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Metal-insulator transition in three-band Hubbard model
Journal of Magnetism and Magnetic Materials 104-107 (1992) 579-580 North-Holland
/Ibm
Metal-insulator transition in three-band Hubbard model J. D u t k a
a,
M. Kamifiski ~ and A . M . Oleg a,b
Institute of Physics, Jagellonian Unit,ersity, Reymonta 4, PL-30059 Cracow, Poland b Max-Planck-lnstitut FKF, Postfach 800665, D-7000 Stuttgart 80, Germany We describe a transition from a metal to an antiferromagnetic (AF) insulator in the three-band Hubbard Hamiltonian for the undoped CuO 2 planes of high-temperature superconductors, including local hole correlations. If the realistic parameters are used, one finds the AF ground states with magnetic moment of = 0.47/zB and -- 0.56/z B for La2CuO 4 and YBa2Cu206, respectively. Correlations and the interoxygen hopping reduce drastically the region of the AF long-range order which disappears for the doping of 0.06 hole per unit celt. Since the discovery of Cu-based high-temperature superconductors (HTS), their common structural elements, CuO 2 planes, have been extensively investigated. Experimental phase diagrams of high-T~ superconductors [1,2] show the existence of the antiferromagnetic (AF) ground state in the undoped systems which vanishes with doping of holes into CuO 2 planes. Any model trying to describe magnetic properties of these compounds must take into account both the fact of strong localization of holes and considerable band formation effect [2]. The aim of this paper is to extend our previous study of the ground state of C u O 2 planes [3], and present new results for the metal-insulator transition (MIT) and the ground state of undoped and doped C u O 2 planes in HTS. We use a three-band Hubbard Hamiltonian of the form [4]
Y=t°En m +CEnp, +v0 E (-i) me,-
io
.....
(ira)o
×(d+m,,ai~+h.c.) +tpp Y'.
(- 1)~'Jai,,aj~ +
(ij)o" + UdEndm?ndm,, m
+Udp
+ UpEt'lpiTrtpi ~ i
E Fldmcrl'lpio"" (im)o'o"
(I)
The undoped state contains one hole per unit cell, + n = 1. Operators d+,~ and ai~, create a hole of a spin oin an atomic Cu(3d~2 y2) and O(2p~) or O(2py) or+ + bital, respectively, ndmo_= dmo.dmo, and nt, i,,= aio.aio. stand for the hole density of spin ~r at a copper site m and at an oxygen site i, respectively. Parameters t ° and tp0 are the atomic Cu(3d) and O(2p) level energies, respectively, and define the charge transfer energy zl = tv° - t °. V0 and tpp stand for hybridization energies resulting from nearest neighhbor C u - O and O - O hopping, respectively, while rrm~ and Yzi are the respective phase factors. Ud and Up are the on-site Coulomb elements at a Cu and O site, respectively. The intersite repulsion between holes placed at the nearest neigbor Cu and O sites is described by Udp. First we perform the H a r t r e e - F o c k (HF) approximation to obtain the ground state with A F long-range order (LRO). We assume that magnetic moment in
sublattice is centered on Cu(3d) orbitals: M = (nd ~ ) (n d ~ ). Strong local correlations between holes resulting from Coulomb repulsion are included by means of a local ansatz (LA) method [5]. Using L A method we can write the correlated groundstate as 1 0 o ) = exp(--En~7~On)I4~HF), where I&HF) is the ground state. The variational parameters "0n are determined by the minimization of the energy of the ground state 100). The operators O n describe local Coulomb correlations within the CuO 2 plane as shown in detail in the previous papers [3,5]. The calculations presented below were performed for the parameters obtained by using local density approximation ( L D A ) [6]: A = Ep0 _ Ea0 = 3.6 eV, V0 = 1.3 eV, tpp = 0.65 eV, Ud = 10.5 eV, Up = 4 eV, Udp = 1.2 eV. They are in a good agreement with the values calculated by many other authors [7], so they may be considered as realistic for HTS. In fig. 1 we present the m e t a l - A F - i n s u l a t o r phase diagram on the ( a , Ud) plane for undoped system (6 = 0). The transition occurs both in the H F approximation and in the LA, but the regions of stability of the A F L R O are strongly reduced in the correlated ground states. Altogether, we note that the M I T in the CuO 2 planes is of chargetransfer type according to the classification of transition metal oxides introduced by Zaanen, Sawatzky and Allen [8]. This is in agreement with the earlier results reported in a similar model, both within the Gutzwiller [9] and variational Monte Carlo [10] method. As already reported [3], the magnetic moment in the Cu sublattice is M = 0.70~B. The quantum fluctuations reduce this mean-field value to M = 0.47~B for La2CuO 4 (39.3% reduction in a two-dimensional Heisenberg model) and to M -- 0.56/x B for YBazCu306 (27% reduction in the system of two layers). These values are not far from the experimental results, being Mex p = (0.55 + 0 . 0 5 ) ~ B and (0.64-0.66)iZB, for La2CuOa_y [11] and for Y B a 2 f u 3 0 6 [1], respectively. In fig. 2 we show the dependence of the A F energy gap GAF on zl in the undoped state. The gap of GAy = 3.1 eV found in the presence of interoxygen hopping is considerably reduced from its too = 0 value, but is still larger than the 2 eV observed experimentally [12]. Using the L D A parameters [6], the A F order in CuO 2 planes disappears for relatively small doping of
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
580
J. Dutka et al. / Metal-insulator transition in three-band model
120
-
I
AF
i
"
AF
I
j/.- /
i
3.0 •
i
B0 LA
<2]--2 0 ~- . z
d
4.0 i tl]'
1.0
NM
M e t.al 1
0 0 -'=~= 0.0
.
.
.
.
. E0
10
. ,t 0
30
0 90
0.95
1.00
1 0f~
1 10
A Fig. 1. Metal-insulator phase diagram in the (Ua, A) plane, obtained in the Hartree-Fock approximation (HF) (dashed line) and in the LA method (solid line) for the LDA parameters [6]. The values of Ud and zl are given in eV. 6 - 0 . 0 6 hole per unit cell. This agrees quite well with the recent M o n t e Carlo study which gave a fast decrease of A F o r d e r for the doping ~ < 0.005 [13], but is larger than the experimental values [2]. O n e reason of this discrepancy is that we do not consider here the spin glass phase with short-range A F order. A n o t h e r follows from fig. 3, where we show the weak d e p e n dence of the A F transition point on the value of A. Therefore, a small inaccuracy in A causes a large 5.0 /"
/
4.0
/
b /"
/
~3.o
/
i /
/
/ /
/
/
/// /
© E20
/
/
10
[
i 00
~,
05
"~T
'
~--T--T~
15
~
7
I' ///
2.5
, , i ,~ , ,
:35
,
~
r
~
i
4.5
Fig. 2. Energy gap GAp (in eV) as a function of A = % -- Ea' as obtained using LA method for the LDA parameters [6] and for (a) tpp = 0.65 eV (solid line), and (b) tpp = 0 (dashed line).
Fig. 3. AF phase diagram in the (zl, n) plane obtained with the LA method. The parameters and the meaning of the lines as in fig. 2. change of the critical concentration 8 c for the disapp e a r a n c e of A F L R O , as well as in the value of GAF. The situation is not symmetrical for the hole (n = 1 + 6) and electron (n = 1 - 8) doping and the interoxygen hopping reduces the region of stability of the A F LRO. This d e m o n s t r a t e s the i m p o r t a n c e of this hopping process for quantitative studies. Summarizing, the p r e s e n t e d results d e m o n s t r a t c the absence of B r i n k m a n - R i c e transition [9] in the threeb a n d H u b b a r d model and give instead a MIT to an A F insulator. The Hamiltonian, eq. (1), gives a qualitatively correct description of the A F phase of HTS. [1] J.M. Tranquada et al., Phys. Rev. B 38 (1988) 2477. [2] K.C. Haas, Solid State Physics, vol. 42, eds. H, Ehrenreich and D. Turnbull (Academic, Orlando, 1989) p. 213. [3] J. Dutka and A.M. Oleg, Phys. Rev. B 43 (1991) 5622. [4] C.M. Varma, S. Schmitt-Rink and E. Abrahams, Solid State Commun. 62 (1987) 681. V.J. Emery, Phys. Rev. Lett. 58 (1987) 2794. [5] G. Stollhoff and P, Fulde, Z. Phys. B 26 (1977) 257; 29 (1978) 231. A.M. Oleg, P. Fulde and J. Zaanen, Physica B 148 (1987) 260. [6] M.S. Hybertsen, M. Schliiter and N.E. Christensen, Phys. Rev. B 39 (1989) 9028. [7] A.K. McMahan et al., Phys. Rev. B 42 (1990) 6268. E.B. Stechel and D.R. Jennison, Phys. Rev. B 38 (1988) 4632. F. Mila, Phys. Rev. B 38 (1988) 11358. [8] J. Zaanen, G.A. Sawatszky and J.W. Allen, Phys. Rev. Lett. 55 (1985) 418. [9] C.A. Balseiro et al, Phys. Rev. Lett. 62 (1989) 2624. [lO] S.N. Coopersmith, Phys. Rev. B 41 (1990)8711. [111 RJ. Birgeneau et al., Phys. Rev. B 38 (1988) 6614. [121 J.M. Ginder et al., Phys. Rev. B 37 (1988) 7506. U. Venkateswaran et al., Phys. Rev. B 38 (1988) 711)5. [131 G. Dopf et al., Phys. Rev. B 41 (1990) 9264.
/Ibm
Metal-insulator transition in three-band Hubbard model J. D u t k a
a,
M. Kamifiski ~ and A . M . Oleg a,b
Institute of Physics, Jagellonian Unit,ersity, Reymonta 4, PL-30059 Cracow, Poland b Max-Planck-lnstitut FKF, Postfach 800665, D-7000 Stuttgart 80, Germany We describe a transition from a metal to an antiferromagnetic (AF) insulator in the three-band Hubbard Hamiltonian for the undoped CuO 2 planes of high-temperature superconductors, including local hole correlations. If the realistic parameters are used, one finds the AF ground states with magnetic moment of = 0.47/zB and -- 0.56/z B for La2CuO 4 and YBa2Cu206, respectively. Correlations and the interoxygen hopping reduce drastically the region of the AF long-range order which disappears for the doping of 0.06 hole per unit celt. Since the discovery of Cu-based high-temperature superconductors (HTS), their common structural elements, CuO 2 planes, have been extensively investigated. Experimental phase diagrams of high-T~ superconductors [1,2] show the existence of the antiferromagnetic (AF) ground state in the undoped systems which vanishes with doping of holes into CuO 2 planes. Any model trying to describe magnetic properties of these compounds must take into account both the fact of strong localization of holes and considerable band formation effect [2]. The aim of this paper is to extend our previous study of the ground state of C u O 2 planes [3], and present new results for the metal-insulator transition (MIT) and the ground state of undoped and doped C u O 2 planes in HTS. We use a three-band Hubbard Hamiltonian of the form [4]
Y=t°En m +CEnp, +v0 E (-i) me,-
io
.....
(ira)o
×(d+m,,ai~+h.c.) +tpp Y'.
(- 1)~'Jai,,aj~ +
(ij)o" + UdEndm?ndm,, m
+Udp
+ UpEt'lpiTrtpi ~ i
E Fldmcrl'lpio"" (im)o'o"
(I)
The undoped state contains one hole per unit cell, + n = 1. Operators d+,~ and ai~, create a hole of a spin oin an atomic Cu(3d~2 y2) and O(2p~) or O(2py) or+ + bital, respectively, ndmo_= dmo.dmo, and nt, i,,= aio.aio. stand for the hole density of spin ~r at a copper site m and at an oxygen site i, respectively. Parameters t ° and tp0 are the atomic Cu(3d) and O(2p) level energies, respectively, and define the charge transfer energy zl = tv° - t °. V0 and tpp stand for hybridization energies resulting from nearest neighhbor C u - O and O - O hopping, respectively, while rrm~ and Yzi are the respective phase factors. Ud and Up are the on-site Coulomb elements at a Cu and O site, respectively. The intersite repulsion between holes placed at the nearest neigbor Cu and O sites is described by Udp. First we perform the H a r t r e e - F o c k (HF) approximation to obtain the ground state with A F long-range order (LRO). We assume that magnetic moment in
sublattice is centered on Cu(3d) orbitals: M = (nd ~ ) (n d ~ ). Strong local correlations between holes resulting from Coulomb repulsion are included by means of a local ansatz (LA) method [5]. Using L A method we can write the correlated groundstate as 1 0 o ) = exp(--En~7~On)I4~HF), where I&HF) is the ground state. The variational parameters "0n are determined by the minimization of the energy of the ground state 100). The operators O n describe local Coulomb correlations within the CuO 2 plane as shown in detail in the previous papers [3,5]. The calculations presented below were performed for the parameters obtained by using local density approximation ( L D A ) [6]: A = Ep0 _ Ea0 = 3.6 eV, V0 = 1.3 eV, tpp = 0.65 eV, Ud = 10.5 eV, Up = 4 eV, Udp = 1.2 eV. They are in a good agreement with the values calculated by many other authors [7], so they may be considered as realistic for HTS. In fig. 1 we present the m e t a l - A F - i n s u l a t o r phase diagram on the ( a , Ud) plane for undoped system (6 = 0). The transition occurs both in the H F approximation and in the LA, but the regions of stability of the A F L R O are strongly reduced in the correlated ground states. Altogether, we note that the M I T in the CuO 2 planes is of chargetransfer type according to the classification of transition metal oxides introduced by Zaanen, Sawatzky and Allen [8]. This is in agreement with the earlier results reported in a similar model, both within the Gutzwiller [9] and variational Monte Carlo [10] method. As already reported [3], the magnetic moment in the Cu sublattice is M = 0.70~B. The quantum fluctuations reduce this mean-field value to M = 0.47~B for La2CuO 4 (39.3% reduction in a two-dimensional Heisenberg model) and to M -- 0.56/x B for YBazCu306 (27% reduction in the system of two layers). These values are not far from the experimental results, being Mex p = (0.55 + 0 . 0 5 ) ~ B and (0.64-0.66)iZB, for La2CuOa_y [11] and for Y B a 2 f u 3 0 6 [1], respectively. In fig. 2 we show the dependence of the A F energy gap GAF on zl in the undoped state. The gap of GAy = 3.1 eV found in the presence of interoxygen hopping is considerably reduced from its too = 0 value, but is still larger than the 2 eV observed experimentally [12]. Using the L D A parameters [6], the A F order in CuO 2 planes disappears for relatively small doping of
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
580
J. Dutka et al. / Metal-insulator transition in three-band model
120
-
I
AF
i
"
AF
I
j/.- /
i
3.0 •
i
B0 LA
<2]--2 0 ~- . z
d
4.0 i tl]'
1.0
NM
M e t.al 1
0 0 -'=~= 0.0
.
.
.
.
. E0
10
. ,t 0
30
0 90
0.95
1.00
1 0f~
1 10
A Fig. 1. Metal-insulator phase diagram in the (Ua, A) plane, obtained in the Hartree-Fock approximation (HF) (dashed line) and in the LA method (solid line) for the LDA parameters [6]. The values of Ud and zl are given in eV. 6 - 0 . 0 6 hole per unit cell. This agrees quite well with the recent M o n t e Carlo study which gave a fast decrease of A F o r d e r for the doping ~ < 0.005 [13], but is larger than the experimental values [2]. O n e reason of this discrepancy is that we do not consider here the spin glass phase with short-range A F order. A n o t h e r follows from fig. 3, where we show the weak d e p e n dence of the A F transition point on the value of A. Therefore, a small inaccuracy in A causes a large 5.0 /"
/
4.0
/
b /"
/
~3.o
/
i /
/
/ /
/
/
/// /
© E20
/
/
10
[
i 00
~,
05
"~T
'
~--T--T~
15
~
7
I' ///
2.5
, , i ,~ , ,
:35
,
~
r
~
i
4.5
Fig. 2. Energy gap GAp (in eV) as a function of A = % -- Ea' as obtained using LA method for the LDA parameters [6] and for (a) tpp = 0.65 eV (solid line), and (b) tpp = 0 (dashed line).
Fig. 3. AF phase diagram in the (zl, n) plane obtained with the LA method. The parameters and the meaning of the lines as in fig. 2. change of the critical concentration 8 c for the disapp e a r a n c e of A F L R O , as well as in the value of GAF. The situation is not symmetrical for the hole (n = 1 + 6) and electron (n = 1 - 8) doping and the interoxygen hopping reduces the region of stability of the A F LRO. This d e m o n s t r a t e s the i m p o r t a n c e of this hopping process for quantitative studies. Summarizing, the p r e s e n t e d results d e m o n s t r a t c the absence of B r i n k m a n - R i c e transition [9] in the threeb a n d H u b b a r d model and give instead a MIT to an A F insulator. The Hamiltonian, eq. (1), gives a qualitatively correct description of the A F phase of HTS. [1] J.M. Tranquada et al., Phys. Rev. B 38 (1988) 2477. [2] K.C. Haas, Solid State Physics, vol. 42, eds. H, Ehrenreich and D. Turnbull (Academic, Orlando, 1989) p. 213. [3] J. Dutka and A.M. Oleg, Phys. Rev. B 43 (1991) 5622. [4] C.M. Varma, S. Schmitt-Rink and E. Abrahams, Solid State Commun. 62 (1987) 681. V.J. Emery, Phys. Rev. Lett. 58 (1987) 2794. [5] G. Stollhoff and P, Fulde, Z. Phys. B 26 (1977) 257; 29 (1978) 231. A.M. Oleg, P. Fulde and J. Zaanen, Physica B 148 (1987) 260. [6] M.S. Hybertsen, M. Schliiter and N.E. Christensen, Phys. Rev. B 39 (1989) 9028. [7] A.K. McMahan et al., Phys. Rev. B 42 (1990) 6268. E.B. Stechel and D.R. Jennison, Phys. Rev. B 38 (1988) 4632. F. Mila, Phys. Rev. B 38 (1988) 11358. [8] J. Zaanen, G.A. Sawatszky and J.W. Allen, Phys. Rev. Lett. 55 (1985) 418. [9] C.A. Balseiro et al, Phys. Rev. Lett. 62 (1989) 2624. [lO] S.N. Coopersmith, Phys. Rev. B 41 (1990)8711. [111 RJ. Birgeneau et al., Phys. Rev. B 38 (1988) 6614. [121 J.M. Ginder et al., Phys. Rev. B 37 (1988) 7506. U. Venkateswaran et al., Phys. Rev. B 38 (1988) 711)5. [131 G. Dopf et al., Phys. Rev. B 41 (1990) 9264.