Observation of FeGe skyrmions by electron phase microscopy with hole-free phase plate

Measuring the size, areal density and periodicity of nanoscale magnetic objects is a key in understanding their behavior and can be useful for practical application. For example, the spin configuration of nanoscale magnetic vortices, called magnetic skyrmion, exhibits novel Hall effects and spin transfer torque at low electric current.^1–3 To investigate skyrmions at the nanoscale, transmission electron microscope (TEM) has proven to be very useful, especially, Lorentz microcopy (LM), small-angle electron diffraction (SmAED) and electron holography (EH). These practical TEM methods allow us to reveal the spin configuration and behavior of skyrmion in detail.^4,5 For example, in Refs. 6–8 we have reported the formation process and stabilization of nanoscale magnetic bubbles as well as Foucault and SmAED imaging in a TEM.^9,10 However, detecting and interpreting the spatial variations in magnetic field in TEM using LM can be difficult. SmAED has an advantage of detecting magnetization in diffraction, but a disadvantage in analyzing skyrmions individually in real space images. Electron holography (EH) enables quantitative interferometric measurement of the electron phase.^11,12 A limitation of EH arises from the disturbance of the reference wave by the investigated magnetic field itself. For these reasons, we investigate the observation of skyrmion with a hole-free phase plate (HFPP).^13–15 A potential advantage of the HFPP, as compared to LM, is that the HFPP images are obtained in focus. In focus HFPP images do not exhibit Fresnel fringes and make it possible to extract the local magnetization from the HFPP images. The HFPP does not require a reference wave as needed in EH. The HFPP imaging was previously used for a ferromagnetic sample, (Pr,Dy)₂Fe₁₄B.¹⁶ The sample in Ref. 16 exhibits high magnetization resulting in strong magnetic contrast. Here we apply the HFPP imaging to the observation of magnetic skyrmion to demonstrate the usability of HFPP imaging of weak magnetic contrast of a nanoscale magnetic structure. A FeGe, helical magnet sample containing skyrmions, was observed and their response to external magnetic fields was investigated.

The phase shift of the HFPP is caused by the charge distribution by the incident electron beam and is not spatially uniform. The relation between image contrast and quantitative phase shift is formulated by Beleggia.¹⁷ Here we discuss a simplified image formation model by deriving an expression of the Zernike type phase plate, i.e. in case of the additional phase shift η is applied by the HFPP to the electron wave in the close vicinity of the unscattered electron beam. The electron wave ψ₀ is described by the axial electron wave which is comprised of the unscattered wave traveling through the sample (reference wave ϕ_ref), and the wave scattered by a phase object ϕ_obj:

A_ref and A_obj are the amplitudes of the respective electron waves. Assuming that the sample is a pure phase object, we can describe the image intensity I(x, y) using the amplitude $A=Aref=Aobj=12$ for normalization and setting ϕ_ref = 0

In Zernike phase contrast mode, a phase plate gives the phase shift η, multiplying the part of the object spectrum by a phase factor exp(iη) as follows:

the last term uses the weak phase object approximation, η >> ϕ_obj, which is applied for the image intensity. Then it can be simplified into the form:

resulting in the linear response of I(x, y) to ϕ_obj. Equation (4) implies that the image contrast in a HFPP image changes linearly with the object phase shift, regardless of the mechanism responsible for the phase shift and the actual phase shift magnitude due to the HFPP.¹⁸ 

The magnetic skyrmion is a phase object originating from magnetic field distribution within a magnetic material, and thus, can be regarded as a phase object. The phase shift due to the mean inner potential of the magnetic material itself is nearly uniform over the observed area. If we can detect the phase shift derived from the skyrmion lattice, the magnetic field distribution can be measured. The magnetic field induced by the skyrmion magnetization adds a phase shift to the phase of electrons transmitting the material:

Here ϕ, e, h, A and B are the electron phase, elementary charge, Planck constant, vector potential and magnetic field, respectively.¹⁹ Equation (5) means that the electron phase is determined by a magnetic flux enclosed within an area. This phenomenon is referred to as the Aharonov-Bhom (AB) effect.²⁰ The magnetization distribution can be extracted from the phase plate images that exhibit the contrast linear with the phase shift. The magnetic field can be obtained by differentiation of the image contrast as follows:

Furthermore the absolute value of the in-plane magnetization can be obtained from the Eq. (6) [Fig. 3(c)] as:

We used a Hitachi HF-3300, a 300 kV TEM equipped with cold field emission gun and highly flexible optics allowing us to arbitrarily adjust magnification for the experiments below. We constructed the optics for HFPP imaging with the objective lens off or only weakly excited as schematically shown in Fig. 1. In this study on the distribution of skyrmions, the image magnification is a few hundred times and the corresponding resolution is about 10 nm. We estimate that the spherical aberration and defocus have negligible effect. (See Fig. 2(b), an in-focus image which has no distortion and no Fresnel fringes.) The optics similar to one used for small-angle diffraction were utilized (see Ref. 21). The HFPP was placed at the selected-area aperture (SAA) plane, and the back focal plane (BFP) of the condenser lens was placed on the HFPP. To obtain images we first focused on the sample using intermediate lenses and adjusted magnification in free lens mode. We then placed the cross-over at the HFPP plane by adjusting the condenser lens current and correcting for the astigmatism of the condenser lens using the Ronchigram method.¹⁸ We installed a phase plate to observe samples, and adjusted the focus on sample again by visually observing when the Fresnel fringes at the sample edge disappear. A 13 nm thick amorphous carbon film prepared by electron beam evaporation was used as a HFPP.¹⁵ The amorphous carbon film HFPP was heated to 200 ∼ 300 °C and kept at this temperature for 5 hours before it was inserted in the TEM in order to remove the contamination on the film.¹⁸ The observed samples were thin films of FeGe known as a helimagnet exhibiting magnetic skyrmions in an external magnetic field.²² A polycrystalline FeGe was synthesized via arc melting and a high-pressure treatment.²² The thin film for TEM observation was prepared by Ar ion milling and the thickness t ≈ 80 nm was measured by electron energy loss spectroscopy (EELS).

Figure 2(a) shows an in-focus image with a HFPP of FeGe skyrmions within the lattice structure, taken using the optical system in Fig. 1. We carefully adjusted the in-focus condition after a HFPP was inserted in the electron beam. An 80 mT magnetic field was applied to the sample in the direction perpendicular to the sample plane. The sample temperature was 216 K. Figure 2(b) shows an in-focus image of the specimen. No magnetic contrast can be observed in Fig. 2(b), which indicates, from Eq. (2), that the magnetic thin film can be regarded a weak phase object for the electron beam. Figures 2(c) shows a defocused (Fresnel) image without a HFPP. When the intermediate lens is defocused to obtain the Fresnel image, a magnetic contrast emerges showing magnetic skyrmions and helical spin structures in Fig. 2(c). The dark halo around the sample edge in focus reversed to a bright halo when the polarity of the external magnetic field was reversed. Therefore, we conclude that the halo contrast is not an effect of the defocus or spherical aberration, but arise from the small phase shift related to the external magnetic field. The dark halo band indicates that the HFPP method can be used to image fields in vacuum. Further investigation is required to develop the quantitative interpretation of the halo contrast around the sample.

The contrast of the skyrmion lattice in Fig. 2(a) is enhanced by the HFPP adjusted to the cross over position. Even though the HFPP phase shift is not known and different from -π/2, the fact that the magnetic phase shift is weak suffices to obtain a linear relation between the image contrast and the magnetic phase shift, see Eq. (4). The image in Fig. 2(a) can therefore be regarded as a phase contrast image. A comparison of Figs. 2(a) and (c) demonstrates the utility of a HFPP to qualitatively visualize the phase shift of the electron beam. Assuming that the electron phase is proportional to the local magnetic flux, Eq. (1), the image implies that the contrast should be proportional to the distribution of sample magnetization.

According to Eq. (5), the phase of the incident beam is related to the magnetic flux of the sample through the magnetic vector potential. Thus, to extract the magnetization distribution of the sample, a differential image of the phase image in Fig. 2(a) was calculated using Python code. Figures 3(a) and (b) show the gradient image of the x and y directions, respectively, obtained by differentiation of contrasts in Fig. 2(a) using Eqs. (6) and (7). The white and black in Figs. 3(a) and (b) indicate positive and negative differential values, respectively. From the analysis of the sign of differential values, a magnetization map can be obtained as shown in Fig. 3(c). Here, the color indicates the spin direction indicated by the color wheel, and the density indicates the in-plane intensity of the sample magnetization. The intensity was calculated from Eq. (7), where I indicates the image intensity of Figs. 3(a) and (b). The figure provides the magnetization distribution map of the skyrmion lattice and is reconstructed from a single HFPP image.

Figure 4(a) is the schematic of magnetization distribution in a skyrmion. Figure 4(b) displays a gray scale image of Fig. 3(c) in which the intensity of the contrast and white arrows indicate the in-plane magnetization. Figure 4(c) is the image intensity profile, corresponding to the distribution of the magnetization calculated using Eq. (7), taken along the white broken line in Fig. 4(b) with the intensity normalized to 0.220 Wb/m². The normalization was obtained by magnetization measurement on a bulk sample and the magnetic moment of iron atoms estimated from Curie-Weiss fitting in the paramagnetic region. The in-plane magnetization is expected to have the maximum value at the median of the red line in Fig. 4(b). The intensity profile agrees with the expected distribution of magnetization and the spin configuration of the skyrmion in FeGe transformed from the helical magnetic structure.

We compare the HFPP result with previous EH work. Park et al. reported the phase shift due to the magnetic flux of a skyrmion in the Fe_0.5Co_0.5Si using EH, which allows the estimation that the phase shift from the skyrmion core to edge is 0.692 rad for 80 nm thick films.⁵ In our experiments the saturation magnetization M_sat of FeGe was taken as 0.190 Wb/m². The modulation of the in-plane magnetization component of the helical magnetic structure is a sinusoidal wave with the periodicity of the skyrmions, a_sk in Fig. 4(a), which we can use to obtain the magnetic flux Φ_mag between the skyrmion core and its edge. Note that a_sk is estimated to be 82 nm by measuring the distance between the maxim in the magnetization distribution in Fig. 4(c). Thus, the magnetic flux Φ_mag around the red line in Fig. 4(b) can be calculated from:

Here, the magnetic flux goes through the area S enclosed by the sine function of the magnetization, M(x) = M_satsin(x) reflecting the magnetic distribution of the skyrmion based on the helical magnetic structure, and the x axis as shown in Fig. 4(a). The amount of Φ_mag is calculated from S =, where $s=2πaskx=0toπ$⁠, which is shown as a blue hatched area Figs. 4(a) and (c). From Eq. (8), using the sample thickness t = 80 nm mentioned above, Φ_mag can be expressed as $Msataskπt$⁠, and the calculated phase shift is $Δϕ=2πeh$Φ_mag = 0.660 rad, which is consistent with published EH results. The phase shift normalized to 0.660 rad obtained from the phase image contrast of Fig. 2(a) is shown in Fig. 4(c). The only assumption used here is that the phase shift changes linearly for the image contrast of Fig. 2(a). Based on this assumption our procedure allows quantitative values of a physical quantity, such as phase shift and magnetization distribution, to be obtained from a single HFPP image. The exact HFPP phase shift does not need to be known as long as the linear relation between the image contrast and the object phase shift holds. This assumption therefore removes a rather difficult step of quantifying the HFPP phase shift, at least as long as weak phase objects are studied.

In summary, we report the application of HFPP to imaging the magnetic distribution of magnetic skyrmions which have attracted attention in spintronics. We quantitatively mapped their magnetization distribution from a single HFPP image by postulating that the image contrast is linear with magnetic phase shift of the skyrmion. This linear relation allows us to extract a value of 0.220 Wb/m² for the magnetization, as expected for skyrmions in FeGe (Fig. 4). Alternatively, if the HFPP phase shift is known, a quantitative magnetization map of an unknown sample can be obtained without any assumptions and without the need to compare the sample magnetization to magnetization calibration data.

Help for the programing using Python code from Jean Nassar is gratefully acknowledged. This work was supported by Graduate Program for System-inspired Leaders in Material Science (SiMS) of Osaka Prefecture University by the Ministry of Education, Culture, Sports, Science and Technology of Japan, and by the National Research Council (NRC) in Canada. The experiments were made possible by outstanding support from Hitachi High Technologies for the Hitachi HF-3300 microscope at NRC-Nano.