734 M.R. Norman / Magnetic anisotropy in UPt~

a b c Fig. 1. Plots of the pair potentials, (Vx~ + ~ . ) / 2 , ( V V)y)/2 and V,~ for q~ = 0. The length of the box is one reciprocal lattice vector (4~r/X/'3a) with the zone center in the middle. The maximum (minimum) values for the pair potentials in units of 60 K are 3.87 (0.49), 2.83 (-1.41) and 2.45 (-2.45), respectively.

Table I Coupling constants from low frequency pairing, with U = 36 K and J = -5 .4 K. Listed is the largest coupling constant for each type of solution. The solution in the last row was obtained by setting V equal to zero in eq. (3).

A ~g 0.20 A 1, (k~£) 0.125 A ,,(kjc + kv.f ) 0.25 A ~,(k~) 0.235

the odd parity gap. Note that two independent odd parity solutions exist, one with d along the hexagonal c-axis, the o ther with d in the basal plane.

The various pair potentials arc shown in fig. 1. Vex + V peaks at q = (0.5, 0) and hexagonally equiva- lent points. Vxx-V~.~, though, peaks only at q = (0 .5 ,0) and Vx~. only at q =0.25(1,X/-3), that is they individually have or thorhombic symmetry (the cou- pled solution for d and d v has full hexagonal symmet- ry, of course).

Equat ion (3) is solved in each group representat ion, with the grid of k vectors (and their appropriate weights) being determined from a band structure cal- culation. The results for the representat ion with the largest coupling constant are shown in table I, with the inverse coupling constant being defined as NFhf I n ( 1 . 1 3 F / T ) , where ~V'hf is the high frequency 11 (renormalized Fermi energy) , equal to 60 K, and N is the density of states at the Fermi energy (NFhf is about 3 for UPt3). The coupling constants were calculated based on a U value of 36 K and a J value of - 5 . 4 K which approximately fit the neutron scattering data

[61.

The interesting point to note from table I is that an odd parity solution has maximal T c despite the anti- ferromagnetic correlations. This solution is totally due to basal plane anisotropy (for an isotropic case, V~x = Vy,. and V y = 0, so no solution with d in the basal plane would be al lowed), and is of the form kx2 + ky.i (note that eq. (3) is solved numerically on a grid in k space, so the " f o r m " only represents how the solution transforms under group operations). It is an axial-like state with nodal points.

A major question to ask at this stage is whether this solution is consistent with experiment . Current phenomenological theory is supportive of an E~ order parameter [7, 8], E l being a two-dimensional repre- sentation. This gap has nodal lines, consistent with various thermodynamic data. The two-dimensional na- ture of the order parameter also allows it to easily explain the observed splitting of the specific heat anomaly at Tc as due to a lifting of the degeneracy caused by the weak magnetic order (which has ortho- rhombic symmetry). One problem with the E l solu- tions, though, is that they imply either an orbital or spin moment of the Cooper pairs along the hexagonal c-axis. This is not consistent with neutron scattering data, which indicates that the moments are locked to the basal plane.

Machida et al. [9] have indicated the possibility that the above findings could also be explained by an odd parity one-dimensional representat ion in the weak spin-orbi t coupling limit. Al though the current case is not in that limit, the author proposes that something analogous to this can happen for the Alu solution found above. The idea is that interaction with the weak magnetic moment can align d in a favorable direction in the basal plane when one is close to T c. Below To, the order parameter becomes sufficiently

M.R. Norman / Magnetic anisotropy in UPt~ 735

strong that the solution relaxes back to the hexagonal solution derived above. One can simulate this effect by setting Vxy to zero in eq. (3). This acts to decouple d x and dy. Note from fig. 1 that Vxx - Vyy selects out a particular x direction. The resulting solution is tabu- lated in table I, and is of the form ky.f. Thus, one expects near T c for the solution to be of the form ky.~, an (0, 1) solution, whereas sufficiently below To, the solution relaxes back to the form kx.f + ky.~, a (1, 1) solution. This is analogous to the E l case where one goes from a (0, 1) solution to a (1, i) solution [8]. It remains to be seen, though, how the analysis of ref. [8] carries over to the current case, and whether the proposal here is a viable explanation. If it indeed turns out that an E 1 solution is the actual ground state, one will need to invoke a more complex microscopic theory.

Acknowledgments

This work was supported by the US Dept . of Ener- gy, Office of Basic Energy Sciences, under Contract No. W-31-109-ENG-38.

References

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