Characterizing Magnetic Skyrmion Ordering and Dis-Ordering in the Presence of Crystalline Dislocations using Lorentz Transmission Electron Microscopy

Topological spin textures, such as magnetic skyrmions, present a novel opportunity to study the relationship between atom-scale details and microscale ordering of spins. In Co-doped Fe5GeTe2 (FCGT), a small change in Co concentration to exactly 50% causes a structural phase change allowing for ferromagnetism and the Dzyaloshinskii-Moriya interaction, leading to the appearance of magnetic Néel skyrmions [1]. Interactions between skyrmions and the surrounding crystal govern temperature- and magnetic field-dependent ordering [2]. These studies describe the long-range ordering of skyrmions, but do not elucidate the mechanics of skyrmion-skyrmion, skyrmion-crystal, or skyrmion-defect interactions. Lorentz transmission electron microscopy (LTEM) is a convenient technique to study magnetic textures under a range of magnetic field and temperature conditions, in the same tool that can measure crystalline defects on an atomic scale. Even without atomic resolution, crystalline defects such as dislocations are apparent in diffraction contrast. Here we investigate skyrmion ordering and dis-ordering in the presence of crystalline defects. LTEM images of skyrmions were taken by cooling FCGT in the TEM under a magnetic field applied by slightly exciting the objective lens. FCGT has Néel skyrmions, so the sample is tilted slightly in the microscope to create contrast. These skyrmions are clearly visible in Fig. 1a, along with several line defects in the crystal. These defects show clear diffraction contrast when imaging on-axis (0001) in conventional TEM mode (Fig. 1b). Tilting the sample into a two-beam condition (Fig. 1c,d) causes certain sets of these defects to become invisible, suggesting that they are dislocations in the crystal lattice. Here we see that various dislocations have Burgers’ vector in the [1-210] directions (Fig 1c) and or the [03-30] directions. LTEM can image both skyrmions and dislocations in the crystal lattice, allowing us to probe the interactions between skyrmions and crystalline defects and the effects on ordering. We collected LTEM images during field cooling and subsequent reheating. FCGT was heated to its Curie temperature, then cooled under an applied field to 210K. The field was removed, then the sample was heated to 250K. Images were collected at several points during cooling and skyrmions were manually identified to analyze their ordering. Under nonzero magnetic field, skyrmions are uniformly small and non-circular (Fig. 2a,j,k). At zero field, skyrmions increase in size but the low temperature prevents equilibration, leading to a non-uniform skyrmion size distribution (Fig. 2d,j). With increased temperature, skyrmions continue to grow but have a slightly more uniform size distribution with more circular shape (Fig. 2g,j,k). Voronoi cells for each skyrmion were calculated and used to assess ordering in proximity to dislocations. In a perfect skyrmion lattice, each skyrmion would have 6 nearest neighbors and each Voronoi cell would be a perfect hexagon with an angle of 60° between the centers of adjacent skyrmions. In Fig. 2b,e,h a map of the number of nearest neighbors of each skyrmion is overlaid on the LTEM image. As FCGT reaches zero field and is heated, larger areas of uniform 6-nearest-neighbor skyrmions appear, as is also suggested by the histogram in Fig. 2m. To assess the angle between adjacent skyrmions, the bond orientational parameter, ψ6, defined by Nelson et al. is measured, following a similar analysis to Huang et al. [3,4]. The magnitude of the ψ6 parameter increases as the magnetic field is removed and FCGT is heated (Fig 2c,f,g,n). This increase in ordering is the opposite of the hexatic melting phenomena predicted by Kosterlitz–Thouless–Halperin–Nelson–Young (KTHNY) theory but is only observed relatively far below the Curie temperature of FCGT. Irregular skyrmions (more or fewer than 6 nearest neighbors, lower |ψ6|) tend to collect along dislocations in the most ordered images (Fig 2h,g), suggesting that skyrmion dis-order is caused by the crystalline dislocations in FCGT. By comparing the location of atomic-scale crystalline defects to the ordering of magnetic skyrmions imaged with LTEM, the dis-ordering effects of dislocations on skyrmions has been observed [5]. Summary of defects in FCGT. A) LTEM image collected at T = 210K, B = 0T showing magnetic skyrmions and crystalline defects. B) On-axis bright field TEM image of the same region taken along the 0001 zone axis. Inset: diffraction pattern (left) and schematic diffraction pattern (right). C) Bright field TEM image collected in a two-beam condition exciting the -3030 diffraction condition as illustrated in the inset. Some defects are not visible. D) Bright field TEM image collected in a two-beam condition exciting the 2-1-10 diffraction condition. A different set of defects have become partially invisible. Analysis of skyrmion ordering. LTEM images collected in the same area as Fig. 1 at (A-C) T = 210K and B = 0.25T, (D-F) T = 210K, B = 0T, and (G-I) T = 250K, B = 0T. A, D, G) show raw LTEM images with skyrmions and dislocations. B, E, H) Voronoi cell map with coloring indicating the number of nearest neighbors of each skyrmion. C, F, I) Voronoi cell map with coloring indicating the magnitude of the ψ6 ordering parameter. J) Histograms showing the nominal skyrmion radius at each of the three conditions imaged. K) Histograms showing the skyrmion circularity at each of the three conditions imaged. M) Histograms showing the number of nearest neighbors of skyrmions at each of the conditions imaged. N) Histograms showing the magnitude of the ψ6 ordering parameter at each of the conditions imaged.