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Two-band hubbard model for copper oxide superconductors
Physica B 165&166 (1990) 1011-1012
North-Holland
TWO-BAND HUBBARD MODEL FOR COPPER OXIDE SUPERCONDUCTORS Kazuhiko KUROKI and Hideo AOKI Department of Physics, University of Tokyo, Hongo, Tokyo 113, Japan Results for the superconductivity specific to strongly correlated two-band systems in a two-band Hubbard model is presented to elucidate effects of doping and band width from both intuitive considerations and quantum Monte Carlo results.
1. Introduction
F
In the present paper we elaborate a new model for the superconductivity in copper oxide, which we have proposed(l) based on a two-band Hubbard Hamiltonian,
~tt #
L - JU
U+F H=:L
:Lt~{ci;cj,,+h.c.}+:L{Unilnn+Fnflnf!}
~=",/3 (i,j)"
+:L{Wni"nL + Vni"nf,,}
(1)
We consider two molecular orbitals for each Cu-O unit, in which doped carriers go into the {3 orbital on top of the (half-filled) ex band. Here (i, j) represents the nearestneighbour Cu-O units, U(F) the Hubbard repulsion energy in the ex({3) orbital and V(W) is the inter-orbital repulsion for parallel (anti-parallel) spins. Extensions to include transfers that hybridize different orbitals can be done. The introduction of two orbitals for cuprates is motivated by our earlier proposal(2) that two molecular orbitals of different symmetries are relevant for the electronic structure. Band structure and cluster ca.lculations(3) also suggest the existence of two orbitals for holes. 2. Cooper pairing We can show from the energy consideration in Fig. 1 that an effective on-site attraction between the j3 band carriers arises when a criterion, 2V > U·
+ F,
(2)
is fulfilled. Here U· = U - O( t) is an energy (chemical potential) required to put an extra carrier onto the ex band, where O(t) correction comes from kinetic energies. The effect of level offset of ex and (3 orbitals can also be incorporated. We have employed the quantum Monte Carlo method for the ground state proposed by Sorella et al( 4) for onedimensional(lD) system to confirm the presence of enhanced two-point correlation of the spin-singlet, on-site Cooper pair. For ID systems, long tail in the pair-pair correlation hallmarks the presence of superconductivity(5). Spin-spin correlation in the ex band is shown to be suppressed in accord with the intuitive picture in Fig.I. The effective attraction between carriers is also re-
(a)
fJ
v+- --------- vr-
a ~
---------+ 2V
(b)
Fig. 1. When 2V > U + F, the configuration with two carriers in the same (3 orbital(a) is eneq~etically more favourable than when they are separated(b).
fleeted in an oscillation in the density-density correlation function. The orbital occupancy correlation, «(nr - (na»(n~ - (n/3»), exhibits no appreciable structure. Important differences from the 'd-p' model reside in the fact that we pick up the most relevant orbitals and we also consider more than one orbital for Cu(3d), for which the Cooper pairing is realized from quite a subtle balance in intra- and inter- molecular-orbital interaction energies. 3. Doping effect The pair-pair correlation is shown to be suppressed for larger doping level of the j3 band. When we dope the half-filled ex band, the correlation does not change appreciably (Fig. 2). Thus, even when we introduce the hybridization between different bands, which will shift the ex band filling, the Cooper pairing can remain. In real materials, increase in doping decreases the Cu-O distance(6). This will change the molecular orbitals, so that the doping dependence of superconductivity should include this effect. 4. Effect of band width Figure 3 shows the effect of decreasing Ittll for the two cases of 2V > U + F and 2V < U + F (with a strong Hund's coupling, V = 0, W > 0, here). When the criterion(Eq.2) is satisfied, the pairing tends to be recovered with narrowing j3 band even when the pairing is suppressed due to finite concentration of carriers, whereas the opposite tendency is observed when the criterion is dissatisfied.
0921.4526/90/$03.50 @ 1990 - Elsevier Science Publishers B.V. (North-Holland)
1012
K. Kuroki, H. Aoki
z o
z o
-
~
....J
~0.1
..... FREE ELECTRON
U=V=W=2.0, F=O.O (,8 band) 0 22 carriers in the Q band x 26 carriers in the Q band
0:::
~
(a)
0.1
2V>U+F FREE ELECTRON
....J
U=V=W=2.0, F=0.5 (,1 band) !::" ........ t,,=-1.0, t p=-1.0 0······ .. t,,=-1.0, tp =-0.75
UJ
0::: 0:::
o U
<.:>
o
z
u
a:: 0.05
<.:>
~
Z
a::
UJ
~
Vl
t-
UJ
Z
t-
o
Vl
Z
o
5
° °
0. '--.L...--.L..--'--....I....--=--....r.........:""'---<::::..-'--~---'---' 5 DISTANCE
DISTANCE
z
o
2V
~
-
....J
..... FREE ELECTRON
U=W=2.0, V=F=O.O (,8 band) !::" t,,=-1.0, t8=-1.0 t,,=-1.0, t3=-0.75
UJ
Fig. 2. Effect of the doping the a band (more-thanhalf- filled with 26 carriers/22 sites( x)).
10
22°. 05
o
o
u
<.:>
z
5. Molecular orbitals For real copper oxides we check whether above pairing criterion can be realizable. An unambiguous starting point before incorporating many-body renormalization would be the simple Hartree-Fock(HF) method. The HF result for a CU06 octahedral cluster with one hole shows that the lowest unoccupied level (by electrons) = la} = anti-bonding 1jJ(Cu3d",._y,) +1jJ(02pa)'s of bIg symmetry, the highest occupied level = 1,8) = 1jJ(02pa)'s + 1jJ(apical 02p.)'s + 1jJ( Cu3d 3z L T ' ) of alg symmetry. For these molecular orbitals, the direct Coulomb integrals are U" = 14.3eV, Uf3 = 7.7eV, U"f3 = 8.2eV. Note that (i)2V - (U + F) is already a few eV("" It!) even for the bare HF orbitals, (ii)V exceeds F, which is a necessary condition for 2V > U' + F when U· > V. Also, the reduction of U into U· is considerable for large U(e.g., U' = 8.31tl for U = lOltl in 10). Many-body calculation using the transition states will be discussed in a forthcoming publication. We would like to thank Mr. M. Eto for providing us with the numerical HF molecular orbitals. Numerical calculations were done at the Computer Centre, University of Tokyo.
a:: ~ ::: 0. a~..L..-..L---L----l.-+~S::fI-=;I~~-::-....L..J 5 zVl DISTANCE o
°
Fig. 3. Effect of decreasing Itlll from 1.0(!::") to 0.75(0) for the case of (a)2V > U+F(U = V = W = 2, F = 0.5) and (b)2V < U + F(U = W = 2, V = F = 0).
References (1) K. Kuroki and H. Aoki, Solid State Commun. 73 (1990) 563. (2) H. Aoki and H. Kamimura, Solid State Commun. 63 (1987) 665. (3) J.F. Annett, RM. Martin, A.K. McMahan and S. Satpathy, Phys. Rev. B 40 (1989) 2620. (4) S. Sorella, S. Baroni, R. Car and M. Parrinello, Europhys. Lett. 8 (1989) 663. (5) V.N. Popov, Functional Integrals in Quantum Field Theory and Statistical Physics (D. Reidel, 1983). (6) J.B. Torrance et aI, Phys. Rev. B 40 (1989) 8873.
North-Holland
TWO-BAND HUBBARD MODEL FOR COPPER OXIDE SUPERCONDUCTORS Kazuhiko KUROKI and Hideo AOKI Department of Physics, University of Tokyo, Hongo, Tokyo 113, Japan Results for the superconductivity specific to strongly correlated two-band systems in a two-band Hubbard model is presented to elucidate effects of doping and band width from both intuitive considerations and quantum Monte Carlo results.
1. Introduction
F
In the present paper we elaborate a new model for the superconductivity in copper oxide, which we have proposed(l) based on a two-band Hubbard Hamiltonian,
~tt #
L - JU
U+F H=:L
:Lt~{ci;cj,,+h.c.}+:L{Unilnn+Fnflnf!}
~=",/3 (i,j)"
+:L{Wni"nL + Vni"nf,,}
(1)
We consider two molecular orbitals for each Cu-O unit, in which doped carriers go into the {3 orbital on top of the (half-filled) ex band. Here (i, j) represents the nearestneighbour Cu-O units, U(F) the Hubbard repulsion energy in the ex({3) orbital and V(W) is the inter-orbital repulsion for parallel (anti-parallel) spins. Extensions to include transfers that hybridize different orbitals can be done. The introduction of two orbitals for cuprates is motivated by our earlier proposal(2) that two molecular orbitals of different symmetries are relevant for the electronic structure. Band structure and cluster ca.lculations(3) also suggest the existence of two orbitals for holes. 2. Cooper pairing We can show from the energy consideration in Fig. 1 that an effective on-site attraction between the j3 band carriers arises when a criterion, 2V > U·
+ F,
(2)
is fulfilled. Here U· = U - O( t) is an energy (chemical potential) required to put an extra carrier onto the ex band, where O(t) correction comes from kinetic energies. The effect of level offset of ex and (3 orbitals can also be incorporated. We have employed the quantum Monte Carlo method for the ground state proposed by Sorella et al( 4) for onedimensional(lD) system to confirm the presence of enhanced two-point correlation of the spin-singlet, on-site Cooper pair. For ID systems, long tail in the pair-pair correlation hallmarks the presence of superconductivity(5). Spin-spin correlation in the ex band is shown to be suppressed in accord with the intuitive picture in Fig.I. The effective attraction between carriers is also re-
(a)
fJ
v+- --------- vr-
a ~
---------+ 2V
(b)
Fig. 1. When 2V > U + F, the configuration with two carriers in the same (3 orbital(a) is eneq~etically more favourable than when they are separated(b).
fleeted in an oscillation in the density-density correlation function. The orbital occupancy correlation, «(nr - (na»(n~ - (n/3»), exhibits no appreciable structure. Important differences from the 'd-p' model reside in the fact that we pick up the most relevant orbitals and we also consider more than one orbital for Cu(3d), for which the Cooper pairing is realized from quite a subtle balance in intra- and inter- molecular-orbital interaction energies. 3. Doping effect The pair-pair correlation is shown to be suppressed for larger doping level of the j3 band. When we dope the half-filled ex band, the correlation does not change appreciably (Fig. 2). Thus, even when we introduce the hybridization between different bands, which will shift the ex band filling, the Cooper pairing can remain. In real materials, increase in doping decreases the Cu-O distance(6). This will change the molecular orbitals, so that the doping dependence of superconductivity should include this effect. 4. Effect of band width Figure 3 shows the effect of decreasing Ittll for the two cases of 2V > U + F and 2V < U + F (with a strong Hund's coupling, V = 0, W > 0, here). When the criterion(Eq.2) is satisfied, the pairing tends to be recovered with narrowing j3 band even when the pairing is suppressed due to finite concentration of carriers, whereas the opposite tendency is observed when the criterion is dissatisfied.
0921.4526/90/$03.50 @ 1990 - Elsevier Science Publishers B.V. (North-Holland)
1012
K. Kuroki, H. Aoki
z o
z o
-
~
....J
~0.1
..... FREE ELECTRON
U=V=W=2.0, F=O.O (,8 band) 0 22 carriers in the Q band x 26 carriers in the Q band
0:::
~
(a)
0.1
2V>U+F FREE ELECTRON
....J
U=V=W=2.0, F=0.5 (,1 band) !::" ........ t,,=-1.0, t p=-1.0 0······ .. t,,=-1.0, tp =-0.75
UJ
0::: 0:::
o U
<.:>
o
z
u
a:: 0.05
<.:>
~
Z
a::
UJ
~
Vl
t-
UJ
Z
t-
o
Vl
Z
o
5
° °
0. '--.L...--.L..--'--....I....--=--....r.........:""'---<::::..-'--~---'---' 5 DISTANCE
DISTANCE
z
o
2V
~
-
....J
..... FREE ELECTRON
U=W=2.0, V=F=O.O (,8 band) !::" t,,=-1.0, t8=-1.0 t,,=-1.0, t3=-0.75
UJ
Fig. 2. Effect of the doping the a band (more-thanhalf- filled with 26 carriers/22 sites( x)).
10
22°. 05
o
o
u
<.:>
z
5. Molecular orbitals For real copper oxides we check whether above pairing criterion can be realizable. An unambiguous starting point before incorporating many-body renormalization would be the simple Hartree-Fock(HF) method. The HF result for a CU06 octahedral cluster with one hole shows that the lowest unoccupied level (by electrons) = la} = anti-bonding 1jJ(Cu3d",._y,) +1jJ(02pa)'s of bIg symmetry, the highest occupied level = 1,8) = 1jJ(02pa)'s + 1jJ(apical 02p.)'s + 1jJ( Cu3d 3z L T ' ) of alg symmetry. For these molecular orbitals, the direct Coulomb integrals are U" = 14.3eV, Uf3 = 7.7eV, U"f3 = 8.2eV. Note that (i)2V - (U + F) is already a few eV("" It!) even for the bare HF orbitals, (ii)V exceeds F, which is a necessary condition for 2V > U' + F when U· > V. Also, the reduction of U into U· is considerable for large U(e.g., U' = 8.31tl for U = lOltl in 10). Many-body calculation using the transition states will be discussed in a forthcoming publication. We would like to thank Mr. M. Eto for providing us with the numerical HF molecular orbitals. Numerical calculations were done at the Computer Centre, University of Tokyo.
a:: ~ ::: 0. a~..L..-..L---L----l.-+~S::fI-=;I~~-::-....L..J 5 zVl DISTANCE o
°
Fig. 3. Effect of decreasing Itlll from 1.0(!::") to 0.75(0) for the case of (a)2V > U+F(U = V = W = 2, F = 0.5) and (b)2V < U + F(U = W = 2, V = F = 0).
References (1) K. Kuroki and H. Aoki, Solid State Commun. 73 (1990) 563. (2) H. Aoki and H. Kamimura, Solid State Commun. 63 (1987) 665. (3) J.F. Annett, RM. Martin, A.K. McMahan and S. Satpathy, Phys. Rev. B 40 (1989) 2620. (4) S. Sorella, S. Baroni, R. Car and M. Parrinello, Europhys. Lett. 8 (1989) 663. (5) V.N. Popov, Functional Integrals in Quantum Field Theory and Statistical Physics (D. Reidel, 1983). (6) J.B. Torrance et aI, Phys. Rev. B 40 (1989) 8873.