**ALBA Synchrotron**

A group of scientists from the University of Oviedo and ALBA has been able to identify and measure for the first time the magnitudes of topological charges inside a thin ferromagnetic film, the most commonly used morphology in nanomagnetic devices. To achieve this result, the 3D magnetization inside the film was determined with magnetic tomography at MISTRAL beamline at ALBA.

A) Perpendicular magnetization of the trilayer film as obtained after reconstructing the magnetization from a dichroic tomogram in MISTRAL acquired at the L3 absorption energy of Fe. Two bifurcation are marked by D1 and D2. The scale bar is 525 nm. B) Emergent magnetic field lines and vectors generated at the at the central part of the film from bifurcation D2 that hosts a magnetic monopole of negative topological charge as indicated by the converging arrows.The different colors of the arrows are for visual clarity. The side of the cube is 100 nm.

Topological charges in magnetic materials

Cerdanyola del Vallès, 15 December 2020

When a **ferromagnetic material** is saturated by applying a relatively intense magnetic field, its magnetization becomes uniform which means that any volume element has the same magnetization. If a field not too strong is applied to reverse the magnetization, then, at some parts the magnetization is as it was originally, whereas at others is inverted. These two opposite magnetizations give rise in some cases to weird magnetic distributions named **magnetic singularities **that have interest by themselves.

Imagine a tropical storm carrying a typhoon which is a singularity in the field of velocities of the wind and that is stable enough to travel many kilometers in strong winds. Something similar exists in magnetic materials. Vortices of magnetization, which have geometrical resemblance to a typhoon, can travel mesoscopic distances while being pushed by applied magnetic fields analogous to the wind of a storm.

The stability of **magnetic singularities** has practical interest since they may be **information carriers in magnetic devices** and, due to this, they are being intensively investigated. Vortices are the simplest singularities but there are many more types. They are characterized by their so-called **topological charges **which are numerical quantities that are associated to the geometrical characteristics of the magnetization.

If the magnetization near the singularity exhibits a rotational distribution around two or more rotation axes, resembling a 2D swirling, then the topological charge is not zero. More precisely, one defines a generalized vectorial curvature which is designed as **Berry curvature** and has quantum origin. The product of the Berry curvature with the Planck constant is identified as an **emergent magnetic field B**e In analogy with electrostatic charges, the flux of **B**e through a closed surface S is proportional to the topological charge enclosed by S which is quantized. In the language of Maxwell equations this reads div **B**e = 4

πℏ

ρ being ρ the volume density of topological charge. Vortex distributions have ±1/2 topological charge due to the magnetization core that is normal to the vortex rotation plane whereas skyrmions, which are 2D singularities, have charge +/-1 (except in some special cases where the magnetization makes several turns). A particularly important singularity is the **magnetic monopole** or Bloch point that is a **3D singularity with charge ±1**. In this case, the field **B**e is radially symmetric around the monopole, as the electric field created by a point charge, and, similar to electrostatics, the flux across a surface including the charge allows to determine its magnitude.

X ray magnetic tomography

In this study, the magnetization was probed with X-rays tuned at specific energies where magnetic dichroism is particularly intense. At , using circularly polarized X-rays, transmission images sensitive to the magnetization of the film were acquired at different angles. **This** **tomographic method allowed determining the 3D magnetization in the film**. The method has limitations due to the large aspect ratio of the sample (lateral dimension >> thickness) and to the limited angular range that could be accessed due to geometrical constrains in the sample holder. In addition, the effective film thickness increases when the angle of incidence of the X-rays departs from the normal of the surface of the film and also parallax distortions complicate the analysis. In spite of all these limitations, the accuracy of the reconstructed magnetization was enough to evaluate the emergent field **B**e mentioned above and to evaluate the surface integrations for localizing the topological charges. The successful results required more than two years of developments and several beamtimes.

The figure depicts the magnetization perpendicular to the film of magnetic domains in a trilayer film Py(80nm)/NdCo(80nm)/Py (80nm) (Py= permalloy) fabricated with DC sputtering. When the magnetization is being reversed, the film spontaneously develops magnetic domains forming stripes with upwards and downwards magnetization as shown in panel A of the figure. Some of the stripes have bifurcations as signaled in the figure by D1 and D2. It is precisely at the bifurcations where the magnetic singularities are hidden.

Two tomographic series of dichroic images allowed to reconstruct in 3D the magnetization and evaluate the emergent field **B**e enabling to localize and identify the topological charges. Panel B corresponds to the field originated by a monopole of charge -1 located at the center of the cube. The dark central part inside the cube locates the position of the singularity which is determined within the resolution of the new method. Interestingly enough, dislocation D1 that looks in panel A similar to D2 is completely different from the monopole since it corresponds to a 2D singularity of charge which is attributed to a meron with nominal charge -1/2.

In conclusion, the results demonstrate **a method based exclusively on experimental data without any a priori assumption on the magnetization, which allows to identify and to determine the nature of magnetic singularities in thin films**. These results have fundamental interest in nanomagnetism and may have applications in spintronics.

**Reference**: A. Hierro-Rodriguez, C. Quirós, A. Sorrentino, L. M. Alvarez-Prado, J. I. Martín, J. M. Alameda, S. McVitie, E. Pereiro, M. Vélez^{ }and S. Ferrer. Revealing 3D Magnetization of Thin Films with Soft X-ray Tomography: Magnetic Singularities and Topological Charges. *Nat Commun* **11,** 6382 (2020). https://doi.org/10.1038/s41467-020-20119-x