- Email: [email protected]
Hole-hole correlations induced by magnetic frustration
Journal of Magnetism and Magnetic Materials 104-107 (1992) 587-588 North-Holland
Hole-hole
correlations i n d u c e d by m a g n e t i c frustration
M.L. Lyra and Solange B. Cavalcanti Departamento de Fisica, Universidade Federal de Alagoas, Maceid-57061, Alagoas, Brazil
A randomly decorated Ising model on a Bethe lattice is considered to study hole-hole correlations of CuO2-based superconducting sheets. Two intrinsically different types of pairing mechanism are observed: one, induced by magnetic fluctuations and the other as a consequence of a free-energy minimisation process due to frustration effects so intense that one might conjecture that frustration is an essential ingredient to the understanding of such high superconducting transition temperature displayed by those materials.
Experimental evidence supports the fact that magnetism is an important ingredient to the understanding of the behaviour of C u - O - b a s e d superconductors [1]. Recently, Aharony et al. [2] have suggested that frustration effects introduced by doping a L a z C u O 4 compound should be the underlying mechanism involved in the h o l e - h o l e pairing and in the subsequent superconductivity observed in this material. Monte Carlo simulations have explicitly shown this pairing tendency of holes [3]. The present approach provides, in spite of its simplicity, a clear overall picture of the magnetic behaviour of the doped CuO. It is based on the use of classical Ising spins S, localised on the vertices of a lattice, to represent the Cu magnetic moments of a CuO 2 sheet, interacting with each other antiferromagnetically via the exchange parameter J(J < 0) [4]. Doping creates holes on the O ions, which are placed on the bonds. The holes, with a net ~1 spin o-, interact with the Cu ions through an aJ coupling constant, which leads to an effective ferromagnetic coupling between the Cu ions. The model does not allow two holes on a single O ion. It is believed that this assumption will not compromise the qualitative aspects of the final results. The itinerancy of the holes is considered here by using a grand-canonical distribution for the hole configuration, which means that one is performing annealed averages. This procedure also takes into account fluctuations over the total hole number owing to the hopping of electrons from one sheet to another. The Hamiltonian of the system is then given by:
H = - J Y ' ~ S i S j - a J Y ' . o ' u ( Si + S i ) n i i - lz Y'~nij,
(1)
where the summation runs over all neares-neighbour sites of a lattice and /x is the chemical potential per hole. H e r e n u is a variable, which gives the hole occupation number in the i-j bond. The hole concentration may be thermodynamically
obtained through the evaluation of the mean number of holes per bond:
x=<%>= Enij e e"/F,e = - k B T ~ In ~ / ~ ,
""
(2)
where ~ is the grand partition function and the sum is to be made over all hole and spin configurations. Equation (2) will be used in order to eliminate the hidden variable g. Some exact results related to the magnetic phase diagram of a quite similar model were presented by dos Santos et al. [5], and connections with superconductivity were suggested. The principal of the present work is to look closely at the pairing mechanism involved in the model outlined. The effective interaction between holes can be directly observed through the h o l e - h o l e correlation function:
C( R ) = (nijnkl> -- (nit) z,
(3)
where R is the spatial distance vector between the i-j and k - l bonds. This correlation function may present three different types of behaviour: (a) C(R) = 0, indicating that the holes are completely decoupled; (b) C(R) < 0, meaning an effective repulsive interaction between them; (c) C ( R ) > 0 indicating an effective attraction, which leads to a pairing tendency. The h o l e - h o l e correlation function defined above is evaluated exactly for two neighbouring bonds of a Bethe lattice with very interesting and surprising results, as we shall see in the following. By using the method of ring-recurrence [6] one is able to write the grand-partition function in terms of an effective field for the sth ring, denoted here by Ps, which is defined through a recursion relation of the form:
Ps-I = f ( P , , K, Kerr, Ix),
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
(4)
M.L. Lyra and S.B. Caualcanti / Hole hole correlations
588 0.016
..... .....
0.012
ALPHA ALPHA
-=
1.0 2.0
z 0 I--.<~ LLJ 0.o08 05 Od 0 £P 0.004
i 0.000 0.00
1.00
2.00
3.00
4.00
TEMPERATURE (kBT/J) Fig. 1. Hole hole correlation plotted against temperature at a hole concentration of x = 0.30, for dilute (a = 1) and frustrated ( a > 1) cases, on a Bethe lattice with coordination number q = 4. Note the appreciable correlation enhancement induced by frustration.
where K = J / k B T a n d Keff is the h o l e - m e d i a t e d effective coupling, that is
Kef f = ½ l n [ c o s h ( 2 a K ) ] .
(5)
In the t h e r m o d y n a m i c limit s ~ oo and p.,. --*p* so that one has to solve the fixed-point version of eq. (4) to find its solutions p * ( K , Ket ¢, /,t) for the various phases. In this way the g r a n d - p a r t i t i o n function is readily obtained and, therefore, the t h e r m o d y n a m i c functions of interest may be found exactly. T h e results o b t a i n e d t h r o u g h this p r o c e d u r e are illustrated in fig. 1 where h o l e - h o l e correlation is plotted against t e m p e r a t u r e , for various a a n d for a c o n c e n t r a t i o n of holes x = 0.30. It is observed that, if the effective h o l e - m e d i a t e d interaction b e t w e e n Cu ions is not strong e n o u g h to m a k e the resulting interaction (Jr = y +Jeff) f e r r o m a g n e t i c ( a < 1), t h e n t h e r e is a positive correlation b e t w e e n holes at finite t e m p e r a tures in the o r d e r e d phase. This m e a n s that a magn e t i c - f l u c t u a t i o n - i n d u c e d pairing m e c h a n i s m exists even w h e n frustration is absent. W h e n the resulting h o l e - m e d i a t e d interactions Jr b e c o m e f e r r o m a g n e t i c
( a > 1), the disorder and c o m p e t i t i o n between these and the pure a n t i f e r r o m a g n e t i c ones introduce frustration cffects, which considerably e n h a n c e the h o l e - h o l c correlation, In this case, the correlation has a feature not observed in the n o n - f r u s t r a t e d case: it persists even at T = 0. It is clear evidence that this correlation is essentially induced by a free-energy minimisation proccss. T h e discrepancy b e t w e e n the m e a n a m p l i t u d e of the correlation in thesc two eases reveals the fundam e n t a l role played by frustration i n h t h e hole pairing mcchanism. The case a = 1 is a pathological one because it m e a n s that the system b e c o m e s diluted at T = 0. In this casc thc free energy is not so m u c h Iowcrcd with pairing as in the frustrated onc and the correlation remains wcak. T h c abscnce of h o l e - h o l e correlation observed in the p a r a m a g n e t i c p h a s e results from the small eonncctivity of the B e t h e lattice. In well c o n n e c t e d lattices this correlation in the disordered p h a s e is believcd to persist and the relative s t r e n g t h of thc fluctuation and the free-energy minimization pairing processes is maintained. In conclusion, the magnctic b e h a v i o u r of the dopcdC u O s u p e r c o n d u c t o r s induces attractive h o l e - h o l e correlations. T h e s e correlations increase substantially w h e n frustration is present, which is the case observed for these c o m p o u n d s [4]. T h e s e strong corrclations induced by the free-energy minimisation pairing mechanism may be a f u n d a m e n t a l ingredicnt n e e d e d to explain the high value of T~ observed. In more realistic models the c o m p e t i t i o n b e t w e e n this pairing mechanism and the incipient localisation effect i n t r o d u c e d by the disorder impurities [7] must be carefully analysed.
References [I] D. Vaknin et al., Phys. Rev. Lett. 58 (1987) 2802. Y.J. Uemura et al. Phys. Rev. Lett. 59 (1987) 1045. [2] A. Aharony et al., Phys. Rev. Lett. 60 (1988) 1330. R.J. Birgeneau et al., Z. Phys. B 71 (1988) 57. [3] P.M.C. de Oliveira and T.J.P. Penna, Physica A 163 (1990) 458. [4] Y. Guo et al., Science 239 (1988) 896. [5] R.J.V. dos Santos et al., Phys. Rev. B 40 (1989) 4527. [6] S.B. Cavalcanti, PhD thesis, University of London (1983). [7] M. Crisan, Physica C 171 (1990) 498.
Hole-hole
correlations i n d u c e d by m a g n e t i c frustration
M.L. Lyra and Solange B. Cavalcanti Departamento de Fisica, Universidade Federal de Alagoas, Maceid-57061, Alagoas, Brazil
A randomly decorated Ising model on a Bethe lattice is considered to study hole-hole correlations of CuO2-based superconducting sheets. Two intrinsically different types of pairing mechanism are observed: one, induced by magnetic fluctuations and the other as a consequence of a free-energy minimisation process due to frustration effects so intense that one might conjecture that frustration is an essential ingredient to the understanding of such high superconducting transition temperature displayed by those materials.
Experimental evidence supports the fact that magnetism is an important ingredient to the understanding of the behaviour of C u - O - b a s e d superconductors [1]. Recently, Aharony et al. [2] have suggested that frustration effects introduced by doping a L a z C u O 4 compound should be the underlying mechanism involved in the h o l e - h o l e pairing and in the subsequent superconductivity observed in this material. Monte Carlo simulations have explicitly shown this pairing tendency of holes [3]. The present approach provides, in spite of its simplicity, a clear overall picture of the magnetic behaviour of the doped CuO. It is based on the use of classical Ising spins S, localised on the vertices of a lattice, to represent the Cu magnetic moments of a CuO 2 sheet, interacting with each other antiferromagnetically via the exchange parameter J(J < 0) [4]. Doping creates holes on the O ions, which are placed on the bonds. The holes, with a net ~1 spin o-, interact with the Cu ions through an aJ coupling constant, which leads to an effective ferromagnetic coupling between the Cu ions. The model does not allow two holes on a single O ion. It is believed that this assumption will not compromise the qualitative aspects of the final results. The itinerancy of the holes is considered here by using a grand-canonical distribution for the hole configuration, which means that one is performing annealed averages. This procedure also takes into account fluctuations over the total hole number owing to the hopping of electrons from one sheet to another. The Hamiltonian of the system is then given by:
H = - J Y ' ~ S i S j - a J Y ' . o ' u ( Si + S i ) n i i - lz Y'~nij,
(1)
where the summation runs over all neares-neighbour sites of a lattice and /x is the chemical potential per hole. H e r e n u is a variable, which gives the hole occupation number in the i-j bond. The hole concentration may be thermodynamically
obtained through the evaluation of the mean number of holes per bond:
x=<%>= Enij e e"/F,e = - k B T ~ In ~ / ~ ,
""
(2)
where ~ is the grand partition function and the sum is to be made over all hole and spin configurations. Equation (2) will be used in order to eliminate the hidden variable g. Some exact results related to the magnetic phase diagram of a quite similar model were presented by dos Santos et al. [5], and connections with superconductivity were suggested. The principal of the present work is to look closely at the pairing mechanism involved in the model outlined. The effective interaction between holes can be directly observed through the h o l e - h o l e correlation function:
C( R ) = (nijnkl> -- (nit) z,
(3)
where R is the spatial distance vector between the i-j and k - l bonds. This correlation function may present three different types of behaviour: (a) C(R) = 0, indicating that the holes are completely decoupled; (b) C(R) < 0, meaning an effective repulsive interaction between them; (c) C ( R ) > 0 indicating an effective attraction, which leads to a pairing tendency. The h o l e - h o l e correlation function defined above is evaluated exactly for two neighbouring bonds of a Bethe lattice with very interesting and surprising results, as we shall see in the following. By using the method of ring-recurrence [6] one is able to write the grand-partition function in terms of an effective field for the sth ring, denoted here by Ps, which is defined through a recursion relation of the form:
Ps-I = f ( P , , K, Kerr, Ix),
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
(4)
M.L. Lyra and S.B. Caualcanti / Hole hole correlations
588 0.016
..... .....
0.012
ALPHA ALPHA
-=
1.0 2.0
z 0 I--.<~ LLJ 0.o08 05 Od 0 £P 0.004
i 0.000 0.00
1.00
2.00
3.00
4.00
TEMPERATURE (kBT/J) Fig. 1. Hole hole correlation plotted against temperature at a hole concentration of x = 0.30, for dilute (a = 1) and frustrated ( a > 1) cases, on a Bethe lattice with coordination number q = 4. Note the appreciable correlation enhancement induced by frustration.
where K = J / k B T a n d Keff is the h o l e - m e d i a t e d effective coupling, that is
Kef f = ½ l n [ c o s h ( 2 a K ) ] .
(5)
In the t h e r m o d y n a m i c limit s ~ oo and p.,. --*p* so that one has to solve the fixed-point version of eq. (4) to find its solutions p * ( K , Ket ¢, /,t) for the various phases. In this way the g r a n d - p a r t i t i o n function is readily obtained and, therefore, the t h e r m o d y n a m i c functions of interest may be found exactly. T h e results o b t a i n e d t h r o u g h this p r o c e d u r e are illustrated in fig. 1 where h o l e - h o l e correlation is plotted against t e m p e r a t u r e , for various a a n d for a c o n c e n t r a t i o n of holes x = 0.30. It is observed that, if the effective h o l e - m e d i a t e d interaction b e t w e e n Cu ions is not strong e n o u g h to m a k e the resulting interaction (Jr = y +Jeff) f e r r o m a g n e t i c ( a < 1), t h e n t h e r e is a positive correlation b e t w e e n holes at finite t e m p e r a tures in the o r d e r e d phase. This m e a n s that a magn e t i c - f l u c t u a t i o n - i n d u c e d pairing m e c h a n i s m exists even w h e n frustration is absent. W h e n the resulting h o l e - m e d i a t e d interactions Jr b e c o m e f e r r o m a g n e t i c
( a > 1), the disorder and c o m p e t i t i o n between these and the pure a n t i f e r r o m a g n e t i c ones introduce frustration cffects, which considerably e n h a n c e the h o l e - h o l c correlation, In this case, the correlation has a feature not observed in the n o n - f r u s t r a t e d case: it persists even at T = 0. It is clear evidence that this correlation is essentially induced by a free-energy minimisation proccss. T h e discrepancy b e t w e e n the m e a n a m p l i t u d e of the correlation in thesc two eases reveals the fundam e n t a l role played by frustration i n h t h e hole pairing mcchanism. The case a = 1 is a pathological one because it m e a n s that the system b e c o m e s diluted at T = 0. In this casc thc free energy is not so m u c h Iowcrcd with pairing as in the frustrated onc and the correlation remains wcak. T h c abscnce of h o l e - h o l e correlation observed in the p a r a m a g n e t i c p h a s e results from the small eonncctivity of the B e t h e lattice. In well c o n n e c t e d lattices this correlation in the disordered p h a s e is believcd to persist and the relative s t r e n g t h of thc fluctuation and the free-energy minimization pairing processes is maintained. In conclusion, the magnctic b e h a v i o u r of the dopcdC u O s u p e r c o n d u c t o r s induces attractive h o l e - h o l e correlations. T h e s e correlations increase substantially w h e n frustration is present, which is the case observed for these c o m p o u n d s [4]. T h e s e strong corrclations induced by the free-energy minimisation pairing mechanism may be a f u n d a m e n t a l ingredicnt n e e d e d to explain the high value of T~ observed. In more realistic models the c o m p e t i t i o n b e t w e e n this pairing mechanism and the incipient localisation effect i n t r o d u c e d by the disorder impurities [7] must be carefully analysed.
References [I] D. Vaknin et al., Phys. Rev. Lett. 58 (1987) 2802. Y.J. Uemura et al. Phys. Rev. Lett. 59 (1987) 1045. [2] A. Aharony et al., Phys. Rev. Lett. 60 (1988) 1330. R.J. Birgeneau et al., Z. Phys. B 71 (1988) 57. [3] P.M.C. de Oliveira and T.J.P. Penna, Physica A 163 (1990) 458. [4] Y. Guo et al., Science 239 (1988) 896. [5] R.J.V. dos Santos et al., Phys. Rev. B 40 (1989) 4527. [6] S.B. Cavalcanti, PhD thesis, University of London (1983). [7] M. Crisan, Physica C 171 (1990) 498.