Demonstration of angular-momentum-resolved electron energy-loss spectroscopy

Abstract

Rotational invariance is a fundamental aspect of symmetry in scattering processes from atomic potentials. Here, we present an approach for measuring orbital angular momentum (OAM), a key descriptor of rotational symmetry, during measurements of atomic transitions. We use an electron optical OAM sorter in combination with electron energy-loss spectroscopy and model-based fitting to separately measure the π* and σ* antibonding transitions in hexagonal boron nitride on the atomic scale. This approach also offers prospects for efficient and atomically-resolved magnetic chiral dichroism measurements.

Fig. 1. Experimental setup for OAM-EELS.

a Schematic diagram of the experimental setup, which includes two sorter elements (S1 and S2) after the sample plane and a spectrometer to realize OAM-EELS double dispersion. b Schematic diagram showing electron beam illumination of h-BN with a finite convergence semi-angle and the effect on the electron wavefunction of single scattering at the B K-edge. c Simulated OAM-EELS measurement for an OAM resolution of ∆ℓ = 1.1ħ.

Fig. 2. Experimental OAM-EELS from the B K-edge.

a Experimental OAM-EELS zero-loss spectrum and b B K-edge spectrum after background removal. c Corresponding experimental OAM profiles at $E_1=191$ eV (dark blue line) and $E_2=198$ eV (olive green line) from the B K-edge compared with the ZL profile (psf) at 0 eV (orange line). d Discretized OAM profiles at $E_1$ (dark blue bars) and $E_2$ (olive green bars) after psf deconvolution.

Fig. 3. Numerical model for simulating scattering delocalization.

a Effect of probe size and scattering delocalization for $m=+1$ for a B atom on the optical axis of the OAM sorter (left column) and for two B atoms off-axis. Each other B atom in the illuminated area (shown) emits its own inelastically-scattered wave. In the far field, the phase of the off-axis atoms is distorted. The OAM sorter records an incoherent sum Γ_m(ℓ) of their contributions (teal bars, the off-axis atom intensities have been multiplied by two). b Simulated OAM profiles Γ_m (ℓ) for m = 0 (dark blue bars) and m = ±1 (m = +1 teal bars, $m=-1$ olive green bars).

Fig. 4. Results of model-based fitting.

a Reconstructed OAM-EELS I(E,m) spectrum after deconvolution. A Gaussian modifier with ∆ℓ ≈ 1.1ħ was used for comparison with Fig. 1c. b EEL spectra ${\text{c}}_m\left(E\right)$ obtained from the experiment. The dashed lines are results of the fit ($m=0$ teal dashed line, $m=+1$ orange dashed line). The full lines are smoothed versions ($m=0$ dark blue line, $m=+1$ olive green line). c Corresponding EELS/XANES spectra for $m=0$ (π*, dark blue line) and $∣m∣=1$ (σ*, olive green line) obtained from ab initio DFT calculations.