Quantification of magnetic interactions in van der Waals heterostructures using Lorentz transmission electron microscopy and electron holography

Abstract

Magnetic van der Waals (vdW) materials are promising for memory and logic applications because of their highly tunable magnetic properties and compatibility with vdW heterostructure devices. However, in conventional plan-view measurements, coupling between magnetic textures in stacked layers is difficult to resolve because the magnetic signal is integrated over the sample thickness. Here, these interactions are quantified in Fe$_3$GeTe$_2$ (FGT)/graphite/FGT heterostructures using cross-sectional Lorentz transmission electron microscopy and electron holography, enabling reconstruction of the local magnetic field within and between the layers. Domain alignment weakens with increasing FGT separation, yielding a dipolar coupling length scale of $λ= 34 \pm 4$ nm for the cross-sectional geometry studied here, corresponding to the average separation at which domain misalignment first emerges. This length scale coincides with an approximately 50\% reduction in the interlayer magnetic field relative to bulk FGT. Surface effects result in canting of the magnetic moments away from the easy axis up to $\sim$100 nm from a surface. Finally, the domain walls are narrow ($\sim$9 nm), while micromagnetic simulations reproduce the observed textures without invoking Dzyaloshinskii-Moriya interaction. These results quantify the internal and stray fields in stacked vdW magnets and guide the design of devices that require controllable coupling between magnetic textures.

Figures (7)

• Figure 1: Sample overview.(a, b) Schematic of the cross-sectional TEM lamella used for Figs. 2, 4, and 6, shown in (a) oblique view and (b) in cross-sectional TEM imaging orientation. The vertical black arrows indicate the orientation of the crystallographic c-axis (the [0001]-direction) of FGT and graphite. The red and green arrow indicate the lamella thickness and FGT flake thickness, respectively.
• Figure 2: Magnetic structure of heterostructure 1 after ZFC.(a-d) Lorentz TEM images acquired at 95 K with a defocus of 0.3 mm. Each image was obtained after an identical ZFC process. The blue arrows indicate the positions of the first defects in domain-wall alignment between the top and bottom FGT layers. Some structural damage is visible in the upper part of the lamella in panels (b-d), which occurred after several temperature cycles between 95 and 300 K.
• Figure 3: Domain width as a function of FGT thickness for FIB-prepared lamellae and as-exfoliated flakes. Domain width plotted for FIB-prepared lamellae and as-exfoliated flakes. The error bars represent the standard error of the mean. The plot includes data for as-exfoliated flakes extracted from Refs. li2018patterning and fei2018two (red and black crosses, respectively). The data point for the lamella with the thickest FGT comes from region 3 of heterostructure 1. The thicknesses of the as-exfoliated flakes were measured by atomic force microscopy, and the corresponding LTEM images are provided in Fig. 5 and Fig. S5. The domain-width data for the as-exfoliated flakes are in reasonable agreement with the analytical stripe-domain model of Lemesh et al.lemesh2017accurate, using a saturation magnetization of $M_s = 2.23 \cdot 10^5$ A/m, a uniaxial anisotropy of $K_u = 0.5 \cdot 10^5$ J/m$^3$ding2022tuning, and an exchange stiffness of $A = 0.95 \cdot 10^{-12}$ J/m leon2016magnetic.
• Figure 4: Magnetic field distribution.(a) Magnetic contour map acquired at 95 K by OAEH. The contour spacing is $2\pi/7$ rad. (b) Magnetic contour map from the cyan rectangle in (a). Arrows indicate the local magnetization direction near the domain corners, with the color denoting the direction. (c, d) In-plane projected magnetic induction, $|\mathbf{B}_{\perp}|\,t$, calculated from the regions indicated by the green and red rectangles in (a). The average values in the upper and lower black rectangles are 30.9 T nm and 50.6 T nm, respectively. The striped white arc highlights a typical near-surface region within a domain with reduced $|\mathbf{B}_{\perp}|\,t$. The horizontal features indicated by white arrows in (d) arise from diffraction contrast, which is also present in the LTEM images in Fig. \ref{['fig:overview']}. (e) Line profiles of the projected in-plane induction along the light- and dark-blue lines in (c) and (d), respectively. Edge effects lead to artificially large values at the specimen boundary; these segments are removed, producing gaps near the vacuum regions. (f) Line profiles of $|\mathbf{B}_{\perp}|$ along the green and red lines in (c) and (d), respectively, assuming a lamella thickness of 230 nm in the vacuum-spacer region. The horizontal dashed lines indicate peak values of $\sim$105 mT (green) and $\sim$150 mT (red). (g, h) Projected in-plane induction calculated from the magnetization distribution ($\boldsymbol{B}_{\perp}=\mu_{0}\boldsymbol{M}_{\perp}$, where $\mu_0$ is the vacuum magnetic permeability and $\boldsymbol{M}_{\perp}$ is the in-plane projected magnetization) recovered using a model-based iterative reconstruction algorithm. (i, j) Corresponding line profiles extracted from (g) and (h), analogous to (e) and (f). The dashed lines in (j) are guides to the eye indicating the approximate peak field strengths.
• Figure 5: LTEM contrast in plan-view samples.(a-d) LTEM images of a 48 nm-thick exfoliated FGT flake, acquired at 95 K with a defocus of 0.8 mm and at tilt angles of 7$\degree$, 13$\degree$, 16$\degree$, and 22$\degree$, respectively. (e) Schematic of the sample geometry. The exfoliated flakes are transferred on top of a silicon TEM grid with a silicon nitride (SiN) membrane with 2 $\mu$m diameter through-holes. See Methods for details. (f) Line profiles of the image intensity along the colored lines in (a-d). The black arrows indicate the positions of the domain walls.