Atomic-resolution imaging of magnetism via ptychographic phase retrieval

Atomic-scale characterization of spin textures in solids is essential for understanding and tuning properties of magnetic materials and devices. While high-energy electrons are employed for atomic-scale imaging of materials, they are insensitive to the spin textures. In general, the magnetic contribution to the phase of high-energy electron wave is 1000 times weaker than the electrostatic potential. Via accurate phase retrieval through electron ptychography, here we show that the magnetic phase can be separated from the electrostatic one, opening the door to atomic-resolution characterization of spin textures in magnetic materials and spintronic devices.

objective lens of a TEM is about 2 T, well below the spin-flop field (order of 10 T) of -Fe2O3 22 .
When a high-energy electron beam transmits through a thin sample, the phase shift induced by the electrostatic and magnetic potentials is 23 : ∆φ = σ  − 2 ℎ   = φ  + φ  (1)   where σ = 2 ℎ ⁄ is the interaction constant,   the projected electric potential,   the path integral of magnetic vector potential along the electron beam direction described in Methods, ℎ the Planck's constant.The two terms on the right side describe the electrostatic and magnetic phase, respectively.The electrostatic potential and spin density for computing magnetic vector potential were obtained by density functional theory (DFT) calculations using the full-potential linearized augmented plane wave (FP-LAPW) method 24 .The electrostatic, magnetic and total phase are derived (see Methods for details) and shown in Fig. 1, along with their diffractograms.The electrostatic phase (Fig. 1c) is by far the largest contribution to the total phase (Fig. 1b).Different from the electrostatic phase that is peaked at atoms, the magnetic phase (Fig. 1d) is distributed in between magnetic atoms.For convenience, the electrostatic and magnetic phases are distinguished as "on-site" and "off-site" phase, respectively.When averaged in the (0001) plane, the magnetic phase has an antiferromagnetic undulation in the [0001] direction, with the undulation amplitude of 0.17 mrad/nm.Note that the peak in the on-site electrostatic phase is 790 mrad/nm, but the peak in the off-site magnetic phase is only 0.47 mrad/nm, three orders of magnitude smaller than the electrostatic one.The ratio of the magnetic to total phases is plotted in Extended Data Fig.S2, indicating that very high accuracy in phase measurement is required to separate the magnetic phase from the electrostatic one.In Fig. 1, the diffractograms of the total, electrostatic, and magnetic phase are also shown.The electrostatic phase contributes to the strong reflections, defined by the reciprocal lattice ge(h,k) = hge1/2 + kge2/2, with the basis vectors ge1 = (1 ̅ 104) and ge2 = (11 ̅ 02) .The magnetic phase contributes to the weak reflections, the positions of which are defined by gm(h,k) = hge1/2 + kge2/2 (both h and k are odd integers), as indicated by the arrows in Fig. 1d.The intensity of the magnetic reflection (0003) is about 2.010 -5 times that of the electrostatic reflection (1 ̅ 104).Because the mean value of the magnetic phase is zero, the magnetic phase has no contribution to the reciprocal origin (0,0).As a result, the electrostatic and magnetic phases are well separated in the reciprocal space.The strong reflections [h and k being even integers, including the origin (0,0)] are contributed by the onsite electrostatic potential, and the weak reflections (h and k being odd integers) by the off-site magnetic potential.Applying masks in reciprocal space (shown in Extended Data Fig.S3c) and performing Fourier transform, the electrostatic and magnetic phases are separated in real space.Hereafter we name the process to extract weak magnetic phase from the total phase as "Fourier separation".The challenge is to obtain the total phase of sufficient accuracy, since the magnetic phase is more than 1000 times weaker than the total phase.
We use the multislice electron ptychography with adaptive propagator to measure the phase of the object.Ptychography is a phase-retrieval method using coherent diffraction imaging in the scanning mode 25,26,27,28 .As an electron probe scans over a two-dimensional grid of an object, two-dimensional coherent diffraction patterns are collected using a pixelated array detector with high dynamic range 29 , as schematically shown in Fig. 2a, forming the so-called 4D-STEM dataset (2D scanning in real space and 2D diffraction in reciprocal space).Various algorithms have been proposed to recover the phase of the object encoded in the datasets 30,31,32,33,34,35 .The experimental challenges like residual lens aberrations, dynamic electron diffraction 36 , sample drift during data collection 37 , and the misorientation between crystal zone axis and electron beam 38 can be corrected, providing accurate phase of the object.Ptychography also offer a higher signal-to-noise ratio than conventional STEM imaging modes at the same level of electron dose 39,40 .The workflow and typical results are schematically shown in Fig. 2. In order to differentiate the defected surface layers from the pristine bulk, multislice reconstruction is necessary, as shown in Fig. 2b.With the contributions from the surface layers removed, the phases of the central slices represent the pristine bulk, as shown in Fig. 2c.In the last step, the Fourier separation is applied to the total phase to obtain the electrostatic phases (not shown) and the magnetic phase (Fig. 2d), which is undulating in the [0001] direction.
Because it is necessary to measure the phases very accurately, we performed simulation tests on the sensitivity of above method on the electron dose, which influences the accuracy of experimental measurements due to the shot noise.The doses of 10 3 e/Å 2 ~ 10 6 e/Å 2 are considered.As reference, a typical high-angle annular dark field (HAADF) or DPC image is recorded with a dose of 10 4 e/Å 2 ~ 10 6 e/Å 2 .The results are shown in Fig. 3   The magnetic phases extracted by Fourier separation are shown in Fig. 3b.Since the magnetic phase is extremely weak, much higher electron dose is required to obtain a reasonable signal-to-noise.As shown in Extended Data Fig.S3b and 3c, the magnetic phase becomes very noisy at the dose of 4.5×10 4 e/Å 2 , and almost indiscernible at a dose of 4.5×10 3 e/Å 2 .It should be noted that, in experimental datasets, there are other sources of noise that are not considered in the simulations shown in Fig. 3, including point defects, dislocations, strain field, etc.Therefore, the simulations like Fig. 3 only suggest a rough estimate of the lower bound of the electron dose for detecting magnetic phase.In experiments, higher electron dose may be needed.
The simulation results show another challenge for the detection of weak magnetic phase, i.e., the disturbance of the strong electrostatic phase.Due to numerical errors inevitable in the reconstruction process, the trace of electrostatic phase appears in the magnetic phase at the atomic columns, even at the infinite electron dose.Fortunately, the electrostatic and magnetic phases are separated in both real and reciprocal spaces, being on-site and off-site, respectively (Fig. 1c and 1d).It means that the onsite phase appeared in the magnetic phase image can be attributed to the trace of the electrostatic phase, and its influence on the interpretation of the magnetic phase is limited.
Fig. 4 shows the experimental imaging of the total and magnetic phases.The electron dose we used in experiment is 9.0×10 5 e/Å 2 .In the diffractograms, the magnetic reflections can be clearly identified.The intensity of the magnetic reflection (0003) is 7.9×10 -5 times that of the electrostatic reflection (1 ̅ 104) .The intensity ratio is the same order of magnitude as the theoretical prediction (~2.0×10 -5 , Fig. 1b).Possible sources of the discrepancy include numerical errors for the measurements of such weak signals, the temperature effect, diffuse scattering due to possible point defects, and the uncertainties in the DFT calculations of a system of strongly-correlated electrons.Fig. 4d and e shows the magnetic phase extracted from the total phase via Fourier separation.The Fe ions are overlaid in the upper left corner, indicating the antiferromagnetism along the [0001] direction.The magnetic phase averaged in the (0001) plane is shown on the right of Fig. 4d, clearly revealing the phase undulation in the [0001] direction.The measured undulation amplitude is 0.40 mrad/nm, consistent with the theoretical prediction (0.17 mrad/nm) considering numerical errors in the measurements of weak signals.As expected from the simulations shown in Fig. 3, the trace of onsite electrostatic phase also appears in the magnetic phase image due to numerical errors.
It should be noted that magnetic reflections occur only in antiferromagnetic materials.For ferromagnetic materials, there is no sharp magnetic reflections but diffuse scattering at magnetic domain boundaries or other magnetic microstructures.Applying Fourier separation to the diffuse scattering would require even higher signal-to-noise ratio in the total phase and in-turn higher electron dose.Another condition for atomic-resolution magnetic imaging of ferromagnetic materials is a sample environment free of magnetic field, otherwise magnetic textures would be washed out easily by the magnetic field in a TEM.High-resolution field-free microscopes have appeared 41 .
In summary, atomic-resolution imaging of magnetism has been realized through phase retrieval of multislice electron ptychography with adaptive-propagator.The phase is accurately measured, enabling direct separation of weak magnetic phase (< 1 mrad/nm) from strong electrostatic phase (~1 rad/nm) in both real and reciprocal spaces.The method would find wide applications first in antiferromagnetic materials/spintronics, then in ferromagnetic ones when high-resolution field-free microscopes are widely available.

Fig. 1 .
Fig. 1.The contributions of electrostatic and magnetic potentials to the phase of the incident electron wave.The potentials are calculated using density-functional theory.a, Spin structure of α-Fe2O3 at room temperature; b, The total phase, it's diffractogram, and the intensity profile along the white dash-dotted line in the diffractogram; c, Electrostatic phase and it's diffractogram; d, Magnetic phase and it's diffractogram.The sample thickness in the simulations is 2.3 nm.The diffractogram intensities are displayed on a logarithmic scale to show the weak magnetic

Fig. 2 .
Fig. 2. Workflow of extracting magnetic phase from total phase.a, Experiment set up for electron ptychography; b, Reconstructed phase images of multiple slices; c, The averaged phase image of the central slices; d, The magnetic phase extracted by Fourier separation.
and Extended Data Fig.S4.The total phases of the central slices reconstructed by ptychography are shown in Fig. 3a.High-quality total phases can be obtained for an electron dose down to 4.5×10 3 e/Å 2 , with all the atomic columns including oxygen are clearly imaged, although single diffraction patterns are quite noisy at this dose, as shown in Extended Data Fig.S5.

Fig. 3 .
Fig. 3.The total and magnetic phases obtained with simulation datasets for different electron doses.a, the total phases via ptychographic reconstruction, b, the magnetic phases via Fourier separation.The sample thickness is 2.0 nm along, [112 ̅ 0] zone axis.

Fig. 4 .
Fig. 4. Experimental magnetic phase imaging of -Fe2O3.a, The total phase averaged over the central slices and b the corresponding diffractogram (on a logarithmic scale).The arrows indicate the magnetic reflections of the lowest order.c, The intensity profile along the line connecting ge1, gm1, and ge2, as indicated in the diffractogram in b by the white dash-dotted line.d, The magnetic phase extracted from the total phase by Fourier separation.The Fe ions are overlaid on the upper left corner with arrows indicating their spin directions.e, The projection of the magnetic phase to the vertical axis.All the phases correspond to a slice of 2.3 nm thick.The electron dose is 9.0×10 5 e/Å 2 .

a,
Magnetic phase of a sample of 2.3 nm thick; b Ratio of the magnetic phase to the total phase; c, The profiles of the magnetic phase and the magnetic/total ratio along the line indicated in b.Extended Data Fig.S3 a, Simulated diffraction pattern of α-Fe2O3 with magnetism considered, in the [112 ̅ 0] zone axis.b, Simulated diffraction pattern of α-Fe2O3 without magnetism considered, in the [112 ̅ 0] zone axis.It is same as Mask I covering the principal reflections in a, which are contributed by the electrostatic potential φ  .c Mask II covering extra reflections in a, which are contributed by the magnetic potential.Extended Data Fig.S4 Simulation test of the total and magnetic phases at different electron doses.The columns from left to right show a the total phases via ptychographic reconstruction, b the magnetic phases via Fourier separation, c the magnetic phases via Fourier separation and subsequent unit-cell-averaging, and tiling.The sample thickness is 2.0 nm along, [112 ̅ 0] zone axis.