CeCu ,(tore)
4. Superconductivity in heavy-fermion metals
4.2. Charge fluctuations
Pairing due to charge fluctuations has been tackled from two angles: (1) pairing via f density fluctuations in a slave-boson scheme (Auerbach and Levin 1986, Lavagna et al. 1987, Zhang and Lee 1987, Newns 1987, Kotliar and Liu 1988), and (2) pairing induced via a quadrupolar Kondo effect (Cox, unpublished).
In the first method, the Landau scattering amplitudes are calculated up to order
1/N
in a slave-boson scheme. The1/N
terms represent hybridization (charge) fluctua- tions. From this, it was found for the Anderson lattice by Lavagna et al. (1987) that a weak instability occurred in the l -= 2 (d wave) channel. Soon after, though, Zhangand Lee (1987) showed that by including the anisotropy of the Coulomb repulsion, the net interaction in the even parity channels was repulsive. Newns (1987) proposed a more generalized model that included more degrees of freedom (fo, fl and f2 charge fluctuations), and estimated a reasonable transition in the s-wave channel for CeCu2Si 2. An alternative approach has been to use a Hubbard model instead of an Anderson model. From this, Kotliar and Liu (1988) found either p-wave or d-wave instabilities dependening on the band filling for a cubic lattice.
At this stage, it should be remarked that, as to be discussed in the spin-fluctuation section, inclusion of 1/N 2 terms in the Anderson lattice theories leads to a qualitatively different picture. Moreover, Millis (1987) has shown that the normal state T 3 In T correction to the specific heat in these theories are two orders of magnitude too small to explain the experimental data in U P t » In the Hubbard model approach, Rasul (1991) has shown that order 1/N 2 corrections actually lead to a repulsive interaction, so conclusions at the 1IN level must be regarded with some suspicion.
Another possible pairing mechanism involves shape fluctuations of the charge.
Quadrupolar interactions are known to be important in many f electron systems, but the investigation of these effects in heavy-fermion metals is still at an early stage.
One possible scenario is an exchange of quadrupolar fluctuations, but such a theory has not been developed to the authors' knowledge. Cox (1987) has suggested that some of the unusual properties of metals such as UBe13 may be understandable in terms of a quadrupolar Kondo effect. Recently, he has suggested that this interaction can induce local on-site pairing (Cox, unpublished). Symmetries of gap functions for various heavy-fermion superconductors were determined, although these assignments depend crucially on the assumed partial waves of the conduction electrons and the assumed crystal field (doublet) ground state of the U ion. Although this theory shows great promise, it is still at too early a stage of development for us to judge its ultimate success.
4.3. Spin fluctuations
Since the discovery of the uranium-based heavy-fermion superconductors, it has been suspected that spin fluctuations are the pairing mechanism (Anderson 1984).
Over the years, this belief has been re-enforced by the observations that followed in U P t » UBel» URuaSi2 and CeCu2Si2, and more recently in UNi2A13 and UPd2AI»:
(1) nearness to a strong magnetic instability, which can be induced by doping, (2) coexistence of small moment magnetism and superconductivity, and (3) anisotropic superconductivity (nodes in the gap function), along with strong evidence that the order parameter is coming from a non-trivial group representation (UPt 3 and UBe 13).
All these properties bear a strong resemblance to 3He, which can be described with a spin fluctuation pairing model (Levin and Valls 1983).
The first approach taken was to describe heavy fermions in an analogous way to 3He, that is by estimating Landau scattering amplitudes and using this to estimate Te, calculate the pressure dependence of normal and superconducting state properties, etc. Early models, based as they were on 3He methods, naturally led to p-ware pairing (Valls and Tesanovic 1984, Bedell and Quadar 1985, Fay and Appel 1985, Pethick
70 M.R. NORMAN and D.D. KOELLING
et al. 1986). In particular, Pethick et al. (1986) were able to demonstrate that their calculated T c decreased with pressure for U P t » just as seen experimentally, even though their effective Fermi energy increases with pressure (which it must since 7 decreases with pressure). Despite this nice success, basic problems exist with these theories: they are effective one band models which (t) ignore the multi-sheeted nature of the Fermi surface, (2) ignore the degeneracy and spatial anisotropy of the f levels, and (3) do not include spin orbit coupling. The problem is quite important, since neutron scattering up to now has not seen the quasiparticle (intraband) contribution to the susceptibility, which for 3He saturates the susceptibility. What is seen is a Van Vleck part which is outside the bounds of these theories. Moreover, in 3He, 7 scales with the logarithm of Z, whereas in heavy fermions, y scales with X. As discussed previously for U P t » such a difference can be accounted for by taking into account the Van Vleck nature of the susceptibility (Konno and Moriya 1987, Norman 1988).
A very lucid discussion of the qualitative differences between 3He and heavy fermions can be found in Varma (1985).
An alternate Fermi liquid approach is to treat the problem in the slave-boson scheme. Spin fluctuation can be recovered either (1) by going to order 1IN 2 (Houghton et al. 1988, Kaga and Yoshida 1988, 1989) in an Anderson lattice model, or (2) by expanding around a saddle point which reproduces the Gutzwiller mean field solution (Kotliar and Ruckenstein 1986) in the Hubbard model. In the first method, the spin fluctuations show up as an R K K Y interaction [related approaches are discussed by Zhang et al. (1987), Ohkawa and Yamamoto (1987), Evans and Gehring (1989) and Coleman and Andrei (1989)]. In the second method, the spin and charge fluctuations both appear at order 1IN. This model does well for both normal-state properties (Rasul and Li 1988) and superconducting-state properties (Rasul et al. 1989) of 3He, so an application to heavy-fermion systems would be highly desirable. The basic conclusion from both methods is that (1) the spin fluctuation part dominates over the charge fluctuation part for pairing, and (2) either p-wave or d-wave pairing is possible depending on parameters. The basic problem with these theories is that they have not been applied to a realistic model of a heavy-fermion superconductor, so their ultimate success remains to be determined.
The final approaeh used for spin fluc~uations is a semi-phenomenological theory which takes the observed dynamic susceptibility, uses this as a model to construct a pair potential, and then tries to solve for To for real systems. This approach was first advocated by Miyake et al. (1986), who observed that the spin fluctuations in heavy fermions were of an antiferromagnetic nature, as opposed to the ferromagnetic nature seen in 3He. This leads to d-wave pairing for a simple lattice [-see Scalapino et al.
(1986, 1987) and Béal-Monod et al. (1986) for related approaches]. A nice review of the physics behind these models can be found in Pethick and Pines (1987), which is highly recommended reading.
The linearized gap equation in this model is A(k) = ~ V(k - k')A(k')/~k, tanh(~k,/2Tc),
where A is the gap function, e the quasiparticle energies, and V the pair potential (assumed to be proportional to )0. The approach was then generalized to include the
frequency dependence of the pair interaction by Norman (1987, 1988) and by Millis et al, (1988), which is necessary to set the temperature scale for To (Millis et al. also discuss the importance of pair-breaking effects in anisotropic superconductors). Using a simplified model of the susceptibility and the Fermi surface, order parameters were calculated for both UPt 3 and UBe13 by Norman. The actual observed momentum dependence of the dynamic susceptibility for UPt 3 (Aeppli et al. 1987, Goldman et al. 1987, Frings et al. 1988) was fit by Putikka and Joynt (1988, 1989) and then used to determine an Elg (d-wave) gap function for hcp UPt 3 (this susceptibility has an energy scale of 5meV and is due to antiferromagnetic correlations between near- neighbor atoms in neighboring planes). Such a gap function has both a nodal structure (line and point nodes) and a group representation (two-dimensional) consistent with a lage body of experimental data for UPt 3 in the superconducting state.
Unfortunately, at this point, the actual complexities of UPt 3 became apparent.
UPt 3 is a non-symmorphic lattice (two atoms per unit cell, separated by a non- primitive translation vector). Because of this, the observed dynamic susceptibility is not invariant under reciprocal lattice translations (ignoring form factors, it has a periodicity of two reciprocal lattice vectors in the c direction and three in the basal direction, due to the non-primitive translation vector). Using this susceptibility in a gap equation, then, gives gap functions which are not properly lattice periodic. Thus both the solutions of Norman and of Putikka and Joynt are invalid.
Several ways have been proposed to get around this problem, none of them completely satisfactory. The susceptibility of Putikka and Joynt was of the form U +JRe(~b) where Ó(q) is equal t o ~ e iq'r with r being vectors connecting near- neighbor atoms to the first U site in the unit cell. Only the real part enters since 4)*
is obtained when summing relative to the second site, and thus the total sum in the cell is 4, + q~* =2Re(~b). Since Re(p) is not invariant under a reciprocal lattice translation, the first proposal was to generalize the gap equations to include the umklapp processes (Norman 1989, Monien and Pethick, unpublished)
A (k) --- - ln(1.13 F / T c ) ~ V ( k - k' - Q ) A ( k ' )
where k and k' are restricted to the first zone, Q are reciprocal lattice vectors, and F is the neutron scattering linewidth (energy scale). The cut-off in the Q sum is achieved by including the form factor of the f electron in the suceptibility. The calculated gap function is of the Axu (p-wave) representation (line nodes), with the second highest T c solution being in the Exù (p-wave) representation (point nodes), where it was assumed that the order parameter d vector ( d ' S = O, S being the spin vector of the Cooper pairs) was orientated along the c axis. An odd parity state is obtained since the effective pair potential peaks at q = 0 (X peaks at a reciprocal lattice vector, which is an umklapp scattering from q = 0). In fact, the susceptibility in such a formalism should be a matrix in reciprocal lattice space, so the above equation has several implicit assumptions in it, whose validity have yet to be established.
A second approach is to treat the susceptibility as a 2 x 2 matrix, with the indices referring to the two uranium sites in the unit cell (Broholm 1988, Norman 1990). In such a model, Xll : X 2 2 : U, and Xx2 =X*I =Jq~. Summing all four terms would
72 M.R. NORMAN and D.D. KOELLING
recover the Putikka and Joynt form. Diagonalizing this matrix would lead to two eigenvalues, U _+ J I ~b 1, both of which are lattice periodic. The proposal, then, was to use either eigenvalue as a pair potential (in the actual theory, it is the m o m e n t - m o m e n t response function, I, which takes this form, and an RPA series is summed to get either )~ or V, as in 3He theory). The decision to use each pair potential separately, of course, is an assumption. In the model of Monien and Pethick, it is the sum that is used [thus for the simple Putikka and Joynt model, the AF correlations between planes drops out of the pair potential, i.e. (U + J]~b]) + (U - J F ~b]) = 2U]. In the model proposed by K o n n o and Ueda (1989), the combination of the two pair potentials depends on the relative phase between the two atoms in the unit cell for each particular energy band and k point involved. Thus, the only way to apply such a model is to take into account the radial/angular character of the quasiparticle wavefunctions over the Fermi surface into account. If one is going to do this, one might as well include them from the beginning when Fourier-transforming the pair interaction from real space. This is a complicated procedure, and so no full solution to this problem exists at present.
If, though, one uses these two pair potentials independently, one finds the same T c ordering of solutions as with the umklapp method. This occurs because the two pair potentials (corresponding to an acoustic and optic branch) are essentially a foldback of the susceptibility into the first zone. Thus, the peak in X at the reciprocal lattice vector Q = (0,0, 1) corresponds to a peak in the optic branch at q = 0 , and so one obtains again odd parity solutions with Alu the highest and Exu the hext highest. An additional feature of these calculations occurs if one takes into account the anisotropy of the susceptibility (i.e. the 3He form of the pair potential, Vs.s, is replaced by Vijsis j where s are Pauli matrices, and i,j are cartesian indices). Since the AF correlations are only present in the )~xx and )~».y components of the susceptibility, it turns out that the order parameter d vector is locked to the c axis since Vxx = Vyy and ~~ drops out of the gap equation since it is a constant. Tbis is equivalent to saying that the spin vector of the Cooper pairs is confined to the basal plane, just as observed for the normal-state spin fluctuations (note, since the pair potential allows only one component of the vector order parameter, line node solutions are possible). In 3He, these effects are only picked up when including higher-order (strong coupling) corrections, but here they are obtained at the weak coupling level because the susceptibility is already anisotropic in the normal stare. One can also get more sophisticated by including the intrinsic ferromagnetic correlations between next-near- neighbor atoms in the plane (Goldman et al. 1987). Including this effect leads to Elu being the highest T c soluton (Norman 1991a). This solution has two desirable properties: (1) it is a two-dimensional group representation, and thus can explain the highly complicated phase diagram in the H T plane of superconducting UPt3, and (2) it is an odd parity solution with the order parameter vector locked to the c axis.
The latter property can be used to explain the observed anisotropy in the H«2 phase line (Choi and Sauls 1991). Unfortunately, the solution has point nodes, whereas thermodynamic data indicate line nodes. The Ezu solution, which has all three desirable properties, has a very low T c in this formalism relative to Elu.
These problems led us to consider pairing via low-frequency fluctuations (Norman
1990), which peak at Q = (0.5, 0, n), with an elastic component connected to the observed weak magnetic ordering (Aeppli et al. 1988, Frings et al. 1988). These fluc- tuations correspond to AF correlations in the plane, and F correlations between planes, as opposed to the high frequency fluctuations, which are F in the plane and AF between planes. The disadvantages of these fluctuations over the high-frequency ones are (1) a smaller spectral weight (Aeppti et al. 1991), and (2) a lower energy scale [F(Q) ~ 0.3 meV]. An advantage is that their peak value at the above q vector is quite strong, being about five times the bulk susceptibility (Broholm 1988), as opposed to a much smaller peak value for the high-frequency ones. To obtain peaks in Z at (0.5, 0, n) wavevectors requires including substantial basal plane anisotropy, taking into account the fact that the moments lie along certain directions in the basal plane (thus Zxx is not equal to Zyy, and a Zxy term taust be included). The isotropic part of the interaction (V~x + Vyy) enters into the expression for singlet and Ms = 0 triplet (d vector along c) solutions, whereas the anisotropic term (Vxx- Vyy, Vxy) determines a new odd-parity solution with the d vector in the basal plane (note, the odd-parity solutions with d along c and d in the basal plane do not mix unless there is a Vxz or Vr z term). The isotropic term gives an even parity solution for the same reason as Miyake et al. (1986), that is the susceptibility peaks at the zone boundary. This solution is of Alg form, which may be surprising since one would expect to get a
"d-wave" solution. Alg turns out to be preferred since it has no symmetry requirements on nodes, and thus the nodes of the gap function (which must be there because of the strong U term) are able to lie in the most favorable location so as to minimize the free energy (note, our gap functions are numerical and not restricted by assumed basis function forms as in earlier models). This is a general feature we have found, that is all models of pair potentials we have used for UPt 3 always gives an Alg solution as the highest of even-parity solutions. It turns out, though, that the anisotropic term (Vxx- Vyy) is so large that an odd-parity Alu solution with the d vector in the basal plane has the highest T« This solution (transforming like k~x + kyy) has neither the group representation, nodal structure, or d vector orientation to explain experimental data. As for the magnitude of T~, it is comparable to that from pairing via the high-frequency spin fluctuations, since it is the bare line-width, F, which enters as an energy cut-off in the T c expression, and not F(Q). F turns out be 5 meV, the same as for the high-frequency case, eren though F(Q) at the AF Q vector is quite small.
In conclusion, the spin fluctuation models have gone a long way beyond their »He counterparts. On the other hand, the theories do not give a gap function with which one is totally happy (see table 3), and the theories are still probably too simplistic for real heavy-fermion metals. First, they replace the complicated quasiparticle matrix elements, of the form ( k, - kl V(r, r')[k', - k' ), by a simplistic V(k - k') as used above (in fact, the ambiguities discussed above about the momentum dependence of the pair potential are all due to this simple replacement). Second, the vertex is treated as a product of Pauli (spin) matrices, as in 3He, and completely ignores any orbital or spin-orbital contributions (in fact, the only thing included at this stage is the directional anisotropy terms in Z). Finally, the frequency dependences of the pair interaction, the quasiparticle dispersions, and the gap function have been treated in
74 M.R. NORMAN and D.D. KOELLING TABLE 3
Calculated symmetry of the gap function for UPt 3 from the semi-phenomenological spin fluctuation theory assuming pairing by either high-frequency spin fluctuations (F ~ 5 meV) or low-frequency spin fluctuations [ F ( Q ) ~ 0.3 meV]. Tabulated is the group representation, the functional form that the order parameter transforms under group operations (the actual order parameter is determined numerically), and the nodal structure. Note that two possible solutions are found for high-o» pairing depending on the ratio of near-neighbor to next-near-neighbor interactions. Also listed are the two most likely order parameters from phenomenological fits to
experimental data.
Rep. Form Nodes
Theory (high m) AI« kzz line
Elu (k x +_ iky)z point
Theory (low co) AI~ k~x + kyy point
Experiment (?) Elg kz(k x +_ iky) line, point
E2u k~(k x + iky)2z line, point
a highly simplistic fashion. Any of these effects could lead to a qualitative change in the solutions found. For instance, the fact that odd-parity solutions are found for AF correlations between planes may be a total artifact of the assumptions made on the matrix elements mentioned above. The problem, of course, is that a simple theory (U on-site, J between sites) may no longer be sutticient once one decides to include matrix element effects, and one is thus forced to work with a truly microscopic model, as opposed to the semi-phenomenological assumptions of the current models.
Thus, the problem of heavy-fermion superconductors is rar from being solved, although we feel at this stage that a spin fluctuation approach has the correct physics for constructing the "ultimate" theory. A likely scenario is one that uses the semi- phenomenological developments as a guide to construct a more realistic K o n d o lattice theory. We might note that the dynamic susceptibility calculated from a slave-boson approach has many things in common with the experimentally observed neutron scattering data (Auerbach et al. 1988), although as emphasized repeatly by Varma, an intrinsic multi-site approach is probably necessary to get an adequate description of the m o m e n t - m o m e n t response function (Varma 1991). Given an adequate micro- scopic theory for the dynamic susceptibility, a pairing self-energy can be constructed and a proper strong-coupling gap equation solved. The resulting solution will then give us some guidance about the validity of a spin fluctuation treatment, and whether it should be pursued or abandoned in favor of a more promising theory, possibly based on quadrupolar or inter-site phonon exchange.