. It is sim-
5 February 1990
ilar to the result by Marsiglio et al. [ 7 ]. From the figure we see that a remains near zero for a large range below the pure negative-U center mechanism. What is most important is that using the value of Uc/ 2 ph = 2 1 estimated from the EELS experiment [ 23 ] for YBa2Cu307_6 system we have a = 0 . 0 4 4 , which is in good agreement with the experiments [24,25]. This shows again the reasonableness o f our model. Relating to this, we derive the values o f the parameters from the experimental data of Tc and ot for different systems. In this process we let COD=300 K be fixed, since the phonon cutoff frequency cannot be very different for these systems [26 ]. The results are shown in table 1. Now, let us consider the isotope effect as a function o f the transition temperature. For simplicity, setting 2ph = 0.052, the average value of that in table 1, we have the solid line shown in fig. 2. What is interesting is that the experimental results (shown by dots) are distributed near the theoretical line in a reasonable way. It reminds us that all the oxide superconductors may be treated as a kind of class of superconducting system. In this system, the higher the To, the smaller the c~ is. This fact can be understood easily within our model. We see from table 1 that the effective attraction Uc of the negative- U center plays an important role in the oxide superconductors. The higher the transition temperature, the stronger the strength of the effective attraction U~ and the relative strength of it compared to the electronphonon coupling are. The former leads to an increase in the transition temperature, and the latter leads to a decrease in the isotope effect, respectively. Besides another prediction can be made by taking table 1 into account, and it is easier to see from fig. 3, T~ as a function o f Uc/2ph, that when Tc is low, it increases rapidly as Uc/2ph increases; when Tc is high, its increasing speed is small. So we can conclude that
Table 1 The values of some parameters obtained from experiments. System
Tc (K)
a
Uc//~ ph
•ph
Ue
BaPbBiO LaSrCuO YBaCuO BiSrCaCuO T1BaCaCuO
11 36 92 110 125
0.22 [ 1] 0.16 [27] 0.043 [24,25 ] 0.026 [3] 0.02 [3]
3.0 4.7 22 37 48.5
0.073 0.079 0.033 0.023 0.02
0.218 0.371 0.727 0.857 0.970 491
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!
I
t e m , the e f f e c t i v e a t t r a c t i o n Uc p r o v i d e d by the nega t i v e - U c e n t e r e n h a n c e s the t r a n s i t i o n t e m p e r a t u r e greatly, the i s o t o p e effect decreases as the r e l a t i v e strength o f the e f f e c t i v e a t t r a c t i o n Uc c o m p a r e d to the e l e c t r o n - p h o n o n c o u p l i n g increases, a n d the transition temperature of some oxide superconductor m a y reach 150 K.
I
B~bBiO 0,4
Z~SrCuO
YgaCuO
I
-p
~o.~
5 February 1990
BiSrCaCuO
o
References
~O.t -1
I
I
I
I
40
0
ao Te (K)
I e
I
120
160
Fig. 2. Isotope effect a as a function of Tc (2 ph= 0.052 ). The dots are the experimental data for different systems.
160
T1BaC a ~iS~-CaCuO
t20
C u O ~
-'---- YBaC~O v
o80
4O
~aSrC~o LBI~bBIO 0
I
I
I
I
I
10
20
"~0
40
50
Uc/xpk Fig. 3. Tc as a function of fc/~-ph.
the t r a n s i t i o n t e m p e r a t u r e o f the o x i d e s u p e r c o n d u c t o r has a s a t u r a t i o n v a l u e (Tcs). H e r e we get T~s~ 150 K. T h a t is to say, the t r a n s i t i o n t e m p e r a ture o f s o m e o x i d e s u p e r c o n d u c t o r m a y reach 150 K. A s i m i l a r result has b e e n o b t a i n e d in a n o t h e r w a y by Eab a n d T a n g [ 2 8 ] . In c o n c l u s i o n , the e x p e r i m e n t a l results o f the isot o p e effect o f the o x i d e s u p e r c o n d u c t o r s h a v e b e e n i n t e r p r e t e d w i t h i n o u r m o d e l . We h a v e s h o w n that all the o x i d e s u p e r c o n d u c t o r s m a y be t r e a t e d as a k i n d o f class o f s u p e r c o n d u c t i n g system. In this sys492
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PHYSICS LETTERS A
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