CRONOGRAMA DE TRABAJO

There have been various attempts to calculate some of the electronic properties of URhGe us- ing density functional theory (DFT), in particular under the local spin density approximation (LSDA). In order to understand and analyse the calculations presented in the literature, a simple understanding of DFT is required, and the following introduction is based on the work of Cottenier[96].

DFT is based on the observation by Hohenberg and Kohn[97] that the energy of the system:

hΦ_|Hˆ_|Φ_i=_hΦ_|Tˆ+ ˆVint+ ˆVext_|Φ_i (4.3)

where ˆT is the kinetic energy, ˆVext the potential energy of interactions with the lattice and ˆVint the potential energy due to electron electron interactions; can be re-written as:

hΦ|Hˆ|Φi=E[n(r)] =FHK[n(r)] + Z

n(r)Vextdr (4.4)

Here, everything is expressed in terms of functionals of the electron densityn(r). Critically, the lattice, and hence the specific details of any particular problem only enter through the second term. The problem of minimising the energy of interacting electrons in a lattice is then simplified to minimising the combined energy of the non-interacting electrons in a lattice and the functional FHK[n(r)] which is independent of the problem considered. Kohn and Sham[98] then went on to show thatFHK[n(r)] could be written as:

FHK[n(r)] =T0[n(r)] +VH[n(r)] +Vxc[n(r)] (4.5)

the Hartree term for electron-electron coulomb repulsion, andVxc[n(r)] includes the complicated exchange and correlation parts. One can then use this to construct a Hamiltonian, known as the Kohn-Sham Hamiltonian, for a system of a non-interacting electron gas in an effective potential which consists of a lattice part and an exchange-correlation part. It is this equation that is solved to obtain a set of eigenstates, and thus the electron density.

This becomes a self consistency problem: The density depends on the eigenstates, the eigen- states on the parameters of the equation and the parameters of the equation on the density. To find the ground state density, one must begin with a guess of the electron density, construct the potentials and thus the Kohn-Sham equation, solve it to get the eigenstates, and construct from them a new density. The process is repeated until the density converges between iterations, at which point the ground state is known.

Practical DFT calculations can be thought of as consisting of three parts. Firstly, one must for- mulate the problem. Whilst this can in theory be exact, it usually involves approximations when constructing the exchange-correlation term. The simplest class, known as local density approx- imation (LDA), usesVXC(n) =R

XC[n(r)]n(r)d3_{r, but these cannot represent any spin-physics.}

The local spin density (LSDA) is the next class, whereVXC(n↑_{, n}↓_{) =}R

XC[n↑_{(r), n}↓_(r)]n(r)d3_r

and other more complex approximations also exist. These might include_∇n terms (GGA), or additional on-site repulsion (U). Another decision which must be made at this stage is the equa- tion to solve: Schr¨odinger’s equation, Dirac’s equation, or an approximation in-between. This has implications for the accuracy at which spin-orbit enters (if at all).

Once the free energy functional has been constructed, it must be solved. In this stage, a suitable basis set is selected, and the densityn(r) constructed from it. In theory, any complete basis set allows the correct density to be found, but in practice, choosing a good basis set is necessary to make the solver converge quickly, or even at all. One logical basis set is the augmented plane wave (APW) set, this uses atomic orbitals within a sphere centred on each ion, and plane waves in the gaps in-between. In practice this set is very slow to work with, and a linearised set is used instead (LAPW). Core electrons for the ions in the lattice can be included in the basis set, at the cost of increased complexity (LAPW+LO), or in the lattice potential around each ion. There are also several variations on this, including RLAPW, where the full relativistic orbitals are used.

Finally, the results can be interpreted. The electron density is exact, insofar as the approx- imations made in constructing the functionals are correct. The eigenstates on the other hand are strictly the eigenstates of the Kohn-Sham equation, not the original system. In practice however, they are usually adequate, and are presented as the band structure of the material, including interactions. This practice is equivalent to saying that the Kohn-Sham effective potential is a good mean-field approximation to the interacting problem, which it usually is. Calculating such properties as the density of states from the resulting band structure is usually quite reliable and

straightforward. Features like Fermi surface topologies can be harder, as they can depend cru- cially on the shape of the band and the precise Fermi energy. In particular, when flat bands such as those found inf-electron materials are present, a tiny rigid shift of a band relative to the others, or a small error on the Fermi energy, can hugely change she shape and size of the resulting Fermi surface pockets.

Early calculations then focused on the simpler properties. The first calculations appeared in 2002[99,100], using LSDA and LAPW. At this time there was still some uncertainty about the magnetic structure[27, 26], but the calculations agree with the structure which is now known to be correct for single crystals. Both of these calculations show that the electron states near the Fermi surface have a substantial U5f weight, with Rh4d also contributing slightly at the Fermi level, and more substantially 2–5 eV below. Both calculations also obtain roughly correct magnetization (0.31µB[99] and 0.25µB[100], compared to experimental values of 0.37–0.42µB[25,

29]). Both papers also note that this arises as a result of larger spin and orbital moments which are directed in opposite directions, with orbital component being slightly larger than spin. Shick[100] also completes the calculation with a Hubbard U term (LSDA+U), which would be appropriate if the U5f electrons had some localised character, but finds the model without U to be more accurate. This is further evidence of URhGe’s itinerant nature. These calculations predict different magnetic anisotropy energies for the a- and b-directions, which qualitatively agree with the observation that the magnetization can be rotated abruptly towards b with a field Bb ≈ 12 T, but the material has very little response to fields in thea direction. Finally, the calculated DOS can be compared with valence band photoemission studies[101], which show general agreement. Between calculations, and compared to the photoemission data, the agreement is generally qualitative, but differences in detail, and in the position of the Fermi level relative to the spectra, remain.

Another more recent paper[102] repeats the calculation, again using LSDA and LAPW, with similar results. They also consider the consequences of changing the unit cell size, through pressure, and its effect on the spin and orbit parts of the magnetization. Their prediction is that at a pressure near 50 GPa they should become equal, leading to a zero moment ferromagnet. Pressure measurements on URhGe extend up to 13 GPa[41], and include resistivity and heat capacity, but not magnetization which would be a useful test of these predictions.

Very recently, two attempts have been made to obtain band structures and Fermi surfaces for URhGe. One is published as part of a paper on ARPES[104]. The other is recently completed by Ed Yelland[103]. Fermi surfaces from these two calculations are presented side by side in figure

4.11. Unfortunately the calculated surfaces are quite different. [104] also includes band structure plots for the SY, XΓ and UZ high symmetry lines, which can be compared to the band structures from [103]. Whilst it is in some places possible to identify bands from one calculation with those from the other, even these also show qualitative disagreements in shape. The two calculations are

kz kx ky FS70 (hole) FS69 (hole) FS71 (electron) FS72 (electron)

X

R

T

Y

S

U

Z

Γ

Figure 4.11: Calculated Fermi surfaces for URhGe. Left: Calculation from [103], using LAPW and LSDA. It is polarised Mkc and the bands are mixed spin due to SOI, which is included at the scalar relativistic level. Right: Calculation from [104], using RLAPW and LDA, for the paramagnetic state. Both calculations treat all U-5f electrons as itinerant. Both calculated structures are oriented the same way, but the right panel is scaled into a cube. The bands which can be degenerate at the zone faces are one above the other in the left panel, the degeneracies appear to be absent in the right panel.

in different regimes, [104] is calculated for the paramagnetic case, using LDA, whereas [103] uses LSDA to calculate both the zero fieldM_kccase (shown in figure4.11) and theM_kb case present above the moment rotation transition. However, given the broadly similar properties (especially those detailed in section 4.3) above and below the transition it seems unlikely that the Fermi surface undergoes such a large reconstruction. The similarity of ARPES spectra at the Γ point and ΓX line above and belowTCurie also suggest that the band structure is not undergoing huge changes[104].

The ARPES data in [104] is also, in theory, a good direct probe of the band structure. In practice however, the large number of flat bands near to the Fermi level can easily blur together given the limited experimental resolution. As the electron leaves the material it can also lose some momentum in the direction perpendicular to the surface. This results in a further blurring of the experimental resolution in this direction — not a problem for a two dimensional material, but URhGe is believed to be fully 3D. This problem is compounded if, as in URhGe, the material is difficult to cleave. A rough surface has many different perpendiculars in different places, so the blurring happens across a range of directions ink-space, not just the one perpendicular to the face. These issues notwithstanding, the ARPES data does show some structure, though the bands themselves, and the points where they cross EF remain unclear. As when comparing the calculations, there is some qualitative agreement between the ARPES bands and both of the calculations, but also some substantial qualitative differences. The ARPES is comparatively close to both of the calculations along the YS line, passable along the UZ line but poor along the ΓX line.

We can also attempt to identify some of the observations with features in the calculated Fermi surfaces. Given the highly complex shapes in both calculated surfaces, and the expectation that neither is exactly correct, it would be possible to find an orbit of more or less any frequency somewhere within one of the models. We shall then not attempt to identify orbits with such detail but rather look for more general characteristics.

The open orbits postulated in sections 4.2 and 4.3 are easily found on the band 71 (green) surface in the calculation from [104]. This calculation does not conform to the expectation of degeneracies outlined in the previous section, though around most of the zone faces there are two bands coming quite close together. Some of the lack of degeneracy could be ascribed to numerical issues when interpolating the band structure across the zone boundary. In particular, the band structures given in the paper do have the expected degeneracies, and these are probably calculated in more detail. In the calculation from [103], the green surface also supports open orbits inka andkc directions provided the green sheet connects to itself (degeneracy lifted). But it does not if the degeneracy remains, when the electron changes between green and red sheets every time it crosses a zone boundary.

the band 69 surface (yellow, at the Y point) in the calculation from [104] is a good candidate. The gap between it and the band 70 (red) surface is due to spin orbit splitting lifting the degeneracy, so will have a small energy gap and magnetic breakdown will be easy.

To conclude then, there are a number of calculations and measurements which further support the established view that URhGe is an itinerant ferromagnet with U5f weight contributing to flat bands near the Fermi surface. The detail of the calculations, including the band structure on the other hand must be considered less reliable, as the calculations differ from each other and the limited experimental data.