2.4.4 CONTAMINACIÓN DEL SUELO

Before discussing the calculations which have been made by various authors on URhGe, it is pertinent to discuss the symmetry constraints of the material. Let us begin with the crystal structure, ignoring for now any considerations related to spin-orbit coupling, magnetic field or magnetism. The group Pnma has eight symmetry operations (in addition to the translation operations of the lattice), given in table 4.1. Of particular interest are numbers 2, 3 and 4: the twofold screw symmetries.

Twofold screw symmetries have some important consequences for the band structure at the zone faces. For the sake of argument, let us consider the face perpendicular tokz, containing the points Z, U, T and R (see figure4.11for the position of these points on an orthorhombic zone). At all points on this plane there are two equivalent crystal momenta, (kx, ky,π

c) and (kx, ky,− π c). As the crystal has spatial inversion symmetry one expects the energy of an kz = π/c electron to be the same as akz =₋π/c one and E(kx, ky, kz) =E(kx, ky,₋kz). At the zone face, both these momenta are the same, and the states would be degenerate. In general however, there is

No. Notation Description

1 1 The identity.

2 2(0,0,1₂) 1₄,0, z Screw, rotate 180◦about the line 1₄,0, z then trans- late alongc by half a unit cell.

3 2(0,1₂,0) 0, y,0 Screw, rotate 180◦_{about the line 0, y,0 then translate} alongbby half a unit cell.

4 2(1

2,0,0) x,

1 4,

1

4 Screw, rotate 180◦about the line x,

1 4,

1

4 then trans-

late alongaby half a unit cell. 5 ¯1 (0,0,0) Inversion, about the origin. 6 a (1

20,0) x, y, 1

4 Glide, mirror about thec =

1

4 plane, then translate

half a unit cell in theadirection 7 m x,1₄, z Mirror about the b= 1₄ plane 8 n (0,1₂,1₂) 1

4, y, z Diagonal glide, mirror about thea=

1 4 plane, then translate by (0,1 2, 1 2).

Table 4.1: Symmetries of the Pnma space group, as listed in the interational tables[89]. These apply to the ion locations in URhGe, and thus the lattice potential. Not listed are the implicit primitive translationst(1,0,0),t(0,1,0) andt(0,0,1). Of course as the lattice is static it also has the time reversal symmetry operationT

some part of the lattice potential which will mix them to provide two new states slightly split; the band crossing becomes an anticrossing.

In the absence of spin or magnetism, the Hamiltonian obeys the space group symmetry. With this, Herring[90] uses group theory considerations to demonstrate that in the presence of a twofold screw symmetry, this mixing does not take place, and the bands remain degenerate. The crux of the argument is that the screw symmetry operation, followed by the time reversal operation, maps the wavefunction of the state at (kx, ky,π_c) to the one at (kx, ky,₋π_c). If the electron fluid does not break these symmetries, the energy of the two states must remain the same. Most of the discussion at the time focused on the hexagonal close packed metals, but the argument applies to any material with both time reversal and a twofold screw. It enforces a degeneracy on the two zone faces perpendicular to the axis of the screw. For URhGe in the paramagnetic state there is a screw along each axis, and time reversal symmetry, so all faces of the BZ are ‘screw degenerate’. This is in addition to the spin-degeneracy which occurs everywhere in the zone in the non-magnetic case.

In the presence of a magnetic field, the Hamiltonian is more complicated because the time reversal symmetry is lost. However, Herring’s argument still applies if the combined operation of screw and time reversal still stands, even if the individual operations do not. By considering the effect of the symmetry operations on spinning spheres∗ with the rotation axis aligned with field,

∗_{Spinning spheres are the simplest object which breaks time reversal symmetry, they are also a good model} for orbital magnetism, but a poor one for electron spins, which are more complicated. Here we only care about a symmetry breaking object.

the following symmetry rules are self evident:

1. A twofold rotation with axis parallel toµ0H preserves the sense of the rotating sphere, so

the symmetry operation is preserved.

2. A twofold rotation with axis perpendicular to µ0H reverses the direction of a spinning

sphere, so this symmetry operation is lost. The combination of the rotation and a time re- versal however preserve the spinning sphere, so this new operation of ‘time reversal rotation’ joins the symmetry group instead.

3. A mirror about a plane which containsµ0H preserves the spinning sphere, so preserves the

symmetry operation.

4. A mirror about a plane perpendicular to µ0H reverses the rotation direction, so the new

operation is the combination of the mirror and time reversal.

And it should be noted that any rule which applies to a rotation also applies to a screw, and any rule that applies to a mirror, also applies to a glide. This is simply because we can separate the space and time parts, and is equivalent to saying the eigenstates are a Bloch wave consisting of the product of an e−ik·r _{part, a part with the lattice periodicity and a spin part.}

By applying the rules above to the space group operations in table4.1, we can obtain a new set of symmetry operations for the magnetic material. This new group is referred to as the magnetic double group of the Pnma space group. Let us take the example of the zero field magnetization below the curie temperature, where the spontaneous field lies along c. Operations 1 and 5, (inversion and identity), and the primitive translations are unaffected, whilst time reversal is lost. Operation 2 is preserved, operations 3 and 4 are replaced with a time reversal screw. Operation 6 is preserved and operations 7 and 8 become time reversal glides/mirrors. If time reversal were not included, the preserved symmetries would be 1, 5, 2 and 6, which form a subgroup of the Pnma group, as they should. If the rules above are applied correctly, the preserved operations must always form a subgroup.

Straight away then, we can see that some of the results from Herring’s work still stand in the magnetic case. In his work he applies the screw and time reversal separately, but both are required and the time reversal no longer exists on its own. Thus the degeneracy is preserved on the BZ faces which lie perpendicular to a time reversal screw. For URhGe, this implies a degeneracy on the BZ faces which are not perpendicular to the field. Whilst the absence of a time reversal screw implies the degeneracy can be lifted on the remaining face, it may still be enforced on some high-symmetry lines. Removing a symmetry enforcement on the degeneracy does not mean the degeneracy is lifted, just that it can be lifted. In practice it normally is, and if it is not there is probably another symmetry enforcing it which has not been considered.

It is also worth considering a case where the field is not aligned with a crystal axis. In this case, the field can be perpendicular to one screw (if it is directed in a plane containing two crystal axes) or perpendicular to none. In the former case that screw is replaced by a time reversal screw, as

above. For screw axes that are neither parallel nor perpendicular, the screw-transformed spinning sphere cannot be transformed back to its original form by a time reversal. In this case, neither the screw nor the time reversal screw enter the double group. For our purposes, only time reversal screws are relevant, so the degeneracy is only preserved if the field is perpendicular to the screw controlling it. Equivalently, the degeneracy is lifted on all faces which have some perpendicular component of magnetic field.

Whilst field can remove the enforced degeneracy, it is not correct to say that a gap will necessarily open up. For a gap to appear, one requires an interaction which splits the energy of the two states. One common choice is spin-orbit interaction (SOI). SOI alone breaks neither the symmetry of the lattice nor time reversal, so cannot lift the screw degeneracy. Rather it enters into the manner in which the degeneracy is lifted by field. In a ferromagnet like URhGe, where the spin-splitting can be assumed to be much larger than the energy scale of SOI, the spin quantization axis is chosen by the ferromagnetic moment. The ferromagnetism also breaks the spin degeneracy on a scale of the exchange energy which is much larger than that of SOI, so we can consider now a single band which is degenerate with itself only at the zone boundary. The SOI is then free to lift that degeneracy. The energy scale of spin orbit is often sufficient to provide a gap between the two bands of order tens to hundreds of millielectronvolts. At low fields, this makes a distinct gap, and an electron driven along the Fermi surface by the field will always reappear on the same band when it crosses the Brillouin zone boundary. This energy gap is still small enough though that at higher fields magnetic breakdown effects are to be expected. One well-known example of SOI causing unusual changes in the Fermi surface of a ferromagnet is in nickel, where small ellipsoidal pockets at the X points show substantial changes in their size as a function of field direction[91]. Another well-known example where some of these effects can be seen is in the divalent hexagonal metals. These crystallise in a hcp lattice which has a screw symmetry perpendicular to the hexagonal face of the zone. None of them are magnetic, so SOI is only able to open a gap with symmetry broken by the application of field. In the light metals, spin orbit is small and the effect is often lost to magnetic breakdown before it is observed at high field. In the heavy metals it is stronger[92]. Elliot[93] calculated the double group for hcp and various other lattices, and Cohen and Falicov[94] used these, together with their own calculations to make predictions about the connectivity of the Fermi surfaces, in particular of Mg. Finally, Josephet al[95] observed the splitting in Zn and Cd directly via quantum oscillations.

We now turn to the practical consequences for URhGe. In the absence of any field or SOI, paramagnetic URhGe would have a fourfold degeneracy on every BZ face. Twofold due to spin, and twofold due to the screw axes. Away from the zone faces, the twofold spin degeneracy would remain, but the screw degeneracy is not enforced. This implies that the bands cross at the BZ face, rather than the more usual anticrossing. When considering the path of a probe electron driven around a Fermi surface sheet by field, the behaviour at the zone edge is different. If there

is an anticrossing, the electron reappears at the other side of the zone on the same sheet it was on when it left. If the degeneracy is not lifted, it will reappear on the other sheet in the degenerate pair. One result of this is that when the Fermi surface reaches the BZ face, the usual extremal neck orbit will not be present.

When splitting is introduced, which can only happen on zone faces with a component of field perpendicular, the crossing changes into an anticrossing. The energy gap between the two bands is limited to the energy scale of the interaction causing the splitting. Whilst atomic spin-orbit interaction can be large, the splitting of bands is usually observed to be very small, this results in a very narrow gap between the bands. There will usually be an extremal orbit on the BZ face, but these small gaps imply it will readily undergo magnetic breakdown. Similarly, the splitting can change the topology of any orbits which cross a zone face, but the original topology can be restored by magnetic breakdown.