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Phonon induced pairing in the two dimensional Hubbard model
Phys~ea C 235-240 (1994) 2143-2144 North-Holland
PHYSICA
Phonon Induced Pairing in the Two Dimensional Hubbard Model M.Mierzejewski *
J.Zielifski*+
P.Entel ?
Instituteof Physics, University of Silesia,40-007 Katowice, Poland We address the problem of electron-phonon interaction in the two dimensional Hubbard model. The grfhlich canonical transformation has been generalized to the case of strongly correlated electrons. The resulting pairing interaction is definitively reduced when compared to the uncorrelated limit. However, among different contributions, the formation of intersite, s-like Cooper pairs is enhanced for small hole concentration. We conclude that within the weak coupling theory phonon mediated pairing interactions play a secondary role and can be dominated by phonon-free mechanisms. This may not be the case for strong electron-phonon coupling. It seems d e a r that correlations between electrons (holes) in Cu09. layers are at least partially responsible for the unusual properties of high Tc systems and may effectively contribute, accompanying phonons, to the formation of a superconducting state. The natural question which arises is about the significance of phonon-induced pairing interaction in the presence of strong local correlations. To discuss this problem we consider the U - . c~ limit of the two dimensional Hubbard model coupled to phonons : H = H0 + H1 ,
has to truncate their many site products in order to eliminate terms linear in gk k+q. The resulting intersite pairing is of the form
~,,,,,, : - ~
1
~ ~ g~,,+~ (u[,q q,k,k t
u+,q)
I,s,~r
¢-,~.-R,)xTo x-"°x,°-"xf "
(4)
and u~q have to be found from the system of equations :
(1) 1
where
x:°x
o=-t
+
,a
1
Hi = -'ff E
q
gk k+qe'(k+q)R*
.I-tkR 1
k,q,i,j,a
X:°Xff" (bq + b+_q) ,
(3)
and X~'" stands for the Hubbard operators. The standard procedure leading to an effectivepalring interactionis to eliminate HI with the help of an appropriate canonical transformation. Due to the projectivecharacter of the Hubbard operators one
P (5) where ~ = 1 - (X °~) Xp = / ~\ '.' p~ o y- -opo ) and ck = -2¢, (cos (k~a) + cos (k~a)) = - t r (k) In what follows we restrict ourselves to the nearest neighbout pairing. It is clear from the above that m the case of strongly correlated system the strength of phonon-induced pairing depends on the band fillit~g n, and has Lo evaiua~eu ~ezt-~u,~,,--~,:, ~,~gether with the chemical potentmt The BCS-hke treatment of Ho + Hpatr leads to the following order parameters for interslte Cooper pairs :
*Supported by the State Committee for ScientificResearch, Grant No 2 P302 018 05 tTheoretische Tieftemperaturphyslk, Universltht-GH-Duisburg, 47048 Duisburg, Germany + Partially supported by the DAAD 0921-4534/94/$07 00 © 1994 - [-lscvlcr Sctcncc B V All nght~ rc~crvcd 33DI 0921-4534(9 I)01637-2
M Mterzejewskt et al. / Physwa C 235-240 (1994) 2143-2144
2144
' E . ( k ) a. = -~
(6)
,
k
a (k)
= =
7,17,~,(, where 2(cos(k..)cos(k~.)) ,
(() (k)
=
2 (sin (kra) + ( - ) sin (kya)) 48)
(7)
To get relation to the uncorrelated case we have neglected any momentum dependence of the electron-phonon interaction and kept ck+q and ~k on the Fermi surface. The Fermi surface averages of phonon-induced coupling functions versus the occupation number are shown in Fig.1. The insertion shows the form-factor independent part of the nearest neighbour pairing interaction G normalized to its value for the uncorrelated case, Go = 292/w0. One can see that correlations pronouncedly diminish the phonon induced pairing and only G7 play some role for small concentration of holes. However, within the assumed decoupling scheme, H0 stands for the source ofnonphononic contribution to the formation of A~Cooper pairs These contributions, definitively dominating for small doping, originate from the spin and charge correlations involving different lattice sites and have been discussed in Ref.[1]. Fig 2 shows the typical behaviour of Tc as a function of the occupation number n As shown in the msemon, the non-phononic contributions lead to repulsive effectwe interaction for n _< 0 7
One has to bear in mind that we have considered weak coupling theory in presence of strong local correlations. There are indications that local correlations present in the Hubbard model can play an opposite role and lead within strongcoupling theory to pronounced enhancement of the coupling function [2,3]. Therefore , a more refined treatment of the role of local correlations for phonon-induced superconductivity (especially in the context of recently discussed d-wave pairing [4-6]) is needed. The generalization of the Eliashberg equations to the Hubbard model is under investigation [7].
REFERENCES
1. J.Zielifiski, M.Matlak and P.Entel, Phys.Lett. A136 (1989) 441. 2. J.Zielifiski, M.Matlak and P.Entel, Phys.Lett. A165 (1992) 285; J.Zielitlski, M.Matlak, Phys.Lett. A172 (1993) 467. 3. M.L.Kuli~ and R.Zeyher, preprint 1993. 4. D.Thelen, D.Pines and Jian Ping Lu, Phys.Rew B47 (1993) 9151. 5. Hyekyung Won and Kazumi Maki, Phys.Rev. B49 (i994) 1397. 6. T.P.Devereaux et al. Phys.Rev.Lett. 7'2 (1994) 396. 7. J.Zielifiski, M.Mierzejewskl, P.Entel and R.Grabowski, J Superconductivity, in press.
0.15
0 20 10
0.15 Ga 0.10
GG7 ~1
06,
0.10
o4. G
kTc -7-
o?,
°°o~o~ o4 o~,o~ ,
!!i 00J0
O.O5
0.05
°°%.b
0.2
fig.1
0.4
0.6 nO.8
.0
0.00 0.7
j /
I 0.8
fig.2
o.b
n
1.o
PHYSICA
Phonon Induced Pairing in the Two Dimensional Hubbard Model M.Mierzejewski *
J.Zielifski*+
P.Entel ?
Instituteof Physics, University of Silesia,40-007 Katowice, Poland We address the problem of electron-phonon interaction in the two dimensional Hubbard model. The grfhlich canonical transformation has been generalized to the case of strongly correlated electrons. The resulting pairing interaction is definitively reduced when compared to the uncorrelated limit. However, among different contributions, the formation of intersite, s-like Cooper pairs is enhanced for small hole concentration. We conclude that within the weak coupling theory phonon mediated pairing interactions play a secondary role and can be dominated by phonon-free mechanisms. This may not be the case for strong electron-phonon coupling. It seems d e a r that correlations between electrons (holes) in Cu09. layers are at least partially responsible for the unusual properties of high Tc systems and may effectively contribute, accompanying phonons, to the formation of a superconducting state. The natural question which arises is about the significance of phonon-induced pairing interaction in the presence of strong local correlations. To discuss this problem we consider the U - . c~ limit of the two dimensional Hubbard model coupled to phonons : H = H0 + H1 ,
has to truncate their many site products in order to eliminate terms linear in gk k+q. The resulting intersite pairing is of the form
~,,,,,, : - ~
1
~ ~ g~,,+~ (u[,q q,k,k t
u+,q)
I,s,~r
¢-,~.-R,)xTo x-"°x,°-"xf "
(4)
and u~q have to be found from the system of equations :
(1) 1
where
x:°x
o=-t
+
1
Hi = -'ff E
q
gk k+qe'(k+q)R*
.I-tkR 1
k,q,i,j,a
X:°Xff" (bq + b+_q) ,
(3)
and X~'" stands for the Hubbard operators. The standard procedure leading to an effectivepalring interactionis to eliminate HI with the help of an appropriate canonical transformation. Due to the projectivecharacter of the Hubbard operators one
P (5) where ~ = 1 - (X °~) Xp = / ~\ '.' p~ o y- -opo ) and ck = -2¢, (cos (k~a) + cos (k~a)) = - t r (k) In what follows we restrict ourselves to the nearest neighbout pairing. It is clear from the above that m the case of strongly correlated system the strength of phonon-induced pairing depends on the band fillit~g n, and has Lo evaiua~eu ~ezt-~u,~,,--~,:, ~,~gether with the chemical potentmt The BCS-hke treatment of Ho + Hpatr leads to the following order parameters for interslte Cooper pairs :
*Supported by the State Committee for ScientificResearch, Grant No 2 P302 018 05 tTheoretische Tieftemperaturphyslk, Universltht-GH-Duisburg, 47048 Duisburg, Germany + Partially supported by the DAAD 0921-4534/94/$07 00 © 1994 - [-lscvlcr Sctcncc B V All nght~ rc~crvcd 33DI 0921-4534(9 I)01637-2
M Mterzejewskt et al. / Physwa C 235-240 (1994) 2143-2144
2144
' E . ( k )
(6)
,
k
a (k)
= =
7,17,~,(, where 2(cos(k..)cos(k~.)) ,
(() (k)
=
2 (sin (kra) + ( - ) sin (kya)) 48)
(7)
To get relation to the uncorrelated case we have neglected any momentum dependence of the electron-phonon interaction and kept ck+q and ~k on the Fermi surface. The Fermi surface averages of phonon-induced coupling functions versus the occupation number are shown in Fig.1. The insertion shows the form-factor independent part of the nearest neighbour pairing interaction G normalized to its value for the uncorrelated case, Go = 292/w0. One can see that correlations pronouncedly diminish the phonon induced pairing and only G7 play some role for small concentration of holes. However, within the assumed decoupling scheme, H0 stands for the source ofnonphononic contribution to the formation of A~Cooper pairs These contributions, definitively dominating for small doping, originate from the spin and charge correlations involving different lattice sites and have been discussed in Ref.[1]. Fig 2 shows the typical behaviour of Tc as a function of the occupation number n As shown in the msemon, the non-phononic contributions lead to repulsive effectwe interaction for n _< 0 7
One has to bear in mind that we have considered weak coupling theory in presence of strong local correlations. There are indications that local correlations present in the Hubbard model can play an opposite role and lead within strongcoupling theory to pronounced enhancement of the coupling function [2,3]. Therefore , a more refined treatment of the role of local correlations for phonon-induced superconductivity (especially in the context of recently discussed d-wave pairing [4-6]) is needed. The generalization of the Eliashberg equations to the Hubbard model is under investigation [7].
REFERENCES
1. J.Zielifiski, M.Matlak and P.Entel, Phys.Lett. A136 (1989) 441. 2. J.Zielifiski, M.Matlak and P.Entel, Phys.Lett. A165 (1992) 285; J.Zielitlski, M.Matlak, Phys.Lett. A172 (1993) 467. 3. M.L.Kuli~ and R.Zeyher, preprint 1993. 4. D.Thelen, D.Pines and Jian Ping Lu, Phys.Rew B47 (1993) 9151. 5. Hyekyung Won and Kazumi Maki, Phys.Rev. B49 (i994) 1397. 6. T.P.Devereaux et al. Phys.Rev.Lett. 7'2 (1994) 396. 7. J.Zielifiski, M.Mierzejewskl, P.Entel and R.Grabowski, J Superconductivity, in press.
0.15
0 20 10
0.15 Ga 0.10
GG7 ~1
06,
0.10
o4. G
kTc -7-
o?,
°°o~o~ o4 o~,o~ ,
!!i 00J0
O.O5
0.05
°°%.b
0.2
fig.1
0.4
0.6 nO.8
.0
0.00 0.7
j /
I 0.8
fig.2
o.b
n
1.o