Physica B 163 (1990) 643-646 North-Holland
H O L E PAIRING IN A N E X T E N D E D H U B B A R D M O D E L *
J.H. J E F F E R S O N Royal Signals and Radar Establishment, St. Andrews Road, Malvern, Worcestershire, WR14 3PS, UK Degenerate perturbation theory is used to transform an extended Hubbard model, which includes on-site and inter-site Coulomb repulsions for Cu and O, into an effective Hamiltonian for the low-energy phsyics. For typical estimates of the energy parameters, it is shown that intermediate states with holes excited onto Cu and O ions are likely to be of comparable importance with competing attractive and repulsive effective hole-hole interactions having similar magnitudes. In particular, the condition for an attractive interaction due to near-neighbour Coulomb repulsion is less stringent than in the atomic limit. Other Coulomb terms tend to inhibit pairing, in agreement with numerical simulations.
Recent work on the mechanism of pairing in high-temperature superconductors has focussed attention on extended Hubbard models, which model electron correlations in the C u O : planes common to all the higher T c materials. Such models, whilst looking deceptively simple, have proved extremely difficult to obtain reliable solutions for, particularly at low energies which is the regime relevant to superconductivity. Amongst the many analytical and numerical techniques which have been brought to bear on this problem, one approach has been to eliminate high-energy states by perturbation theory, thereby generating effective Hamiltonians in various limits. Provided there is sufficiently rapid convergence, these transformed Hamiltonians will, in low order, have essentially the same low-energy spectra as the original. Whilst this technique does not (usually) lead to an explicit solution of the problem, it can give considerable insight into the low-energy physics by generating (or renormalising) effective interactions which in turn might suggest appropriate approximations. From the numerical viewpoint, such a technique has the added advantage of reducing the size of the Hilbert space enabling, for example, larger clusters to be diagonalised explicitly. Examples of this technique which are relevant to the high-T c problem are the equivalence of extended and single band Hubbard models (for the unphysical limit of very deep oxygen levels) , the equivalence of the single band Hubbard model to the Heisenberg model for the half-filled band (Mort insulating) case and the so-called t - J model when the band is not half full . The equivalence of the extended Hubbard model to the Heisenberg or t - J models does, however, remain controversial with both analytical work [4, 5] and numerical simulations [6, 7] suggesting that different behaviour could result for certain parameter ranges, though these may be unphysical. In this paper we shall explore this question further by performing a perturbation expansion about the atomic limit and show that, for recent estimates of parameters, interactions due to both charge polarisation and spin fluctuations are likely to be important. A general extended Hubbard model, which contains all on-site and nearest-neighbour interactions between holes in 3dx2_y2 orbitals on copper ~r-bonded with 2px (2py) orbitals on oxygen, may be written :
~¢(= ~'~ [%n i + Udnitni+ ] + ~ [%nj + Upni?ni,~] + Vdp Z nln] + Z [td+~pj~ + Hc] i
+ ~ [t pPJ~Pr~ + + He] <#,)~
where % and ep are atomic-like energies in localized (Wannier) orbitals of Cu and O; U d, Up and Vdp are matrix elements for on-site and nearest neighbour Coulomb interactions, with t and tp n e a r e s t neighbour ( C u - O and 0 - 0 ) hopping matrix elements. The second quantised operators have their usual meaning with i and j denoting Cu and 0 sites, respectively, the angular brackets restricting the summations to nearest neighbour pairs. In the atomic limit (t = tp = O) the Hamiltonian is diagonal and for the cases of interest there will be a ground manifold of * See ref. . 0921-4526/90/$03.50 ~) Elsevier Science Publishers B.V. (North-Holland)
J.H. Jefferson / Hole pairing in an extended Hubbard model
degenerate states which are sufficiently separated from excited states for a perturbation expansion in the hopping to be valid. Denoting the projection operator for the ground manifold by P and the hopping (last two terms in (1)) by V then, as far as the low-energy physics is concerned, the Hamiltonian 9( is equivalent to the following effective Hamiltonian given by degenerate Rayleigh-Schr6dinger perturbation theory : 2( . = P [ H + V R V + ( V R V R V + VR3VPVPV) + •.. ] P
VRZVPV) + (VRVRVRV-
and Ygeff only operates only in the restricted low-energy subspace defined by P. For the Mott insulating case, in which there is exactly one hole on each site in the atomic limit, the effective Hamiltonian is equivalent to a Heisenberg Hamiltonian (dropping constant energy shifts):
J-(,+vdp) 2 ~ + , + and ff =
£p -- £d "
Note that in this lowest non-trivial order of perturbation theory the nearest neighbour O - O hopping does not contribute and all Coulomb terms inhibit the exchange. The two contributions arise from intermediate states with two holes on Cu and O, respectively. A typical estimate of the energy parameters using density functional theory is : t = 1.3, E = 3.6, Vap = 1.2, U a = 10.5 and Up = 4 (eV), showing that the contributions are of comparable order and that J compares favourably with the temperature (-1000 K) at which 2D antiferromagnetic correlations disappear, as measured by neutron scattering. When further holes are introduced into the CuO 2 planes (which is achieved experimentally by doping or increasing the oxygen content) the energy parameters are such (Uj > e + 2Vap ) that they reside on oxygen sites, as confirmed by XPS and EELS experiments . Now for Vap > E charge clustering can occur in the atomic limit, as has been pointed out by a number of authors [6, 7, 12]. This is easily proved by energy balance arguments which show that two or more holes on neighbouring oxygen sites can lower the Coulomb energy by transferring holes from Cu to O. Such 'polarisation' of the Cu sites was originally suggested by Varma et al. . Numerical simulations on small clusters confirm this effect and further show that the kinetic energy terms can stabilise pairing (rather than charge clustering) provided Vap is not too large . In this paper we shall consider this polarisation effect as virtual (in the sense of perturbation theory) since the evidence is that Vap is unlikely to be sufficiently large to give real polarisation in the atomic limit. For the recent estimate of energy parameters given above, such a perturbation expansion should converge reasonably rapidly. The degenerate ground manifold consists of all states in which the Cu sites are singly occupied with the O sites either unoccupied or singly occupied. The first order effective Hamiltonian restricted to this subspace is, from (1) and (2), Yf(e~f~ = P
[ t p p ; , p r , + Hc] e
which is just a single-band Hubbard model for the oxygen holes in the Up = ~ limit. In second order the O charge and Cu spin degrees of freedom are coupled. Some care is needed in deriving these expressions since the self-energy of Cu-holes is changed by the presence of O holes due to the Coulomb repulsion Vdp. Allowing for the possibility of pairing of holes on near-neighbour O sites, the second-order effective Hamiltonian is, from (2), (2) (2) •(2)elf = P[YC~in~l~ + Y~i;] P
J . H . Jefferson / Hole pairing in an extended H u b b a r d model
A K ( s i • s i - ¼)1 n/n/, +
[dir~ p i ¢ n i ~ p f ¢ - A r 2 p m n i . , p j , (ijj'j")o
+ A r p ; ~ d + _ d,~p/, ~ + H c ] n j n f n f , with
rl = --E '
diE = 2t2[ !
Ud _ 2Vdp _ ~ ,
e + vdp
1 1, ~ --vdp
r = r~ + r 2 ,
K = 2t 2
U d -
1 1 E+ u~- v~ + U.- v~-
] -- E
Ar 2 --
r 2 and
dir = Ar 1 + Ar 2 .
Again the angular brackets mean that all O sites ( j , j ' , j") shall neighbour a central Cu site (i). 2(
E/3, which is a much less stringent condition for attraction than given earlier for real polarisation in the atomic limit. For smaller Vdp, diE is positive and inhibits pairing. Coincidentally, the energy parameters of Hybersten et al.  are exactly on the boundary of attraction and repulsion! The antiferromagnetic exchange is reduced (diK < 0) by the presence of a second hole and is thus an effective repulsion. This can be quite large due to the smallness of the energy denominator U d - 2Vdp -- e, (even though U d is large). It does, however, become negligible for very large Ud, which is consistent with numerical simulations which show that large U d favours pairing. Note also that there will be small corrections to the O Hubbard band in the Up = o0 limit described by )~(eld giving rise to antiferromagnetic exchange and next nearest neighbour hopping of order t p2/ U p . One can draw similar conclusions from a study of higher order terms in the perturbation expansion which also generate new interactions of longer range. For example, in third order there will be Cu self-energy terms of order t Z t p / ( E - Vdp) which again favour nearest neighbour pairing, whilst in fourth order there will be next nearest neighbour pairing interactions such as enhanced superexchange, as discussed by Emery and Reiter . In summary, we have derived a perturbation expansion for the low-energy states of a general Hubbard model and shown that for recent estimates of energy parameters, both on-site and nearest-neighbour Coulomb interactions are likely to be important in determining contributions to real space pairing. It is emphasized, however, that these results are only suggestive since the criterion for pairing (Vdp > e/3) does not take into account the hopping terms in the effective Hamiltonian which are, a posteriori, of the same order of magnitude.
References  A preliminary account of this work was presented at an Institute of Physics meeting, Magnetism and Superconductivity, London, UK, September 1988 (unpublished).  C.M. Varma, S. Schmitt-Rink and E. Abrahams, Proc. Int. Conf. on Superconductivity, S.E. Wolf and V.Z. Kresin, eds. (Plenum, New York, 1987) 355.
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J . H . Jefferson / Hole pairing in an extended Hubbard model
J.E. Hirsch, Phys. Rev. Lett. 54 (1985) 1317. V.J. Emery and G. Reiter, Phys. Rev. B 38 (1988) 4547. J.H. Jefferson, J. Phys. C 21 (1988) I~ 425. C.A. Balserio, A.G. Rojo, E.R. Gagliano and B. Alascio, Phys. Rev. B 38 (1988) 9315. J.E. Hirsch, S. Tang, E. Lob and D.J. Scalapino, Phys. Rev. Eett. 60 (1988) 1668, ibid., Phys. Rev. B 39 (1989) 243. V.J. Emery, Phys. Rev. Lett. 58 (1987) 2794. C. Bloch, Nucl. Phys. 6 (1958) 329. M.S. Hybersten, M. Scluter and N.E. Christensen, Phys. Rev. B 39 (1989) 9028. J.C. Fuggle, P.J.W. Weijs, R. Schoorl, G.A. Sawatzky, J. Fink, N. Nucker, P.J. Durham and W.M. Temmernam, Phys. Rev. B 37 (1988) 123.  S.A. Trugman, Physica Scripta T 27 (1989) 113.  J.H. Jefferson, Ann. Phys. Fr. 13 (1988) 425.  A. Aharony, R.J. Birgeneau and M.A. Kastner, Prog. in High Temp. Supercond. 14 (1989) 113.