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

This study quantifies magnetic interactions in Fe$_3$GeTe$_2$/graphite/Fe$_3$GeTe$_2$ van der Waals heterostructures using cross-sectional Lorentz transmission electron microscopy and electron holography, revealing a dipolar coupling length scale of 34 nm, surface-induced moment canting up to 100 nm, and narrow domain walls that can be modeled without Dzyaloshinskii-Moriya interaction.

The Gist

Imagine you are trying to understand how two magnets talk to each other when they are stacked on top of one another, separated by a thin layer of something non-magnetic (like a piece of paper). In the world of tiny, 2D materials called van der Waals heterostructures, scientists want to build computer memory and logic devices using these magnetic stacks. But there's a problem: usually, when you look at these stacks from the top (like looking down at a sandwich), you can't tell which magnet is doing what because their signals blend together into a blurry mess.

This paper is like taking a knife, slicing that sandwich in half, and looking at the cross-section to see exactly what's happening between the layers.

Here is a simple breakdown of what the researchers discovered, using some everyday analogies:

1. The "Sandwich" Experiment

The scientists built a special sandwich:

• Top Bun: A magnetic material called Fe3GeTe2 (FGT).
• Filling: A layer of graphite (like pencil lead) of varying thickness.
• Bottom Bun: Another layer of FGT.

They used a powerful microscope (Lorentz TEM) and a technique called electron holography (think of it as a super-precise magnetic camera) to slice this sandwich open and look at the magnetic fields inside.

2. The "Whispering" Distance (The Coupling Length)

The main question was: How far apart can these two magnetic layers be before they stop "listening" to each other?

• The Analogy: Imagine two people trying to whisper a secret to each other across a room. If they are close, they hear it clearly. If they move apart, the whisper gets lost in the noise.
• The Discovery: The researchers found that the two magnetic layers stay perfectly synchronized (their magnetic "domains" line up) as long as they are within about 34 nanometers of each other.
• The Tipping Point: Once the gap gets wider than 34 nanometers, the "whisper" breaks. The top layer and bottom layer start doing their own thing, and their magnetic patterns no longer match up.
• The Signal Drop: At this specific distance, the magnetic force felt in the gap drops by about 50%. It's like the signal strength on your phone dropping from "Full Bars" to "One Bar" just as you walk out the door.

3. The "Surface Chill" Effect

The paper also found something interesting happening at the very edges of the material.

• The Analogy: Think of a crowd of people standing in a line, all facing forward (the "easy axis"). If you are in the middle of the crowd, you feel the pressure of the people on both sides and stay straight. But if you are at the very edge of the line, you might lean a little bit because there's no one on your other side to hold you up.
• The Discovery: Near the surface of the magnetic material (up to about 100 nanometers deep), the magnetic atoms start to "lean" or tilt away from their perfect straight line. This is caused by the lack of neighbors on the surface side.
• Why it matters: If you are building a tiny device that is only 100 nanometers thick, the entire thing might be "leaning" because of surface effects, which changes how the device works.

4. The Mystery of the "Wall"

There was a long-standing debate in the scientific community about the "walls" between magnetic regions in these materials. Are they Néel-type (twisting like a corkscrew) or Bloch-type (swirling like a vortex)?

• The Confusion: Previous tests suggested they were corkscrews (Néel) because the image only showed up when the sample was tilted.
• The New Clue: By looking at the cross-section, the researchers found these walls are incredibly thin (only about 9 nanometers wide).
• The Conclusion: They realized that the "tilting" effect seen in old tests might just be because the walls are so narrow, not necessarily because they are corkscrews. Furthermore, they ran computer simulations that showed you don't need a special force (called Dzyaloshinskii–Moriya interaction) to create these patterns; simple magnetic rules are enough. It's like realizing you don't need a complex recipe to bake a cake; sometimes simple ingredients work just fine.

Why Should You Care?

This research is like a blueprint for the next generation of computers.

• Better Memory: By knowing exactly how far apart to stack magnetic layers (around 34 nm), engineers can design devices where the layers talk to each other perfectly, or stay silent when needed.
• Smaller Devices: Understanding how surfaces affect the magnetic "lean" helps engineers design smaller, more efficient chips that don't lose their magnetic properties.
• New Tools: The method they used (slicing the sandwich and looking at the cross-section) is a new tool that can be used to study all sorts of other magnetic materials, potentially leading to faster, greener, and smarter technology.

In a nutshell: The scientists sliced open a magnetic sandwich, measured exactly how far the layers can be before they stop syncing, and realized that the edges of the material behave differently than the middle. This gives engineers the precise "ruler" they need to build the magnetic computers of the future.

Technical Summary

1. Problem Statement

Magnetic van der Waals (vdW) materials, such as $\text{Fe}_3\text{GeTe}_2$ (FGT), are promising for spintronic memory and logic devices due to their tunable properties and ability to form heterostructures without lattice-matching constraints. However, a significant characterization challenge exists:

• Integrated Signal Limitation: In conventional plan-view magnetic imaging, the measured signal is integrated through the entire sample thickness. This makes it difficult to resolve which specific magnetic layer generates a signal or to quantify the coupling between stacked magnetic layers separated by non-magnetic spacers.
• Unknown Interaction Scales: The precise length scale over which dipolar coupling persists between vertically stacked FGT layers, and how surface effects influence local magnetization, had not been quantitatively resolved in cross-sectional geometries.
• Domain Wall Topology Debate: There is ongoing debate regarding whether FGT hosts Néel-type or Bloch-type domain walls, particularly given that bulk FGT is centrosymmetric (lacking intrinsic Dzyaloshinskii–Moriya interaction, or DMI) but often exhibits Néel-like features attributed to defects or surface oxidation.

2. Methodology

The authors employed a combination of advanced electron microscopy techniques on cross-sectional samples to overcome the limitations of plan-view imaging:

• Sample Preparation:
  • Heterostructures were assembled using mechanically exfoliated FGT and graphite flakes on Si/SiO$_2$ substrates.
  • FGT/Graphite/FGT stacks were created with varying graphite spacer thicknesses (from monolayer graphene to 110 nm).
  • Cross-sectional lamellae were prepared using Focused Ion Beam (FIB) milling. Special care was taken to minimize surface damage (final thinning at 2 kV).
• Imaging Techniques:
  • Lorentz Transmission Electron Microscopy (LTEM): Used to visualize magnetic domain structures at 95 K (below the Curie temperature of ~220 K). Images were acquired after Zero-Field Cooling (ZFC) cycles to observe domain configurations.
  • Off-Axis Electron Holography (OAEH): Used to quantitatively reconstruct the magnetic phase shift.
    • Electrostatic contributions were removed by subtracting phase maps acquired above the Curie temperature (room temperature).
    • This allowed for the reconstruction of local magnetic induction ($B_\perp$) and magnetization vectors within and between layers.
• Simulation:
  • Micromagnetic Simulations: Performed using the Excalibur code to model the observed domain structures. The models included exchange energy, uniaxial anisotropy, and magnetostatic interactions. Crucially, simulations were run without invoking DMI to test if the observed textures could be explained by standard magnetostatic coupling alone.

3. Key Contributions and Results

A. Quantification of Dipolar Coupling Length Scale

• Observation: By varying the separation between two FGT layers using graphite spacers, the authors observed that magnetic domains in the top and bottom layers remain aligned at small separations but become misaligned as the gap increases.
• Result: They determined a characteristic dipolar coupling length scale of $\lambda = 34 \pm 4$ nm. This is the average separation distance at which domain misalignment first emerges.
• Field Reduction: OAEH measurements revealed that at this separation ($\lambda$), the magnetic induction in the vacuum spacer is reduced by approximately 50% relative to the bulk FGT value. At larger separations (e.g., 110 nm), interactions persist but are significantly weakened.

B. Surface Effects and Magnetization Canting

• Surface Influence: The study identified a surface effect where local magnetic moments cant away from the magnetic easy axis (c-axis) up to ~100 nm from a free surface.
• Evidence: This was observed as a reduction in the projected in-plane magnetic induction near the sample surfaces and at the FGT/vacuum interfaces.
• Mechanism: Model-based iterative reconstruction confirmed that this reduction is not solely due to demagnetizing fields but indicates a physical canting of the magnetization vector, likely driven by surface defects or oxidation.

C. Domain Wall Topology and Width

• Width Measurement: OAEH data from cross-sectional lamellae revealed that domain walls are extremely narrow, approximately 9 nm wide.
• Contrast Mechanism: In plan-view LTEM, domain wall contrast is only visible when the sample is tilted. While this is often interpreted as evidence for Néel-type walls, the authors argue that the narrow width of Bloch-type walls can also produce tilt-dependent contrast.
• Role of DMI: Micromagnetic simulations successfully reproduced the experimental domain structures (including misalignment at large separations) without including DMI. This suggests that DMI is not strictly necessary to explain the observed textures in these specific samples.
• Conclusion on Topology: The authors conclude that the tilt-dependent contrast in plan-view samples may arise from the narrowness of the walls rather than a definitive Néel topology, highlighting the need for vector field holography or in-plane field application to unambiguously determine the wall type.

D. Impact of FIB Patterning

• The study noted that FIB-prepared lamellae exhibit smaller domain widths compared to as-exfoliated flakes of the same thickness. This is attributed to FIB-induced surface damage reducing the effective magnetic anisotropy or altering stray-field energy due to the proximity of side surfaces.

4. Significance and Impact

• Design Guidelines for Spintronics: The quantification of the coupling length scale ($\lambda \approx 34$ nm) provides critical design parameters for vdW-based spintronic devices (e.g., magnetic tunnel junctions, spin valves). Engineers can now tune interlayer coupling from strong to weak by controlling the spacer thickness.
• Methodological Advancement: The paper demonstrates that cross-sectional electron holography is a superior method for resolving vertical magnetic interactions in vdW heterostructures compared to plan-view techniques, which suffer from signal integration.
• Fundamental Insight: The findings challenge the assumption that Néel-type walls are the default in off-stoichiometric FGT, suggesting that narrow Bloch walls might be present and that surface effects play a dominant role in the magnetization of flakes thinner than 100 nm.
• Future Applications: The methodology is applicable to a wide range of systems, including those with proximity-induced spin-orbit coupling and superconductor/ferromagnet interfaces, enabling the study of flux vortices and complex magnetic textures.

In summary, this work provides the first quantitative map of magnetic interactions in vertically stacked vdW magnets, establishing a clear relationship between physical separation, magnetic field reduction, and domain alignment, while offering a refined perspective on domain wall topology in 2D magnets.