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On the mechanism for cooper-pairs in high-Tc-superconductors
PhysicaC 153-155 (1988) 196-197 North-Holland, Amsterdam
ON THE MECHANISMFOR COOPER-PAIRS IN HIGH-T -SUPERCONDUCTORS C
K.H. BENNEMANN
I n s t i t u t e for Theoretical Physics, Freie U n i v e r s i t ~ t B e r l i n , Arnimallee 14, D-IO00 Berlin 33, FRG I t is argued that pairing due to charge f l u c t u a t i o n s with energy ~c is most l i k e l y dominantly responsible for high-Tc-superconductivity. Despite intensive e f f o r t the mechanism responsible for Cooper-pairing in high-Tc-superconductors has remained unclear (1). Amongst the v a r i ous proposed pairing mechanisms, exchange of anti-ferromagnetic spin f l u c t u a t i o n s (2), electron-phonon coupling (3), and coupling to charge f l u c t u a t i o n s (4) accompanying CuO-bond polarisat i o n s , or Cu3+ ~ Cu2+, 02- ~ O- t r a n s i t i o n s , respectively, are p a r t i c u l a r l y studied. We use the Hubbard Hamiltonian H=ZCini+~tijc~cj+ZUiinicni++ZUijni{njo, +ZV~+ . . . .
(1)
where t i j denotes the hopping integral for elect r o n i c t r a n s i t i o n s between sites i , j and states p,d, and Uii and Uij refer to the i n t r a - and inter-atomic Coulomb interactions between the electrons, c~, cj are the usual electron creat i o n - and a n n i h i l a t i o n operators in Wannierrepresentation. V ~ denotes the repulsive i n t e r atomic i n t e r a c t i o n described by a Born-Mayer potential and necessary for c a l c u l a t i n g phonons, l a t t i c e structure, etc. For pairing due to exchange of phonons, s p i n - f l u c t u a t i o n s , or chargef l u c t u a t i o n s i t is possible to rewrite Eq. ( i ) as H ~ Hef f ; HBCS and then approximately Tc - w exp{-(1+~ ) / X . . . } . w refers to the average energy f l u c t u a t i o n of the pairing f i e l d . The electron-phonon coupling results from the interatomic distance dependence of H, 6H=~S~(3H/3rii)6rii , involving e s s e n t i a l l y the changes 6c i and 6 t i j (3). The r e s u l t a n t electron-phonon coupling has been studied in d e t a i l (3,5) and one obtains for the electronphonon coupling constant X approximately one, phonon energies ranging from Oqzph
TcS3OK. Furthermore, for the pressure dependence of Tc one calculates (3Tc/3p)sO.1K(kbar)-i for La2-S and (~Tc/~p)LO.2K(kbar)-1 for YBa2-S, which disagrees strongly with the experimental results (6). (~Tc/~p-O.29K(kbar)-1 for La2-S, and -0.16K(kbar)-1 for YBa2-S). Magnetic i n t e r a c t actions and s p i n - f l u c t u a t i o n s r e s u l t from Uii and Uij , etc. For a nearly h a l f - f i l l e d hybridized (pd)-band one expects anti-ferromagnetism. Also, Uii and U i j , i f strong enough, give rise to a Hubbard gap in the (pd)-band. The i n t e r p l a y of the Coulomb- and electronphonon coupling determines the Peierls instab i l i t y expected for (quasi-) two dimensional systems l i k e the CuO2-planes. The a n t i - f e r r o magnetic s p i n - f l u c t u a t i o n s with energy wspfl in p r i n c i p l e may give rise to an a t t r a c t i v e coupling between the electrons and thus cause Cooper-pairing and superconductivity (2), simil a r l y as in the case of He3. Eq. ( i ) can be rew r i t t e n as H ÷ Her f ~ HBCS, with H i n t - E J i j ~ i ~ j , i , j r e f e r r i n g to neighboring Cu-atoms andJij.-(4t2/U)+. For ~spfl one estimates (7) nearly the same energy range O/wspfl for both La2-S and YBa2-S (UCu(YBa2-S)
K.H. Bennemann / Mechanism for Cooper-pairs
example, CuO bond p o l a r i z a t i o n is caused by the correlated charge f l u c t u a t i o n s Cu3+ ~ Cu2+ and 02- ~ 0-. Coupling of the electrons in the (p,d) states near CF to these charge f l u c t u a t i o n s may give rise to Cooper-pairing (4,9). S h i f t i n g the center of g r a v i t y of one CuO-band towards Cu may cause an a t t r a c t i o n of the center of gravity of the other CuO-bonds towards O. For the energies of these (intra-bond, etc.) charge f l u c t u a t i o n s one estimates (10) mc ~ nh (nh = number of holes in (p,d) band), as is obvious for 02- m 0-. We estimate mc - UZN(o)+..., where U refers to intra-bond ( U i i ÷ U i j ) Coulomb coupling, and thus ~c = 0.1~O.3eV. Since ~c ~ nh, and approximately Tc ~ mc exp{-(l+~ )/~ - . . . } , one obtains in agreement with experiment Tc ~ nh, the increase of Tc (Tc ~ 40K ÷ 92K) for La2-S ÷ YBa2-S, and for (3Tc/3P)123/(3Tc/3P)LapS the value ~ 0.6 (expt ~ 0.6), i f in (123)-S one has nh 0.2÷ 0.3 in the CuO2-planes. Note, from core-electron spectroscopy (11) and Tc(x) or Tc(y) such d i f f e r e n t values for nh seem possible. From = yo(l+~elph+~c+...) - 11mJ/mole K2 one may estimate ~c ~ i÷2. (~c U2N(o)/mc, U=Uii÷Uij). In view of these results i t seems possible that amongst the three possible pairing f i e l d s discussed here the coupling to charge fluctuations (Cu3+ ~ Cu2+, 02- ~ 0-) causes dominantly high-Tc-superconductivity. I t follows from Eq. ( i ) and the previous discussion, that presumably for a l l pairing mechanisms H ÷ Herf ~HBCS, where corresponding to the l a t t i c e structures of La2-S and YBa2-S the e f f e c t i v e Hamiltonian HBCS yields highly anisotropic superconducting properties, but of BCS-type. From the approximate gap-equation
.
,4~kt+Ak '2)~
~ 2
Ak = Z Vkk,Ak, ~ann ~ / V ~ k , + A k , ) , k
i
_ _
2 (2)
_
with the pairing potential ik'r i -ik'.r. Vkk, ~ .Z.Vij e - - e -J, l,j
(3)
one expects approximately two order parameters &l and A2 r e f e r r i n g to CuO-chains and CuO2planes, respectively, since the pairing potent i a l Vij in Wannier-real space representation separates the contribution to £ r e s u l t i n g from intra-chain and intra-plane Cooper-pairing, at least approximately. Thus, approximat e l y ~I/A2 = Tc/Tc', where Tc' refers to the t r a n s i t i o n temperature in YBa2-S, which would r e s u l t without chain contribution to supercond u c t i v i t y . In view of experimental results for Tc of La2-S and Tc(y) of YBa2Cu307-y we take Tc' ~ 4OK. Then, A1/A2 = 2.3 which compares well with the value ~ 3 obtained from nuclear spin l a t t i c e r e l a x a t i o n experiment (12). Eqs. (2), (3) suggests also that d i f f e r e n t coherence lengths ~ are expected for CuOchains and CuO2-planes.
197
In summary, d i f f e r e n t possible pairing f i e l d s are discussed. Present experimental facts suggest that coupling to intra-bond charge f l u c t u a tions (for example, of type Cu3+ ~ CuZ+, 02- m 0-) may cause dominantly high-Tc-supercond u c t i v i t y . D i f f e r e n t hole number in CuO-chains and CuO2-planes may explain Tc .- 40K ÷ 92K for YBa2-S.
REFERENCES 1) T.M. Rice, Z. Phys. B67 (1987) 141; A.P. Malozemoff and P.M. Grant, Z. Phys. 867 (1987) 275. 2) J.E. Hirsch, Phys. Rev. Lett. 59 (1987) 228; V.J. Emery, Phys. Rev. Lett. 58 (1987) 2794; Proc. 18. Int. Conf. Low Temp. Physics, Kyoto (1987) in Jap. J. of Appl. Phys. 26. 3) W. Weber, Phys. Rev. Lett. 58 (1987) 1371; A.A. A l i g i a , M. Kuli~, V. Z l a t i ~ , and K.H. Bennemann, in p r i n t (1987) (4) K.H. Bennemann, Proc. Int. Conf. Many Body Physics, Aug. (1987), Oulu/Helsinki; C.M. Varma, S. Schmitt-Rink, and E. Abrahams, Sol. St. Commun. 62 (1987) 681; J.E. Hirsch, in: Workshop on Mech. of High-Tc-Superconductivity, Phys. I n s t . , Univ. of Minnesota (Oct. 1987). 5) K.H. 8ennemann, Phys. Lett. 126 (1987) 67. 6) The experimental results for 3Tc/8 p were reported by several groups, s. J. S c h i l l i n g et a l . , (1987); H. WUhl et a l . , (1987); S. Takahashi et a l . , (1987); Tech. Report ISSP No. 1770 (1987). 7) ~spfl < O.05eV, or smaller, since
8)
TN + OT Y. Kitaoka, K. Ishida, S. Hiramatsu, and K. Asayama, in print (1988); Y. Kitaoka, S. Hiramatsu, K. Ishida, K. Asayama, H. Takagi, H. lwabuchi, S. Uchida, and
S. Tanaka, in p r i n t (1988). 9) s. K.H. Bennemann, ref. (4) 10) wc = (DcCuO/Dnh)nh, where cCuo refers to the bond-energy; ~c - U2N(o) - 0.3eV. (11) P. Steiner, S. HUfner, V. Kinsinger, I. Sander, B. Siegwart, H. Schmitt, R. Schulz, S. Junk, G. Schwitzgebel, A. Gold, C. P o l i t i s , H.P. MUller, R. Hoppe, S. Kemmler-Sack, and C. Kunz, preprint (1987). (12) W.W. Warren, R.E. Walstedt, G.F. Brennert, G.P. Espinoza, and J.P. Remeika, Phys. Rev. Lett. 59 (1987) 1860.
ON THE MECHANISMFOR COOPER-PAIRS IN HIGH-T -SUPERCONDUCTORS C
K.H. BENNEMANN
I n s t i t u t e for Theoretical Physics, Freie U n i v e r s i t ~ t B e r l i n , Arnimallee 14, D-IO00 Berlin 33, FRG I t is argued that pairing due to charge f l u c t u a t i o n s with energy ~c is most l i k e l y dominantly responsible for high-Tc-superconductivity. Despite intensive e f f o r t the mechanism responsible for Cooper-pairing in high-Tc-superconductors has remained unclear (1). Amongst the v a r i ous proposed pairing mechanisms, exchange of anti-ferromagnetic spin f l u c t u a t i o n s (2), electron-phonon coupling (3), and coupling to charge f l u c t u a t i o n s (4) accompanying CuO-bond polarisat i o n s , or Cu3+ ~ Cu2+, 02- ~ O- t r a n s i t i o n s , respectively, are p a r t i c u l a r l y studied. We use the Hubbard Hamiltonian H=ZCini+~tijc~cj+ZUiinicni++ZUijni{njo, +ZV~+ . . . .
(1)
where t i j denotes the hopping integral for elect r o n i c t r a n s i t i o n s between sites i , j and states p,d, and Uii and Uij refer to the i n t r a - and inter-atomic Coulomb interactions between the electrons, c~, cj are the usual electron creat i o n - and a n n i h i l a t i o n operators in Wannierrepresentation. V ~ denotes the repulsive i n t e r atomic i n t e r a c t i o n described by a Born-Mayer potential and necessary for c a l c u l a t i n g phonons, l a t t i c e structure, etc. For pairing due to exchange of phonons, s p i n - f l u c t u a t i o n s , or chargef l u c t u a t i o n s i t is possible to rewrite Eq. ( i ) as H ~ Hef f ; HBCS and then approximately Tc - w exp{-(1+~ ) / X . . . } . w refers to the average energy f l u c t u a t i o n of the pairing f i e l d . The electron-phonon coupling results from the interatomic distance dependence of H, 6H=~S~(3H/3rii)6rii , involving e s s e n t i a l l y the changes 6c i and 6 t i j (3). The r e s u l t a n t electron-phonon coupling has been studied in d e t a i l (3,5) and one obtains for the electronphonon coupling constant X approximately one, phonon energies ranging from Oqzph
TcS3OK. Furthermore, for the pressure dependence of Tc one calculates (3Tc/3p)sO.1K(kbar)-i for La2-S and (~Tc/~p)LO.2K(kbar)-1 for YBa2-S, which disagrees strongly with the experimental results (6). (~Tc/~p-O.29K(kbar)-1 for La2-S, and -0.16K(kbar)-1 for YBa2-S). Magnetic i n t e r a c t actions and s p i n - f l u c t u a t i o n s r e s u l t from Uii and Uij , etc. For a nearly h a l f - f i l l e d hybridized (pd)-band one expects anti-ferromagnetism. Also, Uii and U i j , i f strong enough, give rise to a Hubbard gap in the (pd)-band. The i n t e r p l a y of the Coulomb- and electronphonon coupling determines the Peierls instab i l i t y expected for (quasi-) two dimensional systems l i k e the CuO2-planes. The a n t i - f e r r o magnetic s p i n - f l u c t u a t i o n s with energy wspfl in p r i n c i p l e may give rise to an a t t r a c t i v e coupling between the electrons and thus cause Cooper-pairing and superconductivity (2), simil a r l y as in the case of He3. Eq. ( i ) can be rew r i t t e n as H ÷ Her f ~ HBCS, with H i n t - E J i j ~ i ~ j , i , j r e f e r r i n g to neighboring Cu-atoms andJij.-(4t2/U)+. For ~spfl one estimates (7) nearly the same energy range O
K.H. Bennemann / Mechanism for Cooper-pairs
example, CuO bond p o l a r i z a t i o n is caused by the correlated charge f l u c t u a t i o n s Cu3+ ~ Cu2+ and 02- ~ 0-. Coupling of the electrons in the (p,d) states near CF to these charge f l u c t u a t i o n s may give rise to Cooper-pairing (4,9). S h i f t i n g the center of g r a v i t y of one CuO-band towards Cu may cause an a t t r a c t i o n of the center of gravity of the other CuO-bonds towards O. For the energies of these (intra-bond, etc.) charge f l u c t u a t i o n s one estimates (10) mc ~ nh (nh = number of holes in (p,d) band), as is obvious for 02- m 0-. We estimate mc - UZN(o)+..., where U refers to intra-bond ( U i i ÷ U i j ) Coulomb coupling, and thus ~c = 0.1~O.3eV. Since ~c ~ nh, and approximately Tc ~ mc exp{-(l+~ )/~ - . . . } , one obtains in agreement with experiment Tc ~ nh, the increase of Tc (Tc ~ 40K ÷ 92K) for La2-S ÷ YBa2-S, and for (3Tc/3P)123/(3Tc/3P)LapS the value ~ 0.6 (expt ~ 0.6), i f in (123)-S one has nh 0.2÷ 0.3 in the CuO2-planes. Note, from core-electron spectroscopy (11) and Tc(x) or Tc(y) such d i f f e r e n t values for nh seem possible. From = yo(l+~elph+~c+...) - 11mJ/mole K2 one may estimate ~c ~ i÷2. (~c U2N(o)/mc, U=Uii÷Uij). In view of these results i t seems possible that amongst the three possible pairing f i e l d s discussed here the coupling to charge fluctuations (Cu3+ ~ Cu2+, 02- ~ 0-) causes dominantly high-Tc-superconductivity. I t follows from Eq. ( i ) and the previous discussion, that presumably for a l l pairing mechanisms H ÷ Herf ~HBCS, where corresponding to the l a t t i c e structures of La2-S and YBa2-S the e f f e c t i v e Hamiltonian HBCS yields highly anisotropic superconducting properties, but of BCS-type. From the approximate gap-equation
.
,4~kt+Ak '2)~
~ 2
Ak = Z Vkk,Ak, ~ann ~ / V ~ k , + A k , ) , k
i
_ _
2 (2)
_
with the pairing potential ik'r i -ik'.r. Vkk, ~ .Z.Vij e - - e -J, l,j
(3)
one expects approximately two order parameters &l and A2 r e f e r r i n g to CuO-chains and CuO2planes, respectively, since the pairing potent i a l Vij in Wannier-real space representation separates the contribution to £ r e s u l t i n g from intra-chain and intra-plane Cooper-pairing, at least approximately. Thus, approximat e l y ~I/A2 = Tc/Tc', where Tc' refers to the t r a n s i t i o n temperature in YBa2-S, which would r e s u l t without chain contribution to supercond u c t i v i t y . In view of experimental results for Tc of La2-S and Tc(y) of YBa2Cu307-y we take Tc' ~ 4OK. Then, A1/A2 = 2.3 which compares well with the value ~ 3 obtained from nuclear spin l a t t i c e r e l a x a t i o n experiment (12). Eqs. (2), (3) suggests also that d i f f e r e n t coherence lengths ~ are expected for CuOchains and CuO2-planes.
197
In summary, d i f f e r e n t possible pairing f i e l d s are discussed. Present experimental facts suggest that coupling to intra-bond charge f l u c t u a tions (for example, of type Cu3+ ~ CuZ+, 02- m 0-) may cause dominantly high-Tc-supercond u c t i v i t y . D i f f e r e n t hole number in CuO-chains and CuO2-planes may explain Tc .- 40K ÷ 92K for YBa2-S.
REFERENCES 1) T.M. Rice, Z. Phys. B67 (1987) 141; A.P. Malozemoff and P.M. Grant, Z. Phys. 867 (1987) 275. 2) J.E. Hirsch, Phys. Rev. Lett. 59 (1987) 228; V.J. Emery, Phys. Rev. Lett. 58 (1987) 2794; Proc. 18. Int. Conf. Low Temp. Physics, Kyoto (1987) in Jap. J. of Appl. Phys. 26. 3) W. Weber, Phys. Rev. Lett. 58 (1987) 1371; A.A. A l i g i a , M. Kuli~, V. Z l a t i ~ , and K.H. Bennemann, in p r i n t (1987) (4) K.H. Bennemann, Proc. Int. Conf. Many Body Physics, Aug. (1987), Oulu/Helsinki; C.M. Varma, S. Schmitt-Rink, and E. Abrahams, Sol. St. Commun. 62 (1987) 681; J.E. Hirsch, in: Workshop on Mech. of High-Tc-Superconductivity, Phys. I n s t . , Univ. of Minnesota (Oct. 1987). 5) K.H. 8ennemann, Phys. Lett. 126 (1987) 67. 6) The experimental results for 3Tc/8 p were reported by several groups, s. J. S c h i l l i n g et a l . , (1987); H. WUhl et a l . , (1987); S. Takahashi et a l . , (1987); Tech. Report ISSP No. 1770 (1987). 7) ~spfl < O.05eV, or smaller, since
8)
TN + OT Y. Kitaoka, K. Ishida, S. Hiramatsu, and K. Asayama, in print (1988); Y. Kitaoka, S. Hiramatsu, K. Ishida, K. Asayama, H. Takagi, H. lwabuchi, S. Uchida, and
S. Tanaka, in p r i n t (1988). 9) s. K.H. Bennemann, ref. (4) 10) wc = (DcCuO/Dnh)nh, where cCuo refers to the bond-energy; ~c - U2N(o) - 0.3eV. (11) P. Steiner, S. HUfner, V. Kinsinger, I. Sander, B. Siegwart, H. Schmitt, R. Schulz, S. Junk, G. Schwitzgebel, A. Gold, C. P o l i t i s , H.P. MUller, R. Hoppe, S. Kemmler-Sack, and C. Kunz, preprint (1987). (12) W.W. Warren, R.E. Walstedt, G.F. Brennert, G.P. Espinoza, and J.P. Remeika, Phys. Rev. Lett. 59 (1987) 1860.