Elastic Scattering of Electron Vortex Beams in Magnetic Matter

Abstract

Elastic scattering of electron vortex beams on magnetic materials leads to a weak magnetic contrast due to Zeeman interaction of orbital angular momentum of the beam with magnetic fields in the sample. The magnetic signal manifests itself as a redistribution of intensity in diffraction patterns due to a change of sign of the orbital angular moment. While in the atomic resolution regime the magnetic signal is most likely under the detection limits of present transmission electron microscopes, for electron probes with high orbital angular momenta, and correspondingly larger spatial extent, its detection is predicted to be feasible.Rapid developments in nanoengineering call for characterization methods capable to reach high spatial resolution. In this domain, the scanning transmission electron microscope (STEM) provides a broad scale of measurement techniques ranging from Z-contrast [1] or electron energy-loss elemental mapping [2], differential phase contrast (DPC) [3,4], via local electronic structure studies of single atoms [5] to counting individual atoms in nanoparticles [6]. As a specific case of high-spatial resolution electron energy loss spectroscopy, an electron magnetic circular dichroism (EMCD) method has been introduced [7] as an analogue to x-ray magnetic circular dichroism, which is a well established quantitative method of measuring spin and orbital magnetic moments in an elementselective manner [8,9].Recenly, the introduction of electron vortex beams (EVB) [10][11][12], i.e., beams with nonzero orbital angular momentum, aimed at probing EMCD at atomic spatial resolution. It was shown theoretically that EVBs need to be of atomic size in order to be efficient for magnetic studies [13][14][15]. Several methods of generating atomic size electron vortex beams have been proposed [16][17][18][19], yet an experimental demonstration of atomic resolution EMCD has not been presented in the literature.An alternative route to utilizing EVBs for magnetic measurements is based on Zeeman interaction between their angular momentum and the magnetic field in the sample. The Pauli equation for an electron with energy E in an electrostatic potential V (r) and a constant magnetic field B c readswhere −e is the electron charge, m is the electron mass, p = −i ∇ is the momentum operator,L andŜ are the orbital and spin angular momentum operators, and Ψ(r) is a two-component spinor wavefunction. The second term on the left hand side of Eq. 1 manifests a coupling between the magnetic field and the orbital and spin angular momenta of the electron beam. A previous study has indicated that the effect of spin on elastic scattering is very weak [20]. Moreover, generating intense spin polarized electron beams remains a technological challenge [21] and so far magnetic field mapping with spin-polarized electrons in the TEM could not be demonstrated. While the spin angular momentum of electrons in the propagation direction is at most 2 , EVBs can be generated with very high orbital angular momenta (OAM) [12,22,23], which permits an in...

Magnetic effects in the paraxial regime of elastic electron scattering

“…For a beam with size significantly larger than a unit cell, the interaction with the uniform part is expected to dominate and a proportionality is expected between angular momentum and magnetic interaction, while in the atomic resolution limit the behaviour should be different. This is also in agreement with the results of recent numerical simulations 13 . Eq.…”

“…How to obtain a microscopic description of B p and A p from first principles electronic structure theory is discussed in the following. The same procedure was applied in recent preceding work 13 . From electronic structure theory, e.g., using density functional theory (DFT), one can obtain the magnetization density as a vector field from the spin-resolved density matrix ρ(r) according to…”

“…Parameter combinations with α = 6 and l ≥ 20 were excluded as they result in large beam size requiring larger supercell sizes. The calculations for non-zero l were performed with a beam initially spin polarized with spin up electrons in the propagation (z) direction, since the OAM magnetic interaction should be very similar regardless of spin 13 . The l = 0 calculations were performed both with spin up and spin down beams in order to look at spin effects.…”

“…Magnetic effects should, however, also appear in the elastic scattering regime and albeit previous suggestions that such effects are very weak 11,12 , it was recently shown in computational work 13 that electron vortex beams carrying tens of quanta of angular momentum should yield a magnetic signal in elastic scattering, which is deemed feasible to detect with modern technology. This appears promising considering that vortex beams with as much as hundreds of of OAM have been produced 8,14 .…”

“…The case of anti-ferromagnet LaMnAsO is also considered in Sec. III C. The previous work 13 suggested that an exper-imentally measurable magnetic signal was only obtainable with large OAM beams, which unfortunately hinders the possibility of high resolution STEM imaging because large OAM also results in large beam dimensions. One important question to be addressed is therefore whether more beneficial conditions can be found, e.g., by varying the beam parameters, in order to allow for atomic resolution magnetic measurements with the suggested technique.…”