Ley N° 29783 y su modificatoria Ley N° 30222 “Ley de Seguridad y Salud en el

The Landau theory described before can be extended to give a complete phenomenological theory of superconductivity, in which case it is called the Ginzburg-Landau theory of super- conductivity. This theory can also be used to describe the Josephson coupling between two (unconventional) superconductors, that was already briefly mentioned in section 2.5 for the case of conventional superconductors. In the conventional case this coupling is independent of the orientation of the crystals, since the order parameter is isotropic. When an unconventional superconducting state breaks some of the crystal symmetries however, we might expect that this influences the Josephson coupling between superconductors. In particular, we may expect the effect to depend on the crystal orientation. It turns out that a full derivation of the cou- pling term in the Ginzburg-Landau formalism is quite a tour de force, requiring more advanced knowledge of mathematics and solid state physics20, so we will just cite the result here.

The relationship between the critical current in a Josephson junction between two super- conductors, labeled (1) and (2), and the symmetry of the superconducting states is given by [64]

IJ ∝Φ1(n1)Φ2(n2) (6.22)

wherenj denotes the interface normal on sidej and Φj are functions with the same symmetry properties as the order parameter of the superconducting state on sidej. For a single junction the sign of the current does not matter, since just switching the point of view would make it positive again. When we have multiple connected junctions however, it can be meaningful, for if one junction has a positive current and another a negative, they still have opposite signs when changing the point of view. This corresponds to a phase shift of π in the order parameter. In the next section we will make use of this fact when considering experiments to determine the symmetry of the superconducting phase. It is especially important to note again that the basis functions are not unique, so we must be careful when drawing conclusions from this formula.

This formula is particularly easy to work with if one of the superconductors is a conventional superconductor, and the superconducting state of the unconventional superconductor belongs to a 1-dimensional representation. In that case we simply have

IJ ∝Φ(n) (6.23)

where Φ is a basis function of the irreducible representation.

Things become significantly more complicated when dealing with two-dimensional repre- sentations. For the spin-singlet states we can use the symmetry functions that are listed in table 6.2. For the spin-triplet states we could, from the d(k) listed in table 6.2 figure out the coefficients η in the expansion of the order parameter in the basis functions, and use these to construct the appropriate scalar function using table 4.4. This would obvisouly be rather cumbersome.

For singlet-triplet junctions there is another formula that describes the Josephson current [65]:

IJ ∝ hψ(k)d(k)·(k×n)iF S, (6.24) where ψ(k) is the basis function for the singlet superconductor, d(k) is the basis function for the triplet superconductor, and the bracketed term is averaged over the part of the Fermi surface for which the k-vector has a positive projection on the interface normal vector n. For coupling between a conventional superconductor and a triplet superconductor, theψ(k) becomes a constant again and we can fill in thed(k) listed in table6.2.

20

See, for example, [61,62,63]. It is also discussed at lenght in [45], and briefly in [64].

1

2

3

4

5

6

x

y

z

Figure 6.2 – Definition of the x-, y- andz-axis with respect to the symmetry transformations of

D3d.

6.4.1 Directions of Zero Coupling

Symmetry requires that the basis functions vanish in some directions. To understand this, consider a situation in which the 2C3 symmetry, the rotations about the three-fold axes, is

broken, and consider a vector n along the rotation axis. Let Φ(n) be a basis function for the appropriate irreducible representation. Applying a rotation transformation must change the sign of the basis function, but the vector remains invariant as it lies along the rotation axis, so we have

C3a(Φ(n)) =−(Φ(n)) = Φ(n) (6.25)

which can only be true if we have Φ(n) = 0 for this vector.

Another situation in which this occurs, is when two transformation give the same result, but one is broken while the other is not. Suppose for example that inversion symmetry is broken, but that the reflection symmetry is not broken. Then for vectors perpendicular to the plane of reflection, the inversion and reflection transformation have the same effect, but one requires a sign change while the other requires invariance:

−Φ(n) =I(Φ(n)) =σd(Φ(n)) = Φ(n) (6.26) which again can only be true if Φ(n) = 0 for this vector.

Thus from symmetry considerations alone we can identify zeros of the basis function that are independent of the specific form of the basis function. Let our coordinate system be set-up in such a way that the z-axis is the principal axis of the three-fold rotations, that the x-axis is the axis for one of the two-fold rotations, and that the y-axis lies in one of the reflection planes. This is depicted in figure 6.2. The planes and lines along which the basis functions of the different one-dimensional representations must vanish due to symmetry considerations are depicted in figure6.3. According to equation (6.23) the Josephson current thus vanishes when the interface normal vectorn is in any of these directions.

According to equation (6.24), the Josephson current vanishes when the interface normalnis parallel to thed-vector of the triplet superconductor. This means that whether coupling occurs or not depends crucially on the specific form of the realised state. For the A1u representation for example, we have two distinct basis functions, kxxˆ+kyyˆ and kzˆz. The first of these would imply that no coupling is possible in the horizontal plane, while the second implies that no coupling is possible in the z-direction.

−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 z x y (a) A1g −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 y x z (b) A2g −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 z x y (c) A1u −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 y x z (d) A2u

Figure 6.3 – Lines and planes along which the basis functions for the different irreducible repre- sentations ofD3d must vanish due to symmetry considerations.

In particular, the directions for zero coupling predicted by equation (6.23) and (6.24) are not consistent. At this point we dare not say which equation is right as we have too little understanding of the derivations of these formulae. On first sight, equation (6.23) seems to be more realible since it does not depend on the form of d(k), but the predictions from equation (6.24) have been verified for Sr2RuO [65,66].

6.5

Probing the Order Parameter Symmetry using SQUID-like