Nanoscale mapping of ultrafast magnetization dynamics with femtosecond Lorentz microscopy

Novel time-resolved imaging techniques for the investigation of ultrafast nanoscale magnetization dynamics are indispensable for further developments in light-controlled magnetism. Here, we introduce femtosecond Lorentz microscopy, achieving a spatial resolution below 100 nm and a temporal resolution of 700 fs, which gives access to the transiently excited state of the spin system on femtosecond timescales and its subsequent relaxation dynamics. We demonstrate the capabilities of this technique by spatio-temporally mapping the light-induced demagnetization of a single magnetic vortex structure and quantitatively extracting the evolution of the magnetization field after optical excitation. Tunable electron imaging conditions allow for an optimization of spatial resolution or field sensitivity, enabling future investigations of ultrafast internal dynamics of magnetic topological defects on 10-nanometer length scales.

on scales down to few hundreds of nanometers were obtained by various stimuli such as electrical current and light [1][2][3][4][5][6][7][8][9][10] . Optical control of magnetization is particularly appealing due to the absence of an applied external field and the possibility for ultrafast switching speeds. The physical processes involved, such as direct spin-light interactions, opticallydriven spin currents, and spin-flip scattering and the collapse of exchange splitting in a hot electron environment, remain active fields of study [11][12][13][14][15] . For a further progress, it is essential to provide experimental tools to assess the optically-induced magnetization reordering processes at their intrinsic nanometer spatial and femtosecond temporal scales.
Time-resolved implementations of established experimental techniques for mapping magnetic structures at sub-micrometer dimensions have been already accomplished for magneto-optical Kerr effect microscopy 16 , photoemission electron microscopy 17 , scanning electron microscopy with polarization analysis 18 , scanning transmission x-ray microscopy 3;19 , small-angle x-ray scattering 20 , and holography using x-rays 21 . Novel imaging approaches using circularly polarized high-harmonic radiation 22 may even provide access to magnetization dynamics on timescales of few femtoseconds.
Nanoscale mapping of static magnetic structures by Lorentz microscopy in a transmission electron microscope is a powerful method which routinely achieves resolutions down to few tens of nanometers [23][24][25] , with the possibility to correlate magnetization and structural information within the same instrument. Moreover, due to the direct effect of magnetic fields on the imaging electron wave, Lorentz image contrast allows for a quantitative reconstruction of the magnetization field 8;23;26 .
Time-resolved Lorentz microscopy was previously demonstrated for investigations of field-assisted or laser-excited domain wall movement [27][28][29] . Bostanjoglo and coworkers achieved nanosecond temporal resolution by electronically chopping the electron illumination beam 27 . Obtaining shorter electron pulses became readily accessible with the advent of laser-triggered electron sources [30][31][32] , extending Lorentz microscopy into the nanosecond 28 and picosecond 29 regime. However, the spatio-temporal resolution required to investigate ultrafast magnetic processes, such as the optical control of spin structures, necessitates advanced nanoscale photocathode approaches, providing femtosecond electron pulses with high spatial beam coherence 33 .
Here, we introduce real-space nanoscale mapping of light-induced magnetization dynamics by ultrafast transmission electron microscopy (UTEM) with femtosecond tem-poral resolution. We quantitatively track the time-dependent magnetization field in a single vortex structure during laser-driven ultrafast demagnetization, reaching a 700-fs temporal and below-100-nm spatial resolution. These results demonstrate the capability of femtosecond Lorentz microscopy for the imaging of transient magnetization fields on timescales faster than the spin-lattice equilibration.
In our experiments, we study an isolated magnetic permalloy disc (1 µm diameter, 20 nm thickness), prepared by electron-beam lithography on a silicon nitride membrane (50 nm thickness). The ground state texture of the disc consists of a magnetic vortex state, for which an in-plane oriented magnetization field M ( r) of constant magnitude | M ( r)| = M s curls around the center of the structure 24;34;35 . In the vortex core, with a typical diameter on the 10-nm scale 36 , the magnetization turns to an out-of-plane direction.
Transmission electron microscopy under out-of-focus imaging conditions, so-called Fresnel mode Lorentz microscopy 37 , provides for an image contrast which is sensitive to the in-plane magnetization field (Fig. 1a). For the vortex sample, a conically-shaped Aharonov-Bohm phase shift φ 38 is imprinted onto an incident electron wavefront, where A is the magnetic vector potential, e the electron charge,h the reduced Planck constant, and the integral is computed along the electron beam trajectory. In defocused electron images, this spatial phase information is transferred to changes in electron image intensity, which, in general, can be employed to reconstruct the magnetization field by using phase retrieval approaches such as the transport-of-intensity equation 8 In order to perform time-resolved stroboscopic Lorentz microscopy, we developed a pulsed electron source based on nano-localized photoemission from a Schottky field emit- where H(∆t) is the Heaviside step function, C the amplitude for negative delay times,   The illuminating electron wave function ψ 0 is spatially modified in amplitude and phase by electron-sample interactions. Here, we consider ψ 0 to be a plane wave (parallel illumination). In Lorentz microscopy, the most relevant interactions (i.e. with the electromagnetic field inside the sample) induce a phase shift φ on ψ 0 , while other scattering events can be included by a spatially dependent amplitude modulation A( r). In the projection approximation, the exit wave function ψ below the sample plane is given by where r is a vector in the plane perpendicular to the microscope optical axis (xy-plane, see Fig. 1 in main text for the coordinate system). The amplitude modulation is extracted from in-focus images. The phase shift φ is related to the magnetic structure by the Aharonov-Bohm equation 38 where v is the speed of the imaging electrons and V and A are the electrostatic and magnetic potentials inside the sample. The integrals are computed along the electron beam direction z.
The electrostatic component of the phase shift within the disc is calculated by taking into account a thickness t = 20 nm for the permalloy film and a mean inner potential V = 36 V. For the vortex structure, the magnetic component of the phase shift can be closely approximated to exhibit radially-symmetric conical shape, which for r < R is given by 47 where R is the disc radius, φ For a known exit wave function ψ( r), the image I( r) produced by the electron optical system of the Lorentz microscope can be calculated via where T is the transfer function 37 , and q the wave vector (reciprocal of the spatial vector r). For a coherent illumination, the transfer function including the lowest-order coherent aberrations of an astigmatism-corrected microscope is given by 37;48 where C s is the spherical aberration constant of the objective (mini-)lens, ∆f the defocus (positive for overfocus), and λ the electron wavelength.
The limited degree of temporal and spatial coherence of electron sources gives rise to incoherent aberrations, which affect image contrast. Firstly, the spread of electron wavenumbers k = 1/λ (temporal coherence) results in varying focal lengths for different electron energies. The resulting electron image is an incoherent summation of image intensities (as given by Eq. 4) considering a distribution of electron wavenumbers. We include this contribution by multiplying the transfer function with a damping term (temporal coherence envelope) 46 where C c is the chromatic aberration constant of the imaging lens and σ 2 E is the standard deviation of the electron energy distribution. The relativistically modified acceleration potential is given by U * a = U a 1 + eUa 2moc 2 , in which U a is the experimental acceleration voltage, m 0 the electron rest mass and c the speed of light.
Imaging utilizing illumination conditions with partial spatial coherence can be described in a similar manner. Specifically, waves with different incidence angles α il contribute incoherently to the final image I( r), which is given by the sum of the images resulting from illumination with a single α il , weighted by the probability distribution i(k α il ) of illumination directions 46 : For small tilt angles, the transfer function for tilted illumination can be approximated using χ( q + k α il ) = χ( q) + k α il ∇χ( q). Using this Taylor expansion and a rotationally symmetric Gaussian distribution of incidence angles i(k α il ), the image intensity can be calculated by modifying the transfer function by an additional damping term (spatial coherence envelope) 46 : where θ c is the opening half-angle of the illumination. Within this approximation, the complete transfer function is: For the large defocus values employed here, χ( q) exhibits rapid oscillations in q, so that the lowest order Taylor expansion of Eq. 7 is only of limited validity. In a more precise approach, we numerically integrate Eq. 9 to properly account for the illumination with partial spatial coherence. Specifically, we randomly sampled 2000 values for the incidence angle α il according to a gaussian distribution with a 2θ c spread for the numerical integration of Eq. 9. A comparison of both image simulation approaches and an experimental Lorentz profile is shown in Fig. 4c.
In the simulation, we adopt aberration constants of C s    For a hot spin system, the spatial structure of the magnetization field, such as the vortex core diameter, is expected to adapt due to changes, for example, in the dipole interaction. Generally, changes in the spatial structure will depend on the time scales involved. In comparison to the time-dependent micromagnetic simulation, we consider two limiting scenarios: a) the magnetization field M ( r) adapts adiabatically to the minimum energy configuration for the given saturation magnetization, and b) the local magnetization structure stays constant except for a homogeneous change of the length of the magnetization vectors.
The simulated Lorentz micrographs (-0.5 mm defocus) for these cases are shown in For comparison, we calculate the optically induced temperature rise of the permalloy disc from the incident pump fluence and the known material constants (for Ni, approximate composition of permalloy: 80% Ni and 20% Fe). For the laser illumination conditions and not considering near-field effects, we estimate an absorbance of the 20-nm-thick magnetic sample of about 25%. With a heat capacity c p = 0.445 J/g.K and a density ρ = 8.90 g/cm 3 52 , we find a temperature increase of 268 K, in reasonable agreement with the temperature estimate from the Lorentz contrast given above.

SI 7: Electron-pulse temporal duration and its influence on Lorentz image contrast
For synchronizing laser-pump and electron-probe pulses, and for characterizing the electron pulses duration, we make use of electron-optical cross-correlation in laser-induced near-fields 39;53 . At delay times for which optical-pump pulse and electron-probe pulses overlap at the sample surface, the electron energy spectra exhibit sidebands separated by the photon energy, as shown in Fig. 6a. The 700-fs duration of the electron probe pulses can be extracted from the temporal width of the highest-order photon-sidebands 33;53 .
We note that a light-induced change in electron energy width of about 20 eV during the temporal laser-electron overlap does not lead to a significant variation in Lorentz contrast. However, the out-of-focus imaging conditions adopted in Lorentz microscopy are particularly sensitive to modifications in the transverse electron momentum (i.e. perpendicular to the propagation direction), which can be modulated by the interaction of electrons with optical near-fields. Considering the pump photon momenta, electron deflections on the order of 10 µrad are expected, yielding streak-like 50-nm contrast features in Lorentz micrograph at 5-mm defocus (Feist et al., in preparation). In the experiments reported here, such side effects are largely absent, since the laser pump pulse is much shorter than the electron pulse.
In the main text, we demonstrated the use of 700-fs electron pulses for imaging magnetization dynamics with 95-nm spatial resolution, and the possibility to further reduce the spatial resolution to 55 nm. Finally, we discuss the possibilities for improving the temporal resolution by using shorter electron pulses. Figure 6b shows image profiles of the magnetic vortex for a fixed delay time of -5 ps and a pump fluence of 3.7 mJ/cm 2 , using the same imaging conditions as for the time-resolved experiments, when probing the sample with 700-fs and 400-fs electron pulses. Each image contains the same total electron dose of 1.5 × 10 7 electrons, consisting of a sum of 6 individual exposures of 10 s each for the longer electron pulses, and of 12 individual exposures for the shorter electron pulses, respectively. The spatial resolution of the magnetic contrast feature is not visibly affected by the lower electron number per pulse, so that time-resolved Lorentz microscopy experiments with higher temporal resolution are clearly feasible.