Figure 1

Schematic of the principle of probe-vortex coupling. (a) Top view of a narrow incident Airy disk (gray), focused at a distance $R$ in scan direction ($x$) from the atom (red). The scattering kernel $T_E$ is symbolized as a diffuse cloud with a color-coded phase (hereafter called rainbow wheel), increasing from blue to red in clockwise rotation for the $\mu =1$ channel. According to Eq. (10), the outgoing wave (gray circle) has acquired a phase ramp $d\alpha =\mu y/R$ in the chiral transition. (b) The phase ramp translates into a shift of the diffraction disk by $\mu /R$ in direction $q_y$. (c) The $q_x$ extension is squeezed into one pixel on the detector in the $(q,E)$ geometry. Note that the energy-loss axis must be perpendicular to $q_y$, which is proportional to the scattering angle $q_y=k_0\theta$. The $(\theta ,E)$ map is a true image taken on magnetite ranging from the oxygen $K$ edge at $\sim 540$ eV to the Fe $L_{23}$ edge at $\sim 710$ eV.Reuse & Permissions

Figure 2

Fe $L_3$ energy filtered real-space exit wave functions ${\varphi }_{\mu }$ [Eq. (7)] of a 100-keV STEM probe of 0.1 nm diameter scanning across a single atom for the three dipole allowed transition channels. From top to bottom: transition channels $\mu =-1,0,+1$. From left to right: distance $R$ to the atom $-2$,$-1$,0,1,2 atomic units. Brightness codes are for intensity of the image, and color codes for the phase of the wave function (rainbow wheel as given in Fig. 1). When the Airy disk sits on the atom (center column), the outgoing beam is a real vortex with $\mu \in [-1,0,1]$. At a distance, phase ramps in the Airy disks develop, changing sign with $R$ and with $\mu$. Each square has a side length of 5 atomic units ($\sim$0.26 nm).Reuse & Permissions

Figure 3

Fe $L_3$ energy filtered diffraction patterns $\rho {(\mathbf{q},\mathbf{q})}_{\mu }$ [Eq. (8)] of a 100-keV STEM probe of 0.1 nm diameter scanning across a single atom, corresponding to Fig. 2. From top to bottom: transition channels $\mu =-1,0,+1$. From left to right: distance $R$ to the atom $-2$,$-1$,0,1,2 atomic units. Note the shift of the patterns for $\mu =\pm 1$ in vertical direction $q_y$ as predicted in Fig. 1b, depending on the position of the probe, and the inversion of shifts with change of sign. The $\mu =0$ channel does not show any shift because there is no phase ramp in the image. Convergence angle 18 mrad. The images have a side length of $\pm 50$ mrad.Reuse & Permissions

Figure 4

Fe $L_3$ energy filtered $(R,q_y)$ signal distribution of a STEM probe of 0.1 nm diameter scanning across a single atom, corresponding to Fig. 2. (a) Spin-up polarization of the atom; (b)_ spin-down polarization. The top/bottom asymmetry is well visible and can be used to determine the spin polarization. (c) Full line: intensity integrated over the upper detector half (20 to 50 mrad) line scan of a STEM probe of 0.1 nm diameter across a single atom, corresponding to (a). Dashed line: same for the lower detector half ($-50$ to $-20$ mrad).Reuse & Permissions

Figure 5

HAADF image of the border of the magnetite nanoparticle in the [111] zone axis orientation. The scan line is indicated as a box, the axis of energy dispersion (arrow) is almost perfectly parallel to the scan direction, thus $q_y\left|\right|y$. The bright dots are the A columns containing three Fe atoms per unit cell in this projection. Their projected distance is 0.33 nm.Reuse & Permissions

Figure 6

(a) Scan of the Fe $L_3$ white line signal after standard background subtraction, drift correction, and removal of the continuum. The filled curve shows the scan using the upper detector half (20 to 50 mrad), the empty curve is a scan using the lower detector half ($-50$ to $-20$ mrad). The HAADF signal is superposed (dashed) with maxima indicating the positions of the A columns. The B columns are visible as faint subsidiary peaks. (b) Same scans after Fourier filtering. Atom positions (HAADF maxima) are marked with vertical lines.Reuse & Permissions

Figure 7

Hypothetical line scans showing the first derivative $d[{\rho }_{+}\left(R\right)-{\rho }_{-}\left(R\right)]/dr$ over an atomic row as in Fig. 5 for (a) ferromagnetic, (b) antiferromagnetic, and (c) ferrimagnetic ordering. (c) was calculated with a relative strength of $-30$$\%$ for the spin-down moments, thus taking into account the strong channeling at the A columns.Reuse & Permissions

Figure 8

The phase and amplitude of the incident Airy disk of 0.1 nm diameter (left) and for the scattered wave from the $\mu =1$ transition. The atom is displaced from the beam center to the left by 0, 0.01, and 0.05 nm. The images have a side length of 0.2 nm.Reuse & Permissions

Figure 9

Intensity of the incident Airy disk (left) and $\text{Tr}\left[\rho \right]$ for the scattered wave (mixed state from three transition channels) for atom displacements as in Fig. 8.Reuse & Permissions

Figure 10

(a) Radial profiles of the incident 1-Å probe (dashed line) and the outgoing electron density of the mixed state when the atom is exactly centered on the beam. (b) The same for an incident probe of 5 Å diameter.Reuse & Permissions