Date Published: March 01, 2016
Publisher: International Union of Crystallography
Author(s): S. P. Collins, D. Laundy, T. Connolley, G. van der Laan, F. Fabrizi, O. Janssen, M. J. Cooper, H. Ebert, S. Mankovsky.
The possibility of using X-ray Compton scattering to reveal antisymmetric components of the electron momentum density, as a fingerprint of magnetoelectric sample properties, is investigated experimentally and theoretically by studying the polar ferromagnet GaFeO3.
Compton scattering provides a projection of the electron momentum distribution in a target material (Cooper, 1985 ▸). While the exact relativistic form of the differential scattering cross section is complex, the momentum density derived from every measurement, and calculated by every theory, to date, has been symmetric. We argue that this is because all materials investigated so far have been symmetric with respect to time reversal or spatial inversion. Materials whose orbitals possess neither symmetry are said to be magnetoelectric as they play a major role in magnetoelectric phenomena. Of particular interest are toroidal moments, corresponding to time- and parity-odd vectors, that not only play a vital role in magnetoelectric phenomena (Spaldin et al., 2008 ▸) but have been suggested to be implicated in high-Tc superconductivity (Scagnoli et al., 2011 ▸).
X-ray Compton scattering is an inelastic scattering process whereby the energy loss is an almost linear function of a projection of the electron momentum density: where is called the Compton profile (Cooper, 1985 ▸). Here, lies (almost) parallel to the momentum transfer and is the z projection of electron momentum. The momentum density and Compton profile are historically considered to be symmetric with respect to reversal of the momentum variable (or ). We suggest that this need not be the case.
If there is no net flow of electrons in the sample the integral of the flow along positive and negative z directions must cancel, i.e.While this is satisfied trivially for the symmetric Compton profile, it imposes a useful constraint on each half of the antisymmetric profile: thus providing a ‘zero-sum rule’ that can be used to verify experimental results and model calculations. Most trivially, the zero-sum rule dictates that, for each half of the asymmetric profile, the existence of a positive contribution implies the existence of a negative one, and vice versa.
Finding no obvious symmetry arguments for momentum densities and Compton profiles to be symmetric is, of course, far from demonstrating that they are not. However, one can employ a simple thought experiment to see that such an asymmetry is present in classical orbitals. Consider a highly elliptical planetary orbital, observed from within the orbital plane, as shown in Fig. 1 ▸. The orbiting body would be seen to have a very large positive momentum projection (towards the observer) for a short period of time, when the orbiting ‘planet’ is closest to the ‘star’ that it orbits. Conversely, it would exhibit a small negative momentum projection for a long period of time when it is far from the star and moving slowly. The large positive momentum has no negative counterpart and so the momentum projection distribution (analogous to the Compton profile) must be asymmetric. Note also that such an orbital is asymmetric with respect to time reversal and spatial inversion: reversing time would reverse the direction of the orbit, and spatial inversion would reverse its eccentricity.
The electron momentum density is a real-valued function of momentum, , or (equivalently) of its magnitude, p, and direction, . We can expand this density in terms of a complete set of angular functions and prefactors that depend on p. For example, where are real spherical harmonics (also referred to as multipoles or spherical tensors) of rank K and projection Q (Lovesey et al., 2005 ▸), and are the corresponding tensor components and are functions of p. The merit of such an expansion lies in the fact that each non-vanishing multipole must be consistent with the symmetry of the physical system. For example, an isotropic system allows only a single term in the expansion and we find where is the radial momentum distribution (Cooper, 1985 ▸).
It is likely that the antisymmetric part of the Compton profile is small as it depends on a subtle aspect of the anisotropy of the orbital polarization. An ideal experiment to test these ideas is therefore one where the antisymmetric part can be reversed simply and rapidly, inducing the smallest possible systematic error and allowing a sensitive ‘difference’ measurement to be performed. Reversal of the toroidal vector in many magnetoelectric materials requires simultaneous application of electric and magnetic fields, typically applied during ‘field cooling’. An interesting class of materials where the toroidal moments are more easily manipulated are polar ferromagnets. One such material, that has been studied with X-rays for its directional dichroism (Kubota et al., 2004 ▸) and magnetoelectric multipoles (Staub et al., 2010 ▸), is GaFeO3. Large polar single crystals are available and the ferromagnetic moment can be reversed with a modest applied field. GaFeO3 orders magnetically at the relatively high temperature of TC ≃ 210 K. We therefore selected GaFeO3 as a potentially suitable test material and consider next the implications of crystal symmetry on the observable physical phenomena.
GaFeO3 (space-group No. 33, ) is a polar ferromagnet. It possesses both a magnetic (axial, time-odd) and polar (time-even) vector moment. We have discussed the need for time- and parity-odd multipoles in the context of antisymmetric Compton profiles, and the desirability to possess a toroidal (polar, time-odd) moment. We now apply the magnetic crystal symmetry to all four permutations of time/parity odd/even vectors in order to find out (i) if they can exist and (ii) in which direction(s) they may point. The crystal point-group symmetry in the high-temperature paramagnetic phase is mm2. We can denote the symmetry group as (the identity, twofold rotation about y, mirrors normal to x and z). There are several possible magnetic groups that are consistent with this point group, which are formed by taking each spatial symmetry operator and either applying time reversal or not. Four such groups can be generated, with each placing specific constraints on the directions of the possible vectors, or rendering them absent.
Experiments were carried out on beamline I12 (Diamond Light Source), using a linearly polarized monochromatic incident X-ray beam of energy 125 keV and bandwidth of 0.6 keV, selected by controlled bending of a double Laue monochromator (Drakopoulos et al., 2015 ▸). Compton scattering was detected close to back-scattering (2 169°) by a 23-element germanium solid-state detector. The orientation of crystal (a single polar domain – see Appendix A), X-ray beams and magnetic field were as shown in Fig. 3 ▸ and the sample was maintained at a temperature of 100 K with a nitrogen gas-jet cooler. As the aim of the experiment was to observe a small (antisymmetric) difference in the Compton profiles measured with two opposite magnetic field directions, the (0.3 T) field was flipped rapidly and repeatedly (1 s counting time for each direction) while data were accumulated for around 48 h.
To confirm the occurrence of an antisymmetric Compton profile in polar ferromagnets, density functional theory (DFT)-based theoretical investigations have been performed. The Compton profile was calculated from first principles using the Korringa–Kohn–Rostoker (KKR) Green’s function method. This implies the electronic Green’s function is represented by means of the multiple scattering formalism by Here is the scattering path operator with the combined index representing the spin–orbit and magnetic quantum numbers κ and μ, respectively (Rose, 1961 ▸), and and are the four-component regular and irregular solutions, respectively, to the single-site Dirac equation for the atomic site q (Ebert et al., 2011 ▸). The superscript indicates the left-hand-side solution of the Dirac equation. The electron momentum density ρ(p) = is decomposed into its spin-projected components , which are given by the Green’s function represented in momentum space, where represents the spin character. is expressed in terms of the real-space Green’s function as follows: Here Ω is the volume of the unit cell and are the eigenfunctions of the momentum operator, which can be written as , where is a four-component spinor satisfying the equation (Rose, 1961 ▸) Using a Rayleigh-like expression, one obtains the angular momentum expansion for the eigenfunctions (Benea et al., 2006 ▸), where are Clebsch–Gordan coefficients, are complex spherical harmonics, are spin-angular functions and are spherical Bessel functions.
We propose a new class of Compton scattering experiment with the potential to provide the antisymmetric part of the electron momentum density in materials. We show that the antisymmetric Compton profile is a unique fingerprint of time- and parity-odd properties of the underlying orbitals, and thus a sensitive probe of magnetoelectric phenomena. Initial experiments on the polar ferromagnet GaFeO3 demonstrate that our experimental technique is extremely sensitive, leading to very small systematic errors. Our results show that the magnitude of the antisymmetric momentum density, after broadening with the experimental momentum resolution of 0.79 a.u., is not larger than around of the peak in the symmetric part. While the optimistic eye might be tempted to pick out an antisymmetric difference signal (Fig. 4 ▸) above the statistical noise, we cannot claim that the present results are conclusive in this respect.