Three-dimensional magnetization reconstruction from electron optical phase images with physical constraints

The ability to characterize three-dimensional (3D) magnetization distributions in nanoscale magnetic materials and devices is essential to fully understand their static and dynamic magnetic properties. Phase contrast techniques in the transmission electron microscope (TEM), such as electron holography and electron ptychography, can be used to record two-dimensional (2D) projections of the in-plane magnetic induction of 3D nanoscale objects. Although the 3D magnetic induction can in principle be reconstructed from one or more tilt series of such 2D projections, conventional tomographic reconstruction algorithms do not recover the 3D magnetization within a sample directly. Here, we use simulations to describe the basis of an improved model-based algorithm for the tomographic reconstruction of a 3D magnetization distribution from one or more tilt series of electron optical phase images recorded in the TEM. The algorithm allows a wide range of physical constraints, including a priori information about the sample geometry and magnetic parameters, to be specified. It also makes use of minimization of the micromagnetic energy in the loss function. We demonstrate the reconstruction of the 3D magnetization of a localized magnetic soliton — a hopfion ring — and discuss the influence of noise, choice of magnetic constants, maximum tilt angle and number of tilt axes on the result. The algorithm can in principle be adapted for other magnetic contrast imaging techniques in the TEM, as well as for other magnetic characterization techniques, such as those based on X-rays or neutrons.