interactive Lorentz TEM

Magnetic skyrmions are tiny whirls in the magnetization of a material, typically tens to hundreds of nanometers across. Electrons passing through a magnetic film are deflected by the Lorentz force from the in-plane magnetic induction, by only tens of microradians, far less than Bragg scattering. Equivalently, the electron wave picks up a phase shift from the magnetic vector potential (the Aharonov–Bohm effect). Lorentz TEM in Fresnel mode turns the phase shift into visible contrast by deliberately defocusing the microscope: domain walls then appear as bright or dark regions relative to the domains themselves.

Things to try

• Flip the defocus. Going from over- to under-focus inverts the contrast: the skyrmion core switches between bright and dark. At exactly zero defocus the magnetic contrast vanishes.
• Image a Néel skyrmion. Press Néel (or drag the helicity to 0°). Néel skyrmions, whose spins point radially, produce no Lorentz contrast when the film is flat. Tilt the sample and they appear as a pair of bright and dark half-moons, which swap when you tilt the other way. This is how Néel textures, for example in Fe₃GeTe₂, are identified experimentally.
• Lower the dose. Real images are noisy. Watch how shot noise, down to a fraction of an electron per pixel, propagates into the TIE reconstruction. This regime is where machine-learning approaches help.

What’s being simulated

1. Magnetization. A thin film (40 nm thick, saturation induction μ₀M_s = 0.2 T) with the chosen spin texture, uniform through the thickness.
2. Electron phase shift. For magnetization that is uniform along the beam, the magnetic phase shift has a compact closed form in Fourier space as described by the Mansuripur algorithm: it depends only on the in-plane magnetization, and only on its curl. Néel textures are invisible until the film is tilted because their curl lies entirely within the plane.
3. Fresnel imaging. The exit wave is propagated by ±Δf with the microscope’s phase transfer function for a 200 kV beam, including a damping envelope from the small beam divergence, and Poisson shot noise is added.
4. Reconstruction. The two defocused images are fed into the transport-of-intensity equation (TIE), ∇·(I₀∇φ) = −(2π/λ) ∂I/∂z, a Poisson equation linking the change in intensity with defocus to the phase. The gradient of the recovered phase gives the integrated in-plane magnetic induction, (B_x, B_y) ∝ (−∂φ/∂y, ∂φ/∂x).

It is important to note that the TIE reconstruction gives the magnetic induction integrated along the beam, including stray fields above and below the film, not the magnetization itself. For Bloch skyrmions the two look nearly identical, but for many textures (such as Néel skyrmions) they do not. Interpreting LTEM images therefore requires image simulations, such as those performed in my PyLorentz package. It covers arbitrary 3D and tilted magnetizations from micromagnetics, realistic microscope transfer functions, TIE and single-image reconstructions, and ML-based analysis.