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.