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Electronic state and superconductivity in the d-p model
Physica C 185-189 (1991) 1513-1514 North-Holland
ELECTRONIC STATE AND SUPERCONDUCTIVITY IN T H E d-p MODEL D.S. HIRASHIMA, Y. ONO, T. MATSUURA and Y. KURODA Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-01, Japan Electronic state of the d-p hybridization model with infinite repulsion on the Cu site is investigated with the help of the auxiliary bosun technique and the 1IN expansion method. In particular, effects of the repulsion Upd between the holes on the nearest-neighbor Cu and 0 sites are clarified; it is shown that the charge fluctuation is enhanced by U~. Possible relevance to superconductivity is discussed. 1. INTRODUCTION The CuO2 plane is considered to contain the essence of the high temperature superconductivity (HTS). To clarify its characteristics, we investigate the d-p hybridization model with infinite repulsion on the Cu site with the auxiliary bosun technique and the 1/N expansion. We pay special attention to the charge fluctuation. Since Upd, the repulsion between the holes on the adjacant Cu and O sites, plays a vital role as far as the charge fluctuation is concerned, we study its effects on the electronic state. It should be noted that the strong correlation effect due to the repulsion on the Cu site is at the same time incorporated into the problem. We find that the charge fluctuation is enhanced by Upd, which may suggest the relevance of the excitonic mechanism to HTS. 2. MODEL We start from the following hamiltonian,
i,a
i,~
O,f),o"
t - t~,d~ (c.i,,d,,,b it + h.c.) + U~,d ~ (i,~),~
(1) A coherent band satisfying the Luttinger sum rule, i.e., having a large Fermi surface, is formed inside the charge transfer gap (mid-gap state). (2) For a large charge transfer energy A = % - ca, A > Ax = 2.37 (We set 2t~ = 1 in this paper), both of the width and the intensity of the mid-gap state decrease in proportional to the doped hole number 6 as 6 --, 0. On the other hand, for A < AI, they are finite for any n, n being the total hole number, n - 1 + 6. We have a metal-insulator (M-I) transition at A = AI and n = 1. (3) At the critical point of the M-I transition, the charge susceptibility )~c vanishes. 3.2. Effect of tp With a finite tp, O 2p holes have dispersion e~(k); e~(k) = % + 4 I tp I cos(k::/2)cos(k~/2). For tp > 0, due to the wave-number dependence of the mixing, 3d holes mix more strongly with the lower band (e~(k)) than with
t t , d,,,d,,,cj,,,%,,
(i,j)¢,~'
with the local constraint Q, = I!2~d!~d,,~ + b!b, = 1. In the above, di~, cj~ and bi are the annihilation operators of the (pseudo-) Cu 3d hole, the O 2p~ hole and the auxiliary boson, respectively. Due to the constraint the above hamiltonian is equivalent to the Ud ~ oo d-p model. We have included the direct transfer tp between the nearestneighbor O sites and the repulsion Upe. We treat tpe (and Upa) perturbatively resorting to the 1IN expansion 1,2 with N being the spin-orbital degeneracy; N = 2 in the present case. In doing so, we rigorously deal with the constraint. We summarize the results obtained in the leading order of the expansion in the following. We confine ourselves to the case with the absolute zero, T = 0.
3. ELECTRONIC STATE 3.1. Case with t~ = U ~ = 0 3-5
I
I
3 Al
2.5
I
0
0.1
~
!
0.2t, '
FIGURE 1 AI as a function of t v. (We set 2tr~ = 1.)
0921-4534/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved.
1514
D.$. Hlrmhima et at / Superconductivity in the d.p model
the upper one (e+(k)). The charge transfer energy A is thus effectivelyreduced by a positive tv. Conversely, we need a bager Al to stabilizethe insulating phase. In Fig. 1, the dependence of AI on tp is shown. Note that with a negative tp, AI would decrease as a function of Itv I. 3.3. Effects of Uva 6 In the leading order in the 1IN expansion, we can treat U ~ with the Hartree approximation. With finite Upd, we find (We put tv = 0): (I) The condition for the M-I transitionis replaced by A + 2Up~ > AI. The system can become insulating at n = I even for A < AI with the help of Upd. In Fig. 2a), we show the band scheme for A = 1.5 and Upd = 0.5; it is seen that the system is insulating at n - 1 and that there appear the mid-gap state inside the charge transfer gap, A + 2Ups, at n > 1. (2) The charge susceptibility X¢ is enhanced except in the narrow region around the critical point of the M-I transition. It is enhanced by the factor 1/(1 ÷ 4Upax[,a), where X~,d = On~,[Oe~. The point is that X~ becomes negative due to the strong correlation of U~. Figure 2b) shows the n-dependence of X¢ for A = 1.5 and Uva = 0.5. It is the q = 0-component that is the most enhanced; x¢(q) has a sharp peak around q = 0. (3) For a certain Upa, say U~, X¢ diverges; U$d depends on both of A and n. For A = 1.5, U~ takes the minimum value of ,~ 0.55 around n = 1.17. The divergence of X¢ implies an instabitility toward phase separation into 3d-hole-rich and 2p-hole-rich phases: By taking proper account of the long range Coulomb interaction, however, we would never have a phase separation. The enhancement of X¢ implies a tendency of the exciton formation. 4. QUASIPARTICLE INTERACTION The enhanced charge susceptibility induces a large attractive interaction between quasiparticles. At the same time, however, we should note that there occurs a renormalized repulsion originating from the on-site repulsion. In the static limit, we find it difficult for the attractive interaction to overcome the renormalized repulsion except in the immediate vicinity of the instability. To discuss the superconductivity, we need a more refined treatment of U~, a consideration on the frequency dependence of the interaction and so on. In particular, it would be interesting to investigate how the enhanced charge fluctuation modifies the electron-phonon coupling.
!
a)
/1--1.5 U,, = 0 . 5
,,,
|
•
|
•
|
1
,
,
,
n
1.5
•
,
,
!
•
•
•
,
2
,,
A=I.5
.=
b)
4 XC
2
0. 0
=
!
i
,
,
,
I
1.5
.
,
" |
n
i
2
FIGURE 2 a) Band scheme for A = 1.5 and U~ = 0.5. We put ~v = 1.5 and E~ = 0.0. The dotted line stands for the chemical potential, b) Thermodynamic charge susceptibility X¢ as a function of n for A = 1.5 with Up~ = 0.5 (solid line) and with Upd = 0.0 (dotted line).
, . , . , ~ ~ . , ~ , N ,..,, ~S
1. 2. 3. 4. 5.
P. Coleman, Phys. Rev. B29 (1984) 3035. Y. Ono et al., Physica C159(1989) 878. H. Jichu et al., J. Phys. Soc. Jpn. 58(1989) 4280. H. Jichu et al., J. Phys. Soc. Jpn. 59(1990) 2822. D.S. Hirashima et al., J. Phys. Soc. Jpn. 60 (1991) 2269. 6. D.S. Hirashima et al., J. Phys. Soc. Jpn. 60 (1991) 1864. 7. M. Grilli et al., Int. J. Mod. Phys. B5(1991) 309.
ELECTRONIC STATE AND SUPERCONDUCTIVITY IN T H E d-p MODEL D.S. HIRASHIMA, Y. ONO, T. MATSUURA and Y. KURODA Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-01, Japan Electronic state of the d-p hybridization model with infinite repulsion on the Cu site is investigated with the help of the auxiliary bosun technique and the 1IN expansion method. In particular, effects of the repulsion Upd between the holes on the nearest-neighbor Cu and 0 sites are clarified; it is shown that the charge fluctuation is enhanced by U~. Possible relevance to superconductivity is discussed. 1. INTRODUCTION The CuO2 plane is considered to contain the essence of the high temperature superconductivity (HTS). To clarify its characteristics, we investigate the d-p hybridization model with infinite repulsion on the Cu site with the auxiliary bosun technique and the 1/N expansion. We pay special attention to the charge fluctuation. Since Upd, the repulsion between the holes on the adjacant Cu and O sites, plays a vital role as far as the charge fluctuation is concerned, we study its effects on the electronic state. It should be noted that the strong correlation effect due to the repulsion on the Cu site is at the same time incorporated into the problem. We find that the charge fluctuation is enhanced by Upd, which may suggest the relevance of the excitonic mechanism to HTS. 2. MODEL We start from the following hamiltonian,
i,a
i,~
O,f),o"
t - t~,d~ (c.i,,d,,,b it + h.c.) + U~,d ~ (i,~),~
(1) A coherent band satisfying the Luttinger sum rule, i.e., having a large Fermi surface, is formed inside the charge transfer gap (mid-gap state). (2) For a large charge transfer energy A = % - ca, A > Ax = 2.37 (We set 2t~ = 1 in this paper), both of the width and the intensity of the mid-gap state decrease in proportional to the doped hole number 6 as 6 --, 0. On the other hand, for A < AI, they are finite for any n, n being the total hole number, n - 1 + 6. We have a metal-insulator (M-I) transition at A = AI and n = 1. (3) At the critical point of the M-I transition, the charge susceptibility )~c vanishes. 3.2. Effect of tp With a finite tp, O 2p holes have dispersion e~(k); e~(k) = % + 4 I tp I cos(k::/2)cos(k~/2). For tp > 0, due to the wave-number dependence of the mixing, 3d holes mix more strongly with the lower band (e~(k)) than with
t t , d,,,d,,,cj,,,%,,
(i,j)¢,~'
with the local constraint Q, = I!2~d!~d,,~ + b!b, = 1. In the above, di~, cj~ and bi are the annihilation operators of the (pseudo-) Cu 3d hole, the O 2p~ hole and the auxiliary boson, respectively. Due to the constraint the above hamiltonian is equivalent to the Ud ~ oo d-p model. We have included the direct transfer tp between the nearestneighbor O sites and the repulsion Upe. We treat tpe (and Upa) perturbatively resorting to the 1IN expansion 1,2 with N being the spin-orbital degeneracy; N = 2 in the present case. In doing so, we rigorously deal with the constraint. We summarize the results obtained in the leading order of the expansion in the following. We confine ourselves to the case with the absolute zero, T = 0.
3. ELECTRONIC STATE 3.1. Case with t~ = U ~ = 0 3-5
I
I
3 Al
2.5
I
0
0.1
~
!
0.2t, '
FIGURE 1 AI as a function of t v. (We set 2tr~ = 1.)
0921-4534/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved.
1514
D.$. Hlrmhima et at / Superconductivity in the d.p model
the upper one (e+(k)). The charge transfer energy A is thus effectivelyreduced by a positive tv. Conversely, we need a bager Al to stabilizethe insulating phase. In Fig. 1, the dependence of AI on tp is shown. Note that with a negative tp, AI would decrease as a function of Itv I. 3.3. Effects of Uva 6 In the leading order in the 1IN expansion, we can treat U ~ with the Hartree approximation. With finite Upd, we find (We put tv = 0): (I) The condition for the M-I transitionis replaced by A + 2Up~ > AI. The system can become insulating at n = I even for A < AI with the help of Upd. In Fig. 2a), we show the band scheme for A = 1.5 and Upd = 0.5; it is seen that the system is insulating at n - 1 and that there appear the mid-gap state inside the charge transfer gap, A + 2Ups, at n > 1. (2) The charge susceptibility X¢ is enhanced except in the narrow region around the critical point of the M-I transition. It is enhanced by the factor 1/(1 ÷ 4Upax[,a), where X~,d = On~,[Oe~. The point is that X~ becomes negative due to the strong correlation of U~. Figure 2b) shows the n-dependence of X¢ for A = 1.5 and Uva = 0.5. It is the q = 0-component that is the most enhanced; x¢(q) has a sharp peak around q = 0. (3) For a certain Upa, say U~, X¢ diverges; U$d depends on both of A and n. For A = 1.5, U~ takes the minimum value of ,~ 0.55 around n = 1.17. The divergence of X¢ implies an instabitility toward phase separation into 3d-hole-rich and 2p-hole-rich phases: By taking proper account of the long range Coulomb interaction, however, we would never have a phase separation. The enhancement of X¢ implies a tendency of the exciton formation. 4. QUASIPARTICLE INTERACTION The enhanced charge susceptibility induces a large attractive interaction between quasiparticles. At the same time, however, we should note that there occurs a renormalized repulsion originating from the on-site repulsion. In the static limit, we find it difficult for the attractive interaction to overcome the renormalized repulsion except in the immediate vicinity of the instability. To discuss the superconductivity, we need a more refined treatment of U~, a consideration on the frequency dependence of the interaction and so on. In particular, it would be interesting to investigate how the enhanced charge fluctuation modifies the electron-phonon coupling.
!
a)
/1--1.5 U,, = 0 . 5
,,,
|
•
|
•
|
1
,
,
,
n
1.5
•
,
,
!
•
•
•
,
2
,,
A=I.5
.=
b)
4 XC
2
0. 0
=
!
i
,
,
,
I
1.5
.
,
" |
n
i
2
FIGURE 2 a) Band scheme for A = 1.5 and U~ = 0.5. We put ~v = 1.5 and E~ = 0.0. The dotted line stands for the chemical potential, b) Thermodynamic charge susceptibility X¢ as a function of n for A = 1.5 with Up~ = 0.5 (solid line) and with Upd = 0.0 (dotted line).
, . , . , ~ ~ . , ~ , N ,..,, ~S
1. 2. 3. 4. 5.
P. Coleman, Phys. Rev. B29 (1984) 3035. Y. Ono et al., Physica C159(1989) 878. H. Jichu et al., J. Phys. Soc. Jpn. 58(1989) 4280. H. Jichu et al., J. Phys. Soc. Jpn. 59(1990) 2822. D.S. Hirashima et al., J. Phys. Soc. Jpn. 60 (1991) 2269. 6. D.S. Hirashima et al., J. Phys. Soc. Jpn. 60 (1991) 1864. 7. M. Grilli et al., Int. J. Mod. Phys. B5(1991) 309.