Quantitative Magnetic Information from Reciprocal Space Maps in Transmission Electron Microscopy

One of the most challenging issues in the characterization of magnetic materials is to obtain a quantitative analysis on the nanometer scale. Here we describe how electron magnetic circular dichroism (EMCD) measurements using the transmission electron microscope can be used for that purpose, utilizing reciprocal space maps. Applying the EMCD sum rules, an orbital to spin moment ratio of $m_L/m_S=0.08\pm 0.01$ is obtained for Fe, which is consistent with the commonly accepted value. Hence, we establish EMCD as a quantitative element-specific technique for magnetic studies, using a widely available instrument with superior spatial resolution.

Figure 1

A sketch of the diffraction geometry and distribution of EMCD signal in the $k_x\mathrm{\text{−}}{k}_y$ plane. The sample is tilted with respect to the incoming electron beam from a high symmetry orientation to the 2BC geometry where $\alpha \approx 10°$ and $\beta \approx 0.4°$. In this geometry only the transmitted (0) and Bragg scattered beams ($\mathbf{G}$) are strongly excited (large black spots). The optimum detector positions are indicated as $P_{UpR}$ and $P_{DnR}$ for 2BC geometry using a horizontal mirror axis (solid line). For the 3BC geometry, $\alpha \approx 10°$ and $\beta =0°$, an additional vertical mirror axis is available (dashed line). In the experiment, energy filtered diffraction patterns appear in the $k_x\mathrm{\text{−}}{k}_y$ plane; here an EMCD map is shown as a guide to the eye. The black circle indicates detector positions where $\mathbf{q}\perp {\mathbf{q}}^{\prime }$, i.e., the Thales circle.

Figure 2

Reciprocal space maps of the EMCD signal and $m_L/m_S$ ratio for the Fe sample oriented in a 2BC geometry (left) and 3BC geometry (right). Theoretical relative EMCD maps at the $L_3$ edge are shown in (a) and (e). The inset shows the simulated diffraction pattern. In (b) and (c) [(f) and (g) for 3BC] maps of experimentally obtained relative EMCD signal at $L_3$ and $L_2$ edges are shown. The black lines indicate the applied mirror axes and blue spots the positions of the transmitted and Bragg scattered $\mathbf{G}=(2,0,0)$ and $-\mathbf{G}=(\overline{2},0,0)$ beams. The insets in (b) and (f) show the diffraction patterns averaged over an energy interval from 695 to 740 eV. In (d) and (h), the experimental $m_L/m_S$ maps are shown, where the box indicates the region with minimal noise. Values outside the range of color bar are shown as white.

Figure 3

Experimental EELS spectra at Fe $L_{2,3}$ edges and $m_L/m_S$ ratios as a function of collection window size in $m_L/m_S$ ratio maps. The two spectra in (a) were extracted from the experimental data cube with the sample in 2BC geometry at the $P_{DnR}$ (orange squares) and $P_{UpR}$ positions (blue circles), averaging over $0.5G_{200}\times 0.5G_{200}$ in reciprocal space. The components of the fit function for the spectrum at the $P_{UpR}$ position are shown. (b) $m_L/m_S$ ratio obtained for various window sizes in 2BC geometry using the horizontal mirror axis and in 3BC using the double difference method. The inset shows the histogram with a fit of the $m_L/m_S$ ratio for the window size of $0.5G_{200}\times 0.5G_{200}$ in 2BC orientation [indicated in Fig. 2d].