- Email: [email protected]
Band structure, electron correlations, and an interlayer pairing mechanism for high temperature superconductivity
Physica C 153-155 (1988) 1315-1316 North-Holland, Amsterdam
BAND S T R U C T U R E , E L E C T R O N C O R R E L A T I O N S , AND AN I N T E R L A Y E R P A I R I N G M E C H A N I S M F O R HIGH T E M P E R A T U R E S U P E R C O N D U C T I V I T Y J. A S t t K E N A Z I and C.G. K U P E R Physics Department, Technion, Haifa 32000, Israel
We show that the condnction electrons in YBa~CusO~ ~ arise mainly from O2-O3 p-orbitals, and lie in a narrow band (of width ~ 0.1 eV). The Cooper pairs in the superconducting state come from two distinct CuO~ planes, and the "glue" which binds them is an electronic breathing mode of charge transfer within an interplanar O 4 - C u l - O 4 complex. Our picture is consistent with experiment. Very slow specific heat measurements [1] in YBa~CusOr_6 ( Y B C O ) show a ~ divergence at the superconducting (SC) transition temperature T~. Thermoelectric power ( T E P ) data [2] indicate a narrow conduction band (CB), of width ~ 0.1 eV, full for 5 = 1 and half full for 5 = 0. Various measurements [3-4] indicate O-hole carriers at Er, and a fall of the valence of the C u l atoms [5] from +2 to +1, as 6 increases from 0 to 1. We model Y B C O by two CuOz planes and the CuOa chains between them. In the planes (but not in the broken chains) translation is a good symla merry. We define annihilation operators thus: ai~ specifies an orbital a, belonging to a Cu2(3d) or t, specifies an O2-O3(2p) shell in the plane 1 = ±1. ei~ orbital c~ in a Cul (3d) or O1-O4(2p) shell; l = ±1 for the 0 4 orbital adjacent to the +1 plane; otherwise l = o. e and i specify respectively the spin and the unit cell. The nlodel Hanliltonian H = H, + H~ + H;: Ha
=
E ijla[~o"
o¢1 l,x? 113 1 x-~ ,:,0 lal lC~al~lal~ ~ t i - J aia aJ a + 2 2_~ U i - j ala aia j~r' ja'], aa Vila
ijlrnaOo" Hi
Cio" Cio'Cja~ jo" ],
a'
Z ..~afl tat aiaCjo-,Cjo. a ll~l ID V~ij aia ijlc~acr'
O)
includes band structure and correlation effects. tlere i - j stands for/~i -/~i. Replacing tT~ by [7]i - tT-~ + 5ij6.~ E , ( 2 - 5a,)U:" describes hole states. Starting from localized states we diagonalize H, to second order in ti_~. "-"~ Estimating the parameters i n / / f r o m band-structure results [6], we find that the highest electron band (nlainly Cu2(d.,_ u,)) is empty. Below it lies the partly filled CB, comprising mainly O2-O3(p) orbitals. Its dispersion is found by diagonalizing the 6 x 6 matrix:
0921-4534/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
+ Z ( 1 - 5joS,..vn.v)U; ='r ] + [1 - ~1( h a + h ~ ) ] ( l - 5 j 0 5 ~ ) -y ~ 7a~ - ~odd _ U : a. . . . " ) - , x t'~.a e x p i/~. R.~ + ~l ~..,[(t0 id
(2) ttere a, 13, 7 denote p-orbitals, and d are d-orbitals. $.~ - 1(0) if orbitals a and ;3 are (are not) on the same site. na and ha are respectively the hole occupancies of the a orbital and of its site. In Eq. (2) we have neglected the second order hopping terms to "hole-occupied" O sites (i.e. sites singly occupied by electrons). T h e y are considerably smaller than hopping via hole-occupied Cu sites, ttowever, the first order hopping terms to hole-vacant O sites should be of the same order. Both contribute /~dependent terms ~ 0.3 eV to Eq. (2). To diagonalize this matrix we would need the values of the parameters. However, we expect [6] to get six bands within a range of ~ 2 eV, separated by hybridization gaps, and of widths ~ 0.1 eV each. (Slight orthorhonlbic distortion m a y help to prevent gap-closing degeneracies.) The highest band is our CB. Particles whose energy is larger than the hybridization gaps will not "see" these gaps, and will show a s p e c t r u m characteristic of the width of all six bands. A b o n t 1 eV above the CB lies the e m p t y d,,_ u, band. Its width is ~ 1 eV because the energy denominators for hopping to hole-vacant O sites are small. Will these Cu orbitals be antiferromagnetically (AF) ordered? Neutron diffraction [4] shows that AF ordering in Y B C O occurs only for 6 ~ 1. We shall see below that for 6 < 1, H~ + Hi can stabilize an inter-plane singlet pair (IPSP), and thus suppress A F order. An I P S P anlfihilation operator 11 ajt -11~ is defined by P ~ a = ~1t (a1,:,a-1~ iT ~ - ai~ J, satisfying (p~t~)~ = aiTtaalailajl-l~a-la~T" If inter-plane hopping is neglected, the plane electrons can be described in terms of IPSPs only. Eq. (2) gives the band structure for uncorrelated electrons in IPSPs. The band has two degenerate tmcorrelated states (of the two planes) in unit cell. This fits T E P results [2].
1316
J. Ashkenazi and C.G. Kuper / Band structure electron correlations
We treat H~ similarly. The O1 vacancies break the chains into clusters of O4-Cul-O4 (OCO) complexes linked by non-vacant O1 sites. Using an estimate [6] we propose the following picture: in the ground state of a one-OCO cluster the Cu(d) and O(p) shells are full; for a two-OCO cluster a combination of O1-O4(p) hole orbitals gains enough energy by hopping to the Cul sites to introduce a hole in the ground state (which must be symmetric or antisymmetric about the OCO center); for larger clusters, Cul(du,_~ ) hole orbitals gain enough energy by hopping to O1-O4 sites to introduce n - 2 holes (at the cluster edges a hopping channel is missing), and each edge contributes a p-hole. A given OCO cannot hold two holes; however lowlying intra-OCO excitations should exist, consistlug of a d-electron - p-hole, or vice versa. These excitations are electronic breathing modes (EBM) of quadrupolar charge transfer within an OCO. A photon can excite an EBM only if it has a component of polarization along the c-axis. A ~ 0.4 eV transition - reported in polycrystalline samples [7], but not seen in single crystals with polarization in the a-b plane - may possibly be an EBM. The CB is fill when ~ = 1. Each additional O1 atom introduces roughly one hole into the CB for 1 > 5 > 0.5, no holes for 0.5 > ~ > 0.2 and two holes for 0.2 > 5 > 0, making it half full for 5 = o. For 6 > 0.5, the large disorder of the small OCO clusters drives an Anderson transition in the CB, consistent with experiment [8]. This picture explains [9] the behaviour of the Hall coefficient [10] and the dependence of T~ on 6. The interaction Hi between the 0 4 and the plane orbitals leads to an effective attraction between CB electrons in two planes via an EBM. Fig. 1 illustrates a typical diagram contributing to this interaction. We suggest that it makes the IPSP more stable than the AF state when 5 is somewhat < 1. Below T¢ the IPSPs become correlated (i.e. Cooper pairs); evaluation of T~ for this SC mechanism is in progress. The SC coherence length (CL) in the c-direction will be ~ 2/3 of a lattice vector. Perpendicular to c, defects will weaken the phase coherence between the planes and lead to an anomalously small CL. Both are in agreement with experiment [11]. Such a short CL and narrow CB will give critical behaviour at T~ [1]. The inter-plane hopping, neglected so far, will break some IPSPs. The unpaired electrons thus generated will contribute linearly to the specific heat even below T~, in agreement with experiment. We have neglected lattice vibrations. But the EBM, as well as the EBM-CB interaction, may be coupled to 0 4 atomic displacements. This would give an unconventional isotope effect specific to 0 4 atoms. The mechanism proposed here may also apply to other high temperature SCs. In the La2CuO4 - based SCs the role of the OCO com-
plexes is played by inter-plane La-O complexes; we predict slight inter-plane dimerization (at least in the SC phase and on the short range). We acknowledge the contributions of B. Fisher, J. Genossar and S. Barad to the band scheme discussed here. We thank A. Ron, E. Ehrenfreund, A. Peres, M. l~evzen, G. Deutscher, A. Voronel, M. Weger, M.L. Cohen, A.J. Freeman and W.E. Pickett for stimulating discussions. This work was partly supported by BSF grant No. 086-00410. REFERENCES [1] A. Voronel ,t ,,I., these Proceedings; glassy behaviour {G. Deutscher and K.A. Miiller, Phys. Rev. Left. 59 (1987) 1745} slows processes determining the "true" specific heat change at T~. [2] B. Fisher ,t ,1., J. Superconductivity, in press; these Proceedings. [3] H. Oyanagi ~t ol., Jpn. J. Appl. Phys. 26 (1987) L1561; Z. Inoue ~t ol., ibid 26 (1987) L1365; N. Niicker , t ,,1., submitted to Phys. Rev. B. [4] J.M. Tranquada ,t al., preprint. [5] We use the notation of M.A. Beno , l ,I., Appl. Phys. Lett. 51 (1987)57. [6] S. Massidda ~ ,/., Phys. Lett. A 122 (1987) 198. [7] K. Kamar~s ~t ,d., Phys. Rev. Lett 59 (1987) 919. [8] Yu Mei ,t ,,I., Z. Phys. B 69 (1987) 11. [9] S. Barad ,1 o1., in preparation. [10] Z.Z. W a n g , t ,,l., Phys. Rev. n 36 (1987) 7222. [ l l ] A . Umezawa rf al., preprint.
02(p) ," Cu2(d)',
Cu1(d)
,
02(p)
04(p)
I
,,
02(p)
FIGURE 1 A typical effective interaction diagram between electrons of two planes via an electronic breathing mode of an interplanar OCO complex.
BAND S T R U C T U R E , E L E C T R O N C O R R E L A T I O N S , AND AN I N T E R L A Y E R P A I R I N G M E C H A N I S M F O R HIGH T E M P E R A T U R E S U P E R C O N D U C T I V I T Y J. A S t t K E N A Z I and C.G. K U P E R Physics Department, Technion, Haifa 32000, Israel
We show that the condnction electrons in YBa~CusO~ ~ arise mainly from O2-O3 p-orbitals, and lie in a narrow band (of width ~ 0.1 eV). The Cooper pairs in the superconducting state come from two distinct CuO~ planes, and the "glue" which binds them is an electronic breathing mode of charge transfer within an interplanar O 4 - C u l - O 4 complex. Our picture is consistent with experiment. Very slow specific heat measurements [1] in YBa~CusOr_6 ( Y B C O ) show a ~ divergence at the superconducting (SC) transition temperature T~. Thermoelectric power ( T E P ) data [2] indicate a narrow conduction band (CB), of width ~ 0.1 eV, full for 5 = 1 and half full for 5 = 0. Various measurements [3-4] indicate O-hole carriers at Er, and a fall of the valence of the C u l atoms [5] from +2 to +1, as 6 increases from 0 to 1. We model Y B C O by two CuOz planes and the CuOa chains between them. In the planes (but not in the broken chains) translation is a good symla merry. We define annihilation operators thus: ai~ specifies an orbital a, belonging to a Cu2(3d) or t, specifies an O2-O3(2p) shell in the plane 1 = ±1. ei~ orbital c~ in a Cul (3d) or O1-O4(2p) shell; l = ±1 for the 0 4 orbital adjacent to the +1 plane; otherwise l = o. e and i specify respectively the spin and the unit cell. The nlodel Hanliltonian H = H, + H~ + H;: Ha
=
E ijla[~o"
o¢1 l,x? 113 1 x-~ ,:,0 lal lC~al~lal~ ~ t i - J aia aJ a + 2 2_~ U i - j ala aia j~r' ja'], aa Vila
ijlrnaOo" Hi
Cio" Cio'Cja~ jo" ],
a'
Z ..~afl tat aiaCjo-,Cjo. a ll~l ID V~ij aia ijlc~acr'
O)
includes band structure and correlation effects. tlere i - j stands for/~i -/~i. Replacing tT~ by [7]i - tT-~ + 5ij6.~ E , ( 2 - 5a,)U:" describes hole states. Starting from localized states we diagonalize H, to second order in ti_~. "-"~ Estimating the parameters i n / / f r o m band-structure results [6], we find that the highest electron band (nlainly Cu2(d.,_ u,)) is empty. Below it lies the partly filled CB, comprising mainly O2-O3(p) orbitals. Its dispersion is found by diagonalizing the 6 x 6 matrix:
0921-4534/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
+ Z ( 1 - 5joS,..vn.v)U; ='r ] + [1 - ~1( h a + h ~ ) ] ( l - 5 j 0 5 ~ ) -y ~ 7a~ - ~odd _ U : a. . . . " ) - , x t'~.a e x p i/~. R.~ + ~l ~..,[(t0 id
(2) ttere a, 13, 7 denote p-orbitals, and d are d-orbitals. $.~ - 1(0) if orbitals a and ;3 are (are not) on the same site. na and ha are respectively the hole occupancies of the a orbital and of its site. In Eq. (2) we have neglected the second order hopping terms to "hole-occupied" O sites (i.e. sites singly occupied by electrons). T h e y are considerably smaller than hopping via hole-occupied Cu sites, ttowever, the first order hopping terms to hole-vacant O sites should be of the same order. Both contribute /~dependent terms ~ 0.3 eV to Eq. (2). To diagonalize this matrix we would need the values of the parameters. However, we expect [6] to get six bands within a range of ~ 2 eV, separated by hybridization gaps, and of widths ~ 0.1 eV each. (Slight orthorhonlbic distortion m a y help to prevent gap-closing degeneracies.) The highest band is our CB. Particles whose energy is larger than the hybridization gaps will not "see" these gaps, and will show a s p e c t r u m characteristic of the width of all six bands. A b o n t 1 eV above the CB lies the e m p t y d,,_ u, band. Its width is ~ 1 eV because the energy denominators for hopping to hole-vacant O sites are small. Will these Cu orbitals be antiferromagnetically (AF) ordered? Neutron diffraction [4] shows that AF ordering in Y B C O occurs only for 6 ~ 1. We shall see below that for 6 < 1, H~ + Hi can stabilize an inter-plane singlet pair (IPSP), and thus suppress A F order. An I P S P anlfihilation operator 11 ajt -11~ is defined by P ~ a = ~1t (a1,:,a-1~ iT ~ - ai~ J, satisfying (p~t~)~ = aiTtaalailajl-l~a-la~T" If inter-plane hopping is neglected, the plane electrons can be described in terms of IPSPs only. Eq. (2) gives the band structure for uncorrelated electrons in IPSPs. The band has two degenerate tmcorrelated states (of the two planes) in unit cell. This fits T E P results [2].
1316
J. Ashkenazi and C.G. Kuper / Band structure electron correlations
We treat H~ similarly. The O1 vacancies break the chains into clusters of O4-Cul-O4 (OCO) complexes linked by non-vacant O1 sites. Using an estimate [6] we propose the following picture: in the ground state of a one-OCO cluster the Cu(d) and O(p) shells are full; for a two-OCO cluster a combination of O1-O4(p) hole orbitals gains enough energy by hopping to the Cul sites to introduce a hole in the ground state (which must be symmetric or antisymmetric about the OCO center); for larger clusters, Cul(du,_~ ) hole orbitals gain enough energy by hopping to O1-O4 sites to introduce n - 2 holes (at the cluster edges a hopping channel is missing), and each edge contributes a p-hole. A given OCO cannot hold two holes; however lowlying intra-OCO excitations should exist, consistlug of a d-electron - p-hole, or vice versa. These excitations are electronic breathing modes (EBM) of quadrupolar charge transfer within an OCO. A photon can excite an EBM only if it has a component of polarization along the c-axis. A ~ 0.4 eV transition - reported in polycrystalline samples [7], but not seen in single crystals with polarization in the a-b plane - may possibly be an EBM. The CB is fill when ~ = 1. Each additional O1 atom introduces roughly one hole into the CB for 1 > 5 > 0.5, no holes for 0.5 > ~ > 0.2 and two holes for 0.2 > 5 > 0, making it half full for 5 = o. For 6 > 0.5, the large disorder of the small OCO clusters drives an Anderson transition in the CB, consistent with experiment [8]. This picture explains [9] the behaviour of the Hall coefficient [10] and the dependence of T~ on 6. The interaction Hi between the 0 4 and the plane orbitals leads to an effective attraction between CB electrons in two planes via an EBM. Fig. 1 illustrates a typical diagram contributing to this interaction. We suggest that it makes the IPSP more stable than the AF state when 5 is somewhat < 1. Below T¢ the IPSPs become correlated (i.e. Cooper pairs); evaluation of T~ for this SC mechanism is in progress. The SC coherence length (CL) in the c-direction will be ~ 2/3 of a lattice vector. Perpendicular to c, defects will weaken the phase coherence between the planes and lead to an anomalously small CL. Both are in agreement with experiment [11]. Such a short CL and narrow CB will give critical behaviour at T~ [1]. The inter-plane hopping, neglected so far, will break some IPSPs. The unpaired electrons thus generated will contribute linearly to the specific heat even below T~, in agreement with experiment. We have neglected lattice vibrations. But the EBM, as well as the EBM-CB interaction, may be coupled to 0 4 atomic displacements. This would give an unconventional isotope effect specific to 0 4 atoms. The mechanism proposed here may also apply to other high temperature SCs. In the La2CuO4 - based SCs the role of the OCO com-
plexes is played by inter-plane La-O complexes; we predict slight inter-plane dimerization (at least in the SC phase and on the short range). We acknowledge the contributions of B. Fisher, J. Genossar and S. Barad to the band scheme discussed here. We thank A. Ron, E. Ehrenfreund, A. Peres, M. l~evzen, G. Deutscher, A. Voronel, M. Weger, M.L. Cohen, A.J. Freeman and W.E. Pickett for stimulating discussions. This work was partly supported by BSF grant No. 086-00410. REFERENCES [1] A. Voronel ,t ,,I., these Proceedings; glassy behaviour {G. Deutscher and K.A. Miiller, Phys. Rev. Left. 59 (1987) 1745} slows processes determining the "true" specific heat change at T~. [2] B. Fisher ,t ,1., J. Superconductivity, in press; these Proceedings. [3] H. Oyanagi ~t ol., Jpn. J. Appl. Phys. 26 (1987) L1561; Z. Inoue ~t ol., ibid 26 (1987) L1365; N. Niicker , t ,,1., submitted to Phys. Rev. B. [4] J.M. Tranquada ,t al., preprint. [5] We use the notation of M.A. Beno , l ,I., Appl. Phys. Lett. 51 (1987)57. [6] S. Massidda ~ ,/., Phys. Lett. A 122 (1987) 198. [7] K. Kamar~s ~t ,d., Phys. Rev. Lett 59 (1987) 919. [8] Yu Mei ,t ,,I., Z. Phys. B 69 (1987) 11. [9] S. Barad ,1 o1., in preparation. [10] Z.Z. W a n g , t ,,l., Phys. Rev. n 36 (1987) 7222. [ l l ] A . Umezawa rf al., preprint.
02(p) ," Cu2(d)',
Cu1(d)
,
02(p)
04(p)
I
,,
02(p)
FIGURE 1 A typical effective interaction diagram between electrons of two planes via an electronic breathing mode of an interplanar OCO complex.