Physica B 163 (1990) North-Holland
P.B. LITI-LEWOOD AT&T
Murray Hill, New Jersey 07974, USA
Invited paper Some general arguments affecting the likelihood that and a calculational framework is presented. Simple candidates, as are intraband charge fluctuations nearby are argued to involve strongly-localised interband, or A,,-symmetry mode is discussed in detail
In the last few years, innumerable mechanisms have been proposed for the copper-oxide superconductors. Many of these mechanisms may be termed “conventional”, in that they involve an exchange of a boson (usually an electronic pair excitation) inducing attraction between quasiparticles near the Fermi surface. The boson may be exotic, but the pairing is conventional. In the light of increasing experimental evidence that the superconducting state is BCS-like, and that the normal state possesses a Fermi surface, the “conventional” mechanisms look increasingly attractive. However, should the pairing be conventional, and also weak-coupling, there are no “smoking-gun” experiments on the superconductor that will explicate the source of the attraction. Also, many of the more prosaic excitations (magnons, plasmons, excitons etc.) are likely to be present anyway, but may be mere spectators to the pairing. As many different collective electronic excitations can be envisaged, it is worthwhile considering why high-temperature superconductivity is not ubiquitous. However, generating a plausible collective excitation is far from enough. While the use of second-order perturbation theory guarantees that there will be some attraction, there is usually enough repulsion to go around; conditions of crystalline stability will always enforce the “net” interaction (in a sense that will be defined more carefully below) to be repulsive. Even if this can be surmounted and some attractive channel found, the failure of Migdal’s theorem (since the energy scale of the collective modes is likely comparable to the quasiparticle bandwidth) means that vertex corrections (usually injurious) are not negligible. In this paper, I shall attempt to give some general arguments related to charge fluctuation mechanisms 0921-4526/90/$03.50 (North-Holland)
charge fluctuations can promote superconductivity are discussed, exciton and plasmon mechanisms are suggested to be unlikely a charge-density wave instability. The most successful mechanisms charge-transfer excitations, and a specific model of a low lying
within this conventional approach to superconductivity. We envisage a basis with several atomic orbitals per cell (labelled cr, /?, etc.) and an interaction V between Bloch states PCs,, that can be written (see fig. 1).
V(0, ak + q; CQk; u$‘k’; =
u‘$,a’k’ + q)
k + q)&.(k’> k’ + 4)
Here we have introduced a set of basis functions gb, for particle hole pairs in real space, and also separated the interaction into charge VP and spin V, channels (a)
(b) u,ak+q ma8k
dOd k Fig. 1. (a) T-matrix equation for the renormalized interaction F within the time-dependent Hartree-Fock approximation. (b) Definition of the bare interaction V. (c) Particle-hole polarizability matrix entering eq. (4).
P. B. Littkwood
I Charge fluctuation
[l-3]. Equation 1 is completely general, but the factorization is useful principally if the interactions are short-ranged, so that the dimension of the representation is small . A typical scheme now involves determining the effective interactions I’ at low energy by including virtual excitations to higher energy states. r must take the same form as V in eq. (l), and can therefore be separated into both charge and spin parts, and there is then a relation I’:,(&
w) = c
Equation 2 is no more than a definition of a matrix dielectric function D( q, co). If we had chosen a planewave representation, this would have been the more familiar r(q+G,w)=CF-‘(q+G,q+G’,w)V(q+G’), G’
where we have suppressed the indices and the q = (q, w) dependence for clarity. P is simply the bare particle-hole susceptibility in the basis of the g’s (see fig. 1). The matrix D determines all the particle-hole collective modes of the system, since the condition D = 0 allows for free oscillation. Thus, if we diagonalize D by solving the eigenvalue equation DIV=A’J’
the electronic where structure changes. (For example A<,(q, w = O)+ 0 implies a spin-density-wave instability, or itinerant antiferromagnet, at wave vector q.) We note also that D is not necessarily Hermitian, except at w = 0, so the eigenvalues may be complex and the collective modes have a finite lifetime. With some scheme for calculating D, we then take the renormalized interactions r in the particle-hole channel, and use them as the bare interactions in the particle-particle channel. The neglect of interference between the particle-hole and particle-particle channels should be a good approximation provided the superconducting transition temperature T_ is much smaller than the quasiparticle band width, and also much smaller than the typical energies of particle-hole fluctuations. Summing ladders for the T-matrix in the particle-particle channel gives an instability at the BCS transition temperature T, and a solution for the gap function 4
involving a sum over reciprocal lattice vectors G. The advantage of the local representation we use is simply that the interaction is truncated in real space rather than reciprocal space, and the former seems more natural. Moreover, the simplest approximation for D, namely time-dependent Hartree-Fock (or RPA + ‘*ladders”) comes out simply as [2, 31 D ,>.‘T= 1-2vp <,P
then the collective mode dispersion relation w(q) is given by h( q, o(q)) = 0. The symmetry of the collective mode is determined by the eigenvector ly and the basis functions g’; there may of course be several different such modes, depending on the problem. There is an important stability condition that must be satisfied, namely that at zero frequency, all eigenvalues of Do,<, must be positive; a vanishing eigenvalue in the static limit implies a “soft mode” instability
The where w,,, v,, are fermion Matsubara frequencies. subscripts S, T refer to the singlet and triplet channel, respectively, and L’, = I’, - 3$,, f, = I;, + $, . One may also easily extend this approach to develop Eliashberg equations, where the quasiparticle selfenergies are self-consistently included. Equation 6 admits of solutions of different symmetries, compatible with the space group of the underlying crystal. These equations are general, and not specific to any particular mechanism. We now turn to a discussion of some specific examples, concentrating specifically on charge fluctuation mechanisms. The simplest case to consider neglects local field corrections entirely (so that the above equations are all scalar) and leads us to consider a single dielectric function with the model form 
For example, for the electron gas in the RPA, we have and wq = V%+,qiq,, fj,, = wp, the plasma frequency, marks the upper edge of the particle-hole continuum ( qTF is the Thomas-Fermi screening wave vector).
P. B. Littlewood
I Charge fluctuation mechanisms for high-temperature superconductivity
For w < oq the dielectric function has also an imaginary part. The effective interaction is proportional to E-’ (eqs. (2), (3)) and this is drawn in fig. 2; the effective interaction is attractive for frequencies in the range wq < w < (0: + fi;)“‘, and repulsive at low frequencies. Only at q = 0 is the interaction attractive down to the lowest frequencies. A similar formula to eq. (7) will apply if there is a transverse exciton, where now wq = w_ the exciton frequency. Of course, to have such an excitation present will require more than the single band that we have been considering so far; in principle we should deal with a matrix dielectric function at this stage. However, the exciton is often envisaged as arising from transitions not involving the electronic states that participate in the pairing, as in models of two-dimensional metallic sheets separated by semiconducting regions where virtual excitons abound . The separation of the excitons from metallic regions is generally thought appropriate because the particle-hole pair will then be bound by the long range part of the Coulomb interaction, which must therefore be not too heavily screened in order for the bound excitonic state to exist. While this is indeed the case, it is of secondary importance and we assume here that the system conspires to generate such a bound state. We do note one thing in particular ; the effect of a transverse exciton is to make the low energy interactions more repulsive than in the RPA case. A transverse (i.e., optically active) excitation corresponds to a pole in the dielectric function and the effective interaction is thus always repulsive at frequencies < w,,; since attraction at low energies is expected to be favorable for superconductivity it would seem that dipole-active modes are to be avoided. These arguments discriminate against low-q transverse modes only; at large
Fig. 2. Renormalized interaction dielectric function of eq. (7).
w) with the model
momentum, where the strict selection rules break down anyway, there can indeed be coupling to these excitations so as to yield attraction. However, such a coupling will appear only if local field corrections are kept, i.e., the matrix character of the dielectric function is retained. This physics is absent from eq. (7) where the transverse modes lead only to repulsion. All the attraction in eq. (7) is in fact coming from overscreening by the plasmon. The plasmon mechanism was considered in detail (for a three-dimensional metal) by Grabowski, Rietschel and Sham  who found that the simple RPA approximation does indeed yield a large transition temperature. However, the inclusion of vertex, self-energy and spin-fluctuation corrections was found to lower Z’, drastically; for the specific plasmon mechanism used, no superconductivity was found . Part of the reason for the ineffectiveness of plasmons is that the static interaction is invariably repulsive, and T, is determined as much by the frequency structure of the interaction as by its magnitude. The deleterious vertex corrections are negligible for low boson energies, but in this regime (L?JE, < 1) the RPA does not yield a positive T,. (This arises partly because the low frequency repulsion enforces a node in the gap function 4(w), and partly because the Fermi energy and the plasma frequency are not independent variables.) We doubt that the situation will be much different for an array of two-dimensional electrostatically coupled sheets; although there are here some regions of momentum space where the plasmons soften, the interaction will be mostly dominated by the large volume of q-space where the plasmons are hardly different from the three-dimensional case. The simple cases of plasmons and dipole-active excitons are not the only possibilities in a single band model; but in a general multi-band case more complicated collective excitations can arise, providing the full local-field character of the dielectric matrix is kept. These modes will in general involve redistribution of charge from one orbital (or atom) to another in the unit cell, and they are the direct analog of optical phonons in crystal structures with a basis. The electron-hole pairs are bound by the short range parts of the Coulomb interaction, so the long-range metallic screening need not be a problem. The existence of such modes will be favored in cases when there is strong covalency, so that there is some density of states on two different atoms close to the Fermi surface, and hence the charge transfer energy is low. A large band width will reduce the screening so that near-neighbor Coulomb interactions may be large;
P. B. Littlewood
I Charge fluctuation
thus there will be also a considerable ionic character to the bonding. These characteristics are found in both the copper-oxide superconductors and the BaBiO, family, where there is a broad band of strongly hybridized 0 2p and Cu 3d or Bi 6s states. The simplest model Hamiltonian for CuO, planes that includes this physics is an extended Hubbard model defined in a tight-binding basis of dXz_,’ orbital for Cu and the (q)p, orbital on one of the two 0 atoms in the unit cell and the (a)p y orbital on the other [lo]:
. E,(x) t
+ H.c. + x+
+V c Snd,(Sn*, (‘1)
Fig. 3. Different collective model of CuO, planes.
+x-y) + $,)
Here E = 4 (E, - E,), tz, = k t, and we have included a repulsive U on Cu and 0 as well as the nearest neighbour repulsive V. The notation (ij) specifies nearest neighbor summation and we have both a direct Cu-0 overlap t as well as a nearest neighbor O-O overlap t’. The band parameters E and t are defined in the Hartree-Fock approximation, and 6n, = n, ~ (n,). The rewriting of the interactions according to eq. (1) is given elsewhere, along with detailed calculations using the time-dependent Hartree-Fock approximation for the collective modes [2,3] and superconductivity [3, 111. Similar studies to this have been reported by others , and this model has been studied by other techniques [ 131. The symmetry of the allowed collective modes is easy to determine at q = 0, by analogy with phonons, and some of the allowed modes are shown pictorially in fig. 3. We show a mode of A,@ symmetry, which involves charge transfer between Cu and 0 preserving the full symmetry of the square and there lattice; the B,, mode has d,z_,,Z character are two optically active Eu symmetries. The energies of these modes, and their existence as well-defined oscillations depend in detail on the parameters of the Hamiltonian; in a metal there are always decay channels for these modes (for example a 4 = 0 mode may decay into two particle-hole pairs involving intraband excitations at large and opposite momenta). Con-
+ c (t,,c& (f, 1
for a simple three-band
sequently, the modes may become resonances, but their oscillator strength is conserved. The spectral function for these charge transfer resonances (CTR) is obtained by calculating Im[D,,( q, co)] projected onto the various symmetries, and this decomposition is given elsewhere [2,3]. It was found that for moderate values of V, charge fluctuations of A,r symmetry were pulled down to low energy; in the starting band structure, these would be at an energy -2(E? + 4t2)‘!2 that is the splitting between the bonding and antibonding dpo states near half-filling. An equivalent approach is to calculate the static charge transfer energies within Hartree-Fock, and argue from this that there will be a low energy mode of A,, symmetry if V is large enough. In the Hartree-Fock Hamiltonian, the energy levels are already renormalized so that F is given by E = F,, + $n[U,, - 8V] + n,(&)[8V-
U ~ ;U,,]
Here we are counting holes so that n = (1 + 6 ) is the total number of holes per unit cell (counting from the d”‘ph configuration) and F(, refers to the bare energies for holes. nd is the fraction of holes on Cu, which depends implicitly on the renormalized F; to a good approximation this can be parametrized by the function
depends only weakly on n and t’ (prowhere Ed -2t vided the latter is not too large). Notice that nd(s-+ - m)+O, n,(~-+m)+ II, and at F = 0. close to
P. B. Littlewood
I Charge fluctuation mechanisms for high-temperature
the value extracted from band structure, we have n,(O) = in. Because eq. (9) is nonlinear, it may admit of more than one solution; near E = 0, there will be two solutions if
(11) This instability corresponds to a critical point for E = 0 and the phase diagram is shown in fig. 4 for Ed = 2 and n = 1.2; the dashed lines are spinodals, within which there are two solutions for E, and the solid line marks a first order transition where the energies of the two solutions are the same. As one approaches the critical point, there is a soft mode of A,, symmetry whose frequency vanishes continuously, whereas within the spinodal region the total energy has a double-well potential and the assumed valence becomes unstable toward charge transfer from Cu to 0 or vice versa. A phase diagram with identical structure is also found
6 s 4
\ \ \ \ \
Fig. 4. Phase diagram of the Hamiltonian of eq. (8) calculated within the Hartree-Fock approximation, for E,, = 2, t’ = 0 and n = 1.2 at zero temperature. The solid line marks a first order transition, and the dashed lines are “spinodals” which mark the limits of local stability of the metastable valence state.
within the Gutzwiller approximation  although here the phase boundary is not pushed to large values of V when the onsite interactions U are large , as implied by eq. (11). Both Hartree-Fock and Gutzwiller are mean-field theories, and thus we cannot be sure that the predicted first order transition is real; alternatively it may imply a regime of strongly-fluctuating valence. Close to the critical point it is unlikely that a quadratic theory for the fluctuations is appropriate, but we can nevertheless extract the relevant physics by simple arguments. By inspection of fig. 3 it is easy to see that a low energy A,, mode is an attractive candidate for mediating superconducting pairing. We assume an incoming particle on a Cu atom, which will then excite the A,, symmetry mode; thus charge will be locally depleted from the Cu site, and pushed symmetrically onto the 0 neighbors as in fig. 3. The interaction of a second particle of like charge with the charge redistribution induced by the first gives a contribution to the renormalized interaction Sra, between orbitals (r, /3. The second particle will experience extra attraction to the Cu site (6r,, 0, i.e., an increase in the already repulsive V). This argument can be reworked with the initial particle on the 0 atom, and will yield ST,, < 0 (p = x or y) and Sr,, < 0, so that there is attraction induced between quasiparticles on the same, and neighboring, oxygen atoms. Although this argument is here given explicitly only for a q = 0 mode, the localized character of the collective modes suggests that the physics is not strongly q-dependent. These results for the effective interaction (which are confirmed by detailed calculations [3,11]) demonstrate an important point; the attraction developed in some channels comes only at the expense of enhanced repulsion in others. This conclusion seems to be unavoidable within charge fluctuation models, because of charge conservation in the planes and also the requirement that the dielectric matrix be positive definite at zero frequency. Because of the competition between repulsion and attraction in different channels, whether the net interaction is attractive or repulsive depends on the relative weights of different orbital character near the Fermi surface and demands a detailed calculation. Superconductivity of “extended-s” character (i.e., a pair wave-function of A,, symmetry but with a real-space node on the Cu atom) is found [ll] to be favored here when there is a large 0 character to the states near E,; the node in the pair wave-function supresses the energy cost of both the repulsive U on
P. B. Littlewood
I Charge fluctuation
Cu as well as the Q-0 repulsion V. However, the near-cancellation of repulsion and attraction is so close that sizeable 7’,‘s are only obtained close to the regime where the mode is soft; here though the transition temperatures are found to be large, with T,l t - 0.1 in the RPA. A side-effect of the near-cancellation of attraction and repulsion in mechanisms of this type is that selfenergy effects are naturally large (interactions of either sign give additive corrections to the self-energy, whereas only the net interaction is important for superconductivity). The self-energy corrections will reduce T, by at least an order of magnitude. However, since there is indeed static attraction at short distances, Migdal-violating vertex corrections should be less deleterious than for the plasmon mechanism, where the ony attraction is found at finite o , although no calculations have addressed this point. The CTR model [lo] illustrates the difficulties that are likely to be encountered by most charge fluctuation mechanisms, and that will be fatal for many of them. For example a likely instability in some models is to a charge-density wave ground state; setting aside the coupling to phonons. this will appear as a zero eigenvalue of D,( q, 0) at some non-zero wave vector, often controlled by nesting of Fermi surfaces. For a weak CDW induced by nesting, there is strong qdependence of the soft modes. Since the superconductivity kernel in eq. (6) averages over all momenta (a Fermi-surface average alone is inappropriate because the interaction scale is comparable to the band width) the attraction, though present for some q, is weak and easily overwhelmed by the majority of momentum space where the interactions are unambiguously repulsive [S]. For the doped BaBiO, systems, there is clear evidence for CDW order, but the energy gap is large and the transition is clearly not of the nesting variety. Both here and in the models for CuO, it is clearly necessary to develop treatments within the regime of strong fluctuations close to the critical point of the HartreeFock solution. We remark that expanding about the unstable solution within the mean field theory, but now treated as a local approximation, leads to a “negative-U” model, whose relevance to the hightemperature superconductors has been suggested by Varma . We conclude that the constraints on charge fluctuation mechanisms of superconductivity are severe, as already noted some time ago , and while it is indeed possible to generate a model interaction with the
it is not an easy business. necessary characteristics, The use of strong local field corrections and the short range part of the Coulomb interaction appears crucial in generating attraction for quasiparticles on the energy shell, while at the same time maintaining the structural integrity of the solid. The most successful models involve localized charge fluctuations and a strongly-hybridized broad single-particle band; such a situation arises rarely, but may be realised in both the copper-oxide and BaBiO, family of superconductors.
Acknowledgements This work owes much to collaborations and conversations with E. Abrahams, S.N. Coppersmith, S. Schmitt-Rink and C.M. Varma.
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I Charge fluctuation mechanisms for high-temperature
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