Mapping the magnetization fine structure of a lattice of Bloch-type skyrmions in an FeGe thin film

Magnetic skyrmions are topologically protected spin textures that are attracting considerable attention due to their interesting physical properties and potential applications in energy-efficient spintronic devices for information technology.^1,2 Particular interest in magnetic skyrmions arose after the discovery of their current-induced motion at very low current densities,³ suggesting that they could be used as building blocks in new generations of recording media. The spin structure of a magnetic skyrmion originates from the Dzyaloshinskii-Moriya interaction (DMI), a strong spin-orbit interaction that appears in crystals with broken inversion symmetry, including B20-type structures,⁴ as well as at interfaces in thin magnetic films.⁵ In thin films of B20 chiral magnets, a regular hexagonal skyrmion lattice (SkL) of Bloch-type skyrmions is formed in the presence of a small magnetic field applied parallel to the surface normal direction.⁶ 

Experimental observations of magnetic skyrmions were first made using neutron diffraction.⁴ Subsequently, several different phase contrast techniques in the transmission electron microscope (TEM) have been used to image skyrmions in real space. The phase shift of the electron wave that has passed through a sample in the TEM can be expressed in the following form:

where x and y are directions in the plane of the sample, z is the incident electron beam direction, C_E is an interaction constant (6.53 $×106$ rad V⁻¹m⁻¹ at an accelerating voltage of 300 kV), V₀ is the mean inner potential of the sample, e is the electron charge, $ℏ$ is the reduced Planck constant, and A_z is the component of the magnetic vector potential $A$ along z. The Fresnel mode of Lorentz TEM,^7,8 which has been used to identify the formation of skyrmions and to follow their dynamics,^9,10 records a signal that is approximately proportional to the second derivative of the phase. Differential phase contrast (DPC) imaging¹¹ in the scanning TEM, which has been used to image a magnetic SkL at a domain boundary in B20-type FeGeSi¹² and to study the internal structures of skyrmions,¹³ records a signal that is approximately proportional to the first derivative of the phase. Off-axis electron holography¹⁴ (EH) provides direct access to the phase shift with high spatial resolution and high phase sensitivity. The technique has previously been applied to study the structures of helical and skyrmion phases in B20-type FeCoSi¹⁵ and the internal and lattice structure of skyrmions in FeGe as a function of applied magnetic field and temperature.¹⁶ 

All phase contrast techniques in the TEM are sensitive to the projected in-plane components of the magnetic induction $B=μ0(H+M)=∇×A$ inside and outside the specimen, where $M$ is the magnetization, $H$ is the auxiliary (or demagnetizing) field, and μ₀ is the permeability in vacuum. Until now, only the measurement and mapping of the projected in-plane components of the $B$-field was possible and the $M$-field was not accessible. Recently, we developed an iterative model-based algorithm, which can be used to determine the in-plane magnetization M_xy, projected along the z direction, from the magnetic contribution to the phase $ϕM$ recorded using off-axis EH by making use of a discretized numerical implementation of a forward model based on trial solutions for the magnetization. A short description of the approach is given in the supplementary material.

Here, we study the shapes and fine structures of Bloch-type skyrmions in B20-type FeGe thin films using off-axis EH. We determine the projected in-plane magnetization M_xy of individual skyrmions from electron-optical magnetic phase images and compare our results with micromagnetic simulations. Our measurements of magnetization allow local variations in skyrmion shape and core diameter with applied magnetic field, as well as the magnetic texture in inter-skyrmion regions, to be determined quantitatively.

Single crystalline B20-type FeGe was synthesized using a chemical vapor transport method.¹⁷ A 100-nm-thick lamella with a ⟨110⟩ surface normal was prepared for TEM examination by focused ion beam milling with $Ga+$ ions using a dual-beam scanning electron microscope (Hitachi NB 5000). Off-axis electron holograms were recorded using a positively charged biprism in an image spherical aberration (C_s) corrected FEI Titan 60–300 TEM operated in Lorentz mode at an accelerating voltage of $300 kV$⁠. With the C_s of approximately $0.2 mm$⁠, the point resolution in Lorentz mode is estimated to be $1 nm$⁠. The electron holograms were recorded with a fringe spacing of $2.4 nm$⁠, a fringe contrast of $15 %$⁠, resulting in a spatial resolution of the reconstructed phase of $7.2 nm$⁠, and a phase resolution of $0.15 rad$⁠. A pre-calibrated out-of-plane magnetic field was applied to the sample during off-axis EH using the conventional microscope objective lens. The magnetic contribution to the phase [the second term in Eq. (1)] was separated from the mean inner potential contribution [the first term in Eq. (1)] by taking the difference between phase images recorded at $200 K$ and at room temperature, when FeGe is non-magnetic (⁠$TcFeGe=278.7 K$⁠), on the assumption that there are no changes in diffraction contrast, specimen charging, or mean inner potential between the two temperatures.

Figure 1 shows experimental magnetic phase images of skyrmions in FeGe for different values of the magnetic field applied parallel to the electron beam direction. The sample was field-cooled in the presence of a $100 mT$ magnetic field to $95 K$ in order to generate a SkL and then heated to $200 K$⁠. The SkL did not form a perfect arrangement but contained a dislocation, whose core is marked in Fig. 1(a). The presence of the dislocation results in local geometrical changes to skyrmion shape, as reported recently using off-axis EH.¹⁸ As the applied magnetic field is increased, the hexagonal SkL changes progressively to a more random distribution of individual skyrmions, as shown in Figs. 1(b) and 1(c). The distance between adjacent skyrmions increases with applied magnetic field, until they are eliminated in the saturated ferromagnetic state. The magnetic field dependence of the internal structures of skyrmions is described in detail in Ref. 16. The step in the magnetic contribution to the phase between the centre and the edge of each skyrmion for the present specimen thickness is measured to be approximately 0.6 rad. A color-contour map, which was generated directly from a magnetic phase image and provides a visual representation of the projected in-plane magnetic induction, is shown in Fig. 1(d). The magnetic field of the skyrmions rotates counterclockwise.¹⁹ The measured saturation magnetization is approximately $100 kA m−1$ at 200 K, which is somewhat lower than the expected value²⁰ of $384 kA m−1$ at 0 K, although other experimental studies have also reported reduced magnetization values for FeGe thin films.^21,22 The difference is likely to be due to the temperature of 200 K used in our experiment, the surface chiral twist of the spin structure of each skyrmion, the presence of a non-magnetic surface layer resulting from specimen preparation and possible inaccuracy in the determination of the specimen thickness.

Individual skyrmions from the SkL and the random skyrmion distribution were selected for analysis by determining M_xy from the recorded magnetic phase $ϕM$ using the model-based iterative algorithm, as shown in Fig. 2. The resulting maps of the magnitude of M_xy in Fig. 2 reveal that at $100 mT$ the skyrmions in a SkL have hexagonal shapes, while at $400 mT$ they have circular shapes. A shear in the hexagonal skyrmion shape by approximately $5 nm$ results from the presence of the dislocation in the planar skyrmion arrangement, as marked in Fig. 1(a). The distance between the dislocation core and the chosen skyrmion is approximately $250 nm$⁠, highlighting the extent of the distortion in the skyrmion lattice and the flexibility of the skyrmion shape. As the magnetic field is increased, the skyrmion core diameter decreases from $13.3 nm$ in the SkL to $9.5 nm$ in individual skyrmions, in agreement with theoretical predictions²³ and previous experimental observations.^13,16

Figures 3(a) and 3(b) show the magnitude of the projected in-plane magnetization and a corresponding vector map for an inter-skyrmion region in the SkL. Intensity linescans between neighboring skyrmions are shown in Figs. 3(c) and 3(d). Unexpected peaks in the magnetization distribution are visible in Fig. 3(d).

The experimental results were compared with a classical micromagnetic model for an isotropic chiral magnet that includes the following energy terms: the Heisenberg exchange interaction, the DMI, the interaction with the applied field (the Zeeman term), and the interaction with the demagnetizing field, in the form

where the integration extends over the volume $Vs$ of the sample, $m̂≡m̂(x,y,z)$ is a continuous unit vector field $m̂=M/Ms$⁠, and the saturation magnetization of B20-type FeGe $Ms=384 kA m−1$⁠. In Eq. (2), $Bext$ is the externally applied magnetic field and $Bd$ is the demagnetizing field generated by the divergence of the magnetization in the sample and at its surface. The simulations were performed using MuMax3.²⁴ For a correct description of the demagnetizing field, we used periodic boundary conditions with a repetition of the simulated domain in the x and y directions, in order to describe the extended film.¹⁶ The values of micromagnetic exchange stiffness $A=8.78$ pJ/m and DMI constant $D=1.576$ mJ/m² were chosen to fit the experimentally measured value of the equilibrium period of the helical spin spiral $LD$ in FeGe of 70 nm, according to the well-known analytical expression $LD=4πA/D$⁠. The equilibrium period of a hexagonal unit cell of the SkL, corresponding to the lowest energy density, was found by direct energy minimization using the method of conjugate gradients implemented in MuMax3.

Figures 4(a) and 4(b) show the simulated magnetization distribution within the hexagonal unit cell and the projected magnetization averaged through the film thickness, respectively. Line profiles of the magnetization along the directions marked in Fig. 4(b) are shown in Figs. 4(c) and 4(d). In the line profile shown in Fig. 4(c), which joins neighboring skyrmions, the chain of spins exhibits a Bloch-like domain wall with a fixed right-handed vorticity. Remarkably, in Fig. 4(d), the vorticity of the transition region between more distant skyrmions shows a violation of chirality. In between the points that are indicated by red arrows, the modulation has a left-handed vorticity. Figures 4(e) and 4(f) show the values of the simulated in-plane component of the average magnetization between the skyrmions along the two orthogonal directions in the hexagonal SkL indicated in Figs. 4(a) and 4(b). The $|〈Mxy〉|/Ms$ component [Fig. 4(e)] is in a good agreement with the experimentally measured in-plane magnetization (Fig. 3). This behavior can be explained by a competition between different energy terms, which are associated with a local decrease in energy due to the DMI, and result in a reduction in the total energy of the system. A detailed analysis of such an interplay between energy terms, supported by a visualization of DMI and exchange energy density maps within a skyrmion unit cell, has been presented in Ref. 13. In the present study, such a non-trivial character of the modulation within the skyrmion lattice is responsible for the appearance of two maxima in the projected in-plane magnetization shown in Fig. 4(f). The observed peaks are only approximately 20% of the height of the maximum in the line profile and could not have been identified or measured readily from standard Lorentz TEM images. Intensity analyses of Lorentz TEM images are given in the supplementary material.

In summary, we have measured and interpreted the local magnetization distribution in a lattice of Bloch-type skyrmions in a B20-type FeGe thin film using off-axis electron holography, in combination with the use of a model-based inversion algorithm and micromagnetic simulations. Skyrmions in a close-packed honeycomb lattice possess a hexagonal magnetization distribution that is sensitive to the presence of lattice defects. Upon magnetic field saturation, the skyrmion lattice breaks down into a random distribution of skyrmions that have circular shapes and reduced core diameters. The magnetization distribution in the interstitial region between adjacent skyrmions in a lattice arrangement shows a Bloch-wall-type transition between neighboring skyrmions. Between more distant skyrmions, the magnetization distribution shows a more complex modulation with a change in chirality. This newly resolved magnetic fine structure raises questions about the magneto-transport properties of electrons in skyrmion lattices. Off-axis electron holography combined with magnetization calculations provides a high spatial resolution, quantitative technique for the measurement of local magnetic properties.

See supplementary material for the description of the model-based iterative reconstruction method and for the image intensity analysis of Lorentz defocused images.

R.D.-B., J.C., A.S., and A.K. are grateful to the European Research Council for financial support under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 320832. K.S., N.K., and Y.T. acknowledge financial support from the Grant-in-Aid for Scientific Research (Grant Nos. 24224009 and 14J09358) from MEXT and the FIRST Program in Japan. K.S. was supported by the Japan Society for the Promotion of Science through a Program for Leading Graduate Schools (MERIT). A.S. is grateful for support from the Ukrainian State Fund for Fundamental Research (Project No.: F71/59-2017). Z.-A. L. acknowledges the financial support from the NSF of China (No.11774403).