Three-Dimensional Study of the Vector Potential of Magnetic Structures

The vector potential is central to a number of areas of condensed matter physics, such as superconductivity and magnetism. We have used a combination of electron wave phase reconstruction and electron tomographic reconstruction to experimentally measure and visualize the three-dimensional vector potential in and around a magnetic Permalloy structure. The method can probe the vector potential of the patterned structures with a resolution of about 13 nm. A transmission electron microscope operated in the Lorentz mode is used to record four tomographic tilt series. Measurements for a square Permalloy structure with an internal closure domain configuration are presented.

Figure 1

(a) The vector potential (red arrows) of a square particle with a closure magnetic domain configuration (magnetization is represented by blue arrows); black lines show the locations of 90° magnetic domain walls. (b) shows the basic setup of a VFET experiment in a TEM. Electrons travel along the direction of the green arrows, and experience a phase shift when they traverse the region around the magnetized thin foil (gray square prism). (c)–(e) show under-, in-, and over-focus Fresnel images of a square Py island. The vortex location is indicated with white arrows in both out-of-focus images.

Figure 2

(a) Schematic of the square island and coordinate axes; (b) line profile along the line AB of (a) of the theoretical (solid line) and reconstructed (crosses) $A_z$ component of the vector potential; (c) gray-scale plots of the theoretical (upper row) and reconstructed (bottom row) $A_z$ component of the vector potential along the planes indicated in (a).

Figure 3

Line profiles of the theoretical solid line) and reconstructed (symbols) $B_x(0,y,0)$ (a) and $B_y(x,0,0)$ (b) components of the magnetic induction.

Figure 4

3D visualization of all the physical quantities that can be recovered from a single VFET experiment, namely, (a) the 3D magnetic induction, and (b) the 3D vector potential. The gray outline in (a) and (b) indicates the shape of the square island.