Dynamical diffraction effects in STEM orbital angular momentum resolved electron energy-loss magnetic chiral dichroism

Matteo Zanfrognini,Jan Rusz,Rafal E Dunin Borkowski,Stefano Frabboni,Vincenzo Grillo,Enzo Rotunno,Ebrahim Karimi

doi:10.1103/physrevb.102.184420

Abstract

In this work we derive sum rules for orbital angular momentum(OAM) resolved electron magnetic chiral dichroism (EMCD) which enable the evaluation of the strength of spin and orbital components of the atomic magnetic moments in a crystalline sample. We also demonstrate through numerical simulations that these rules appear to be only slightly dependent from the dynamical diffraction of the electron beam in the sample, making possible their application without the need of additional dynamical diffraction calculations.

Highlights

Since the work of Schattschneider 1 and collaborators, electron magnetic circular dichroism (EMCD) has stimulated the attention of many researchers in the field of electron microscopy because of its potential of providing information about magnetic properties of materials with sub-nanometric resolution
As in the case of X ray circular dichroism (XMCD)2,3 sum rules for EMCD, independently derived by Calmels4 and Rusz5 permit in principle to quantify the orbital and the spin components of the magnetic moment per atom in the sample, even if a practical application of these rules is made complicated by dynamical diffraction effects, which introduce thickness dependent factors to be evaluated by separate dynamical calculations
In this work we derive sum rules for the orbital and spin components of the magnetic moments of the atoms in a crystalline sample for the specific case of zone axis orbital angular momentum (OAM) resolved STEM-EMCD, a technique recently proposed in Ref. 6: in such a proposed experiment, both the energy and the OAM spectra7 of the electrons, having experienced a core-loss process, are measured and differences among the l=±1 spectra are expected in the case of magnetic materials

Summary

INTRODUCTION

Since the work of Schattschneider 1 and collaborators, electron magnetic circular dichroism (EMCD) has stimulated the attention of many researchers in the field of electron microscopy because of its potential of providing information about magnetic properties of materials with sub-nanometric resolution. We underline that these conclusions have general validity, independently from the chosen material and its symmetry: the only requirements are orienting the crystal along an high symmetry direction and using an electron probe characterized by strong channeling properties along the chosen atomic column Under these approximations, we can write the inelastic signal experimentally observed at energy ∆‫ ܧ‬and at OAM lħ = ±ħ as. (in log scale) as a function of the semi-collection angle ߚ, computed for different STEM probe convergences This ratio points out that ‫ܨ‬௫௫(l = +1, ߚ) is at least an order of magnitude larger than ‫ܨ‬௭௭(l = +1, ߚ), and such a ratio decreases by increasing the probe convergences, i.e. the channeling capability of the incoming electron beam: this can be justified remembering that the contribution of the atoms (0,0,az) to the function ܲ௭௭(l = +1, ݇) (and so to ‫ܨ‬௭௭(l = +1, ߚ)) is zero by symmetry as clarified by Eq 4.1.

DERIVATION OF SUM RULES FOR OAMRESOLVED EMCD

Findings

CONCLUSIONS