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A pair potential and self-energy model for Hubbard systems
Physica C 235-240 (I 994) 2205-2206 North-ltolland
PHYSICAg
A Pair Potential and Self-Energy Model for Hubbard Systems J. Costa-Quintana, E. Gonzhlez-Le6n, F. L6pez-Aguilar and L. Puig-Puig ~ " aGrup d'Electromagnetisme, Edifici Cn, Universitat Aut6noma de Barcelona, 08193 Bellaterra (Barcelona), Spain A model for the high-T~ superconductors which takes into accoun~ two O's and one Cu atoms with one orbital in each atom is set up. The dynamically screened hole-hole interaction is deduced from the random-phase approximation. We calculate the pair potential, the spectral function and the density of states of the interacting system, as a function of the quasimomentum of the bubble polarization.
The superconducting mechanism for the highT~ superconductors can be approached considering coupling models obtained just from an effective fermion-fermion interaction. We consider a three-band Hubbard model [1] in the planes of Cu2, 0 2 and 0 3 for YBa2CuaOr, given by [2]
H = HO+Ud Enidlrtidl + E Up E njplnjpi(1)
where x ° ( q , w) : ' ~ 7
k w+e'~--e~q+k )+i0 +
For the d-band we approximate the polarization by the mean value, i.e. 1
n~(~) ~- ~ Z n~(q,~) •
(6)
q where nidl (nivT) is the number operator of Wannier states corresponding to d~_v~ (P~:/Pu) orbital of Cu2 ( 0 3 / 0 2 ) atoms [3]. tIo is the noninteracting hamiltonian, ~k ko k~ ko
where a is the spin-band index (d, p . or pu). The dynamically screened hole-hole interaction is deduced from a recurrent random-phase approximation (RPA) equation as W(q,w) = U + UH(q,w)W(q,w)
(3)
where the matrix elements of II are electron-hole polarizations between different p and d bands, and the U matrix is defined in a d ® p-space. In our ease it is a 6-dimension space (one d-band and two p-bands, each with both spin directions). To calculate this effective interaction we must evaluate the polarizationi2 ], II'~(q,w) = xO(q, w) + X " ( q , - w )
(4)
*We acknowledge fh' .,,tim supp,,rt fr,ml CICY'I pr,,jcct MAT91-0955. L.ILP. ackn,,wh:dge.s ~,upp,,rt fr,,m the .Ministerio de Educaci3n y Cic:rtciaof Spain.
For the p-oxygen band we take a tight-binding band considering only the nearest neighbours. As the nearest neighbours of 0 3 (02) atoms are two Cu2 in the z (y) directio~, we consider an energy dispersion only in the k, (ku) direction. T h a t is, A 4 ~ = e , , - y c o , ak~;
(7)
where A is the bandwidth and a (b) the distance to the Cu2 atoms. In this way the integral of the polarization is one dimensional and we can evaluate it analyticaly. In two o three dimensions no analytical evaluation appears to be possible, except perhaps in a few limiting cases (half-filling and nesting: vector). The polarization for pd mixing is small, and we do not consider it[2]. In this way the 6 × 6 matrices are decoupled in three 2 × 2 matrices. The effective interaction in RPA for parallel and antiparallel spins is
u/2 WT"T( q ' ~ ) ,,
Wl~(q'~)
0921-4534/94/$07.00 © 1994 - F.lsevierScience B.V. All rights reserved.
SSI?IfY)21-4534(94)01668-2
A 4 ~ - ~,,- ~co.,bku
1 - V I],'(q,~)g/2
v/2 1~ Un"(q, ~) (8) U/2
1 - V n . ' ( q . ~ ) + 1+ un..(q.~,) (9)
J. Costa.Quintana et ai./Physica C 235-240 (1994) 2205-2206
2206
For the parallel spins one takes the Feynman diagrams with odd bubble polarizations and for antiparallel spins the even bubble polarizations. Similar expressions result for the d-bands. As the last term of the sums is much smaller than the first we do not consider it in a first approximation. Given the large number of parameters in these expressions, henceforth all enert~iesare considered in units of the bandwith (A).
n.q
't~ -- cq +
1 - nq
F ~ k - q -- iO+ + o¢ -- eq -- F [ 3 k - q
]
+
iO +
where only a numerical integration in q is needed. From this expression we calculate the spectral function and the density of states (DOS) JV'(t#)= Z
Ak(w).
(13)
k
The results of the DOS in a p-band are shown in Fig. 2 for several values of the parameters.
100 •
"
"
I
I
I
I
I
(a)
"
"
k
"~
0
"-'.: ...........
,
I
....q
e~
L : : 7
I
r
-50
"._ I
-
_ ~
.L..
!
(c)
I~
"
(b)-
-
(d)"
tn o c~
-
-ioo I
o
2
3
4
5
(bandwidt.h units) -5
Figure 1. Effective interaction. Solid line q=fi.5, dashed line q=l.5, dotted line q=Tr. All lines with Co=0.4 and U=20. The effective interaction shown in Fig. 1 is well fitted, for U values greater than about five bandwidth (U > 5), by a polarization given by IIP(q'to) ~-
(1.9/a') sin kF/3q2
(10)
W2 __ ~ 2
where s~q = Isin(q/2)l and kF is the momentum corresponding to the Fermi energy. Thus the effective interaction can be approximated by WP(q,~ ) ,~ U
(11)
where
F-" = 1 + __1"9U sin k F r¢
(12)
The first order self-energy, summing the direct and the exchange terms is U F 2-
2
2F
1 q
~k-q
5
-5
0
5
(bandwidth units) Figure 2. DOS: (a) co---0, U=20; (b) ~o=0.4, U=20; (c)¢o=0, U=40; (d) Co=0.4, U=40. In conclusion, considering a three-band hamiltonian and the q-dependence we have obtained the lower, the middle and the upper Hubbard bands. For greater values of U the splitting between the lower and upper bands increases and also does their bandwidth, ~hile the width of the middle band becomes smaller. REFERENCES
~v2 - ~
-- 2 w2 _ F 2 ( t 3 q _ iO +)2
SP(k,~)=
0
×
1. P. Fulde and P Horsch, Europhys. News 24, 73 (1993). 2. F. L6pez-Aguilar, J. Costa-Quintana and L. Puig-Puig, Phys. Rev B 48, 1128 (1993). 3 • K. ~. I" .nass, . . . oouo " . . .orate .... r'nyslcs, 42, 2i3 (1989).
PHYSICAg
A Pair Potential and Self-Energy Model for Hubbard Systems J. Costa-Quintana, E. Gonzhlez-Le6n, F. L6pez-Aguilar and L. Puig-Puig ~ " aGrup d'Electromagnetisme, Edifici Cn, Universitat Aut6noma de Barcelona, 08193 Bellaterra (Barcelona), Spain A model for the high-T~ superconductors which takes into accoun~ two O's and one Cu atoms with one orbital in each atom is set up. The dynamically screened hole-hole interaction is deduced from the random-phase approximation. We calculate the pair potential, the spectral function and the density of states of the interacting system, as a function of the quasimomentum of the bubble polarization.
The superconducting mechanism for the highT~ superconductors can be approached considering coupling models obtained just from an effective fermion-fermion interaction. We consider a three-band Hubbard model [1] in the planes of Cu2, 0 2 and 0 3 for YBa2CuaOr, given by [2]
H = HO+Ud Enidlrtidl + E Up E njplnjpi(1)
where x ° ( q , w) : ' ~ 7
k w+e'~--e~q+k )+i0 +
For the d-band we approximate the polarization by the mean value, i.e. 1
n~(~) ~- ~ Z n~(q,~) •
(6)
q where nidl (nivT) is the number operator of Wannier states corresponding to d~_v~ (P~:/Pu) orbital of Cu2 ( 0 3 / 0 2 ) atoms [3]. tIo is the noninteracting hamiltonian, ~k ko k~ ko
where a is the spin-band index (d, p . or pu). The dynamically screened hole-hole interaction is deduced from a recurrent random-phase approximation (RPA) equation as W(q,w) = U + UH(q,w)W(q,w)
(3)
where the matrix elements of II are electron-hole polarizations between different p and d bands, and the U matrix is defined in a d ® p-space. In our ease it is a 6-dimension space (one d-band and two p-bands, each with both spin directions). To calculate this effective interaction we must evaluate the polarizationi2 ], II'~(q,w) = xO(q, w) + X " ( q , - w )
(4)
*We acknowledge fh' .,,tim supp,,rt fr,ml CICY'I pr,,jcct MAT91-0955. L.ILP. ackn,,wh:dge.s ~,upp,,rt fr,,m the .Ministerio de Educaci3n y Cic:rtciaof Spain.
For the p-oxygen band we take a tight-binding band considering only the nearest neighbours. As the nearest neighbours of 0 3 (02) atoms are two Cu2 in the z (y) directio~, we consider an energy dispersion only in the k, (ku) direction. T h a t is, A 4 ~ = e , , - y c o , ak~;
(7)
where A is the bandwidth and a (b) the distance to the Cu2 atoms. In this way the integral of the polarization is one dimensional and we can evaluate it analyticaly. In two o three dimensions no analytical evaluation appears to be possible, except perhaps in a few limiting cases (half-filling and nesting: vector). The polarization for pd mixing is small, and we do not consider it[2]. In this way the 6 × 6 matrices are decoupled in three 2 × 2 matrices. The effective interaction in RPA for parallel and antiparallel spins is
u/2 WT"T( q ' ~ ) ,,
Wl~(q'~)
0921-4534/94/$07.00 © 1994 - F.lsevierScience B.V. All rights reserved.
SSI?IfY)21-4534(94)01668-2
A 4 ~ - ~,,- ~co.,bku
1 - V I],'(q,~)g/2
v/2 1~ Un"(q, ~) (8) U/2
1 - V n . ' ( q . ~ ) + 1+ un..(q.~,) (9)
J. Costa.Quintana et ai./Physica C 235-240 (1994) 2205-2206
2206
For the parallel spins one takes the Feynman diagrams with odd bubble polarizations and for antiparallel spins the even bubble polarizations. Similar expressions result for the d-bands. As the last term of the sums is much smaller than the first we do not consider it in a first approximation. Given the large number of parameters in these expressions, henceforth all enert~iesare considered in units of the bandwith (A).
n.q
't~ -- cq +
1 - nq
F ~ k - q -- iO+ + o¢ -- eq -- F [ 3 k - q
]
+
iO +
where only a numerical integration in q is needed. From this expression we calculate the spectral function and the density of states (DOS) JV'(t#)= Z
Ak(w).
(13)
k
The results of the DOS in a p-band are shown in Fig. 2 for several values of the parameters.
100 •
"
"
I
I
I
I
I
(a)
"
"
k
"~
0
"-'.: ...........
,
I
....q
e~
L : : 7
I
r
-50
"._ I
-
_ ~
.L..
!
(c)
I~
"
(b)-
-
(d)"
tn o c~
-
-ioo I
o
2
3
4
5
(bandwidt.h units) -5
Figure 1. Effective interaction. Solid line q=fi.5, dashed line q=l.5, dotted line q=Tr. All lines with Co=0.4 and U=20. The effective interaction shown in Fig. 1 is well fitted, for U values greater than about five bandwidth (U > 5), by a polarization given by IIP(q'to) ~-
(1.9/a') sin kF/3q2
(10)
W2 __ ~ 2
where s~q = Isin(q/2)l and kF is the momentum corresponding to the Fermi energy. Thus the effective interaction can be approximated by WP(q,~ ) ,~ U
(11)
where
F-" = 1 + __1"9U sin k F r¢
(12)
The first order self-energy, summing the direct and the exchange terms is U F 2-
2
2F
1 q
~k-q
5
-5
0
5
(bandwidth units) Figure 2. DOS: (a) co---0, U=20; (b) ~o=0.4, U=20; (c)¢o=0, U=40; (d) Co=0.4, U=40. In conclusion, considering a three-band hamiltonian and the q-dependence we have obtained the lower, the middle and the upper Hubbard bands. For greater values of U the splitting between the lower and upper bands increases and also does their bandwidth, ~hile the width of the middle band becomes smaller. REFERENCES
~v2 - ~
-- 2 w2 _ F 2 ( t 3 q _ iO +)2
SP(k,~)=
0
×
1. P. Fulde and P Horsch, Europhys. News 24, 73 (1993). 2. F. L6pez-Aguilar, J. Costa-Quintana and L. Puig-Puig, Phys. Rev B 48, 1128 (1993). 3 • K. ~. I" .nass, . . . oouo " . . .orate .... r'nyslcs, 42, 2i3 (1989).