Spin polarisation with electron Bessel beams?

Despite the statement of Bohr and Pauli that Stern‐Gerlach based spin separation for electrons cannot work [1], it has been argued that spin separation or filtering of electrons is possible in particular geometries [2,3]. The argument has been debated, see e.g. [4], and it seems that the effect exists but is too small to be exploited with present day technology. As of now, no Stern‐Gerlach design of a spin polarizer for free electrons was successful. On the other hand, an unexpected intrinsic spin‐orbit coupling (SOC) in relativistic vortex electrons was discovered, and it was proposed to use this effect to construct a spin filter for free electrons [5]. Recently, it has been shown [6] that crossed electric and magnetic quadrupole fields correspond to so‐called q‐plates which are used in laser optics for spin‐to‐orbital moment conversion (STOC). In combination with electron vortex beams, this opens the possibility to couple the spin of free electrons to the spatial degree of freedom, and so design a spin filter [7]. However, the realisation of such devices is hampered by severe geometric constraints. Here, we propose a different approach exploiting the magnetic fields created by the lenses already present in conventional TEMs. The vector potential of a round magnetic lens in the TEM has cylindrical symmetry over the propagation axis. This is equivalent to an optical q‐plate. Such a field can be used as a STOC device quite similar to the optics case because the total angular momentum J= L + S is a constant of motion. Thus, it seems that electron microscopes are intrinsic spin polarizers. Basic considerations show that a vortex beam of order one passing a standard magnetic round lens (the objective lens in the present case) is intrinsically spin polarized. As shown in Fig. 1, the vortex in plane A can be seen as a continuous line of point sources (red dot) on the ring aperture, each of which results in a tilted plane wave in B. Classically, the momentum p of the particle in A is tilted by the Lorentz force to p’ at B (grey arrows). The spin vector (red arrows) performs a precession in the magnetic field when going from A to B. Conservation of the total angular momentum J=L+S creates small contributions of Bessel beams J 0 or J 2 , depending on the original spin polarisation in plane A, which are superimposed onto the dominant J 1 beam in plane B. This spin‐to‐orbit coupling allows spin filtering because J 0 and J 2 have different radial profiles. In the limit of infinitely small detectors on axis, the spin polarisation tends to 100 %. Increasing the detector size, the polarisation decreases rapidly, dropping below 10 −5 for standard settings of medium voltage microscopes. For extremely low voltages, the figure of merit increases by two orders of magnitude, approaching that of existing Mott detectors (Fig. 2). Our findings may lead to new desings of spin filters, an attractive option in view of its inherent combination with the electron microscope, especially at low voltage.

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