Complex free-space magnetic field textures induced by three-dimensional magnetic nanostructures | Nature Nanotechnology
Three-dimensional geometrical parameters of the cobalt double helices
In curved three-dimensional systems, it is useful to define the geometrical properties in terms of the κ and τ of the system, which are defined as
where P = pH/2π, C is the chirality and the radii of curvature and torsion are defined as κ−1 and τ−1, respectively. It has been shown analytically that the radii of curvature and torsion determine the effective anisotropy and spin wave dynamics of a single magnetic helix20. The helices investigated experimentally in this work have the geometrical parameters defined in Table 1.
The three-dimensional cobalt double helices were fabricated using focused electron beam induced deposition combined with a program compatible with computer-aided design software that allows for the deposition of three-dimensional architectures. Specifically, a single-pixel double-helix model was designed using the open-source program FreeCAD. Following the growth of calibration structures to account for the growth rate of the cobalt precursor according to ref. 23, a stream file to direct the scanning electron beam was created.
The double helices were fabricated on Omniprobe transmission electron microscopy sample holders, which were premilled with the focused ion beam to prevent shadowing of the sample holder during the laminography scan. Both the focused ion beam milling and focused electron beam induced deposition were performed in a Helios 660 NanoLab focused ion beam microscope at the Kelvin Nanocharacterisation Centre of the University of Glasgow. Specifically, for the growth of the three-dimensional magnetic nanostructures, an acceleration voltage of 5 kV and a current of 86 pA were used, in combination with the precursor Co2(CO)8. The growth times varied between 20 and 30 min per double-helix structure.
Following the deposition, the samples were annealed for 40 min at 250 °C, which results in an increase in the spontaneous magnetization of the cobalt, without a large increase of crystalline size or texture formation42,43. In particular, the annealing leads to a nanocrystalline microstructure, a cobalt composition of ~80 at.% and a saturation magnetization of 800–900 kA m−1. This treatment also leads to the formation of a protective carbon shell around the surface of the three-dimensional structure. The annealing procedure has the additional advantage of reducing carbon deposition—and in turn any deformation of the structure—during the X-ray imaging experiments.
X-ray magnetic laminography
Soft-X-ray magnetic laminography was performed at the PolLux beamline at the Swiss Light Source, Switzerland13. X-ray magnetic laminography is a recently developed three-dimensional imaging technique, which involves the measurement of projections of the magnetization of a sample for many different orientations of the sample with respect to an X-ray beam12.
Sensitivity to the magnetization is obtained by probing the XMCD. In this measurement, the XMCD was probed at the Co L2 edge with a photon energy of 796 eV. Although the XMCD contrast is weaker at the L2 edge than at the L3 edge, measurements were performed at the L2 edge due to the high absorption of the double helices, providing a balance between the transmission and magnetic contrast to optimize image quality. We note that, due to the higher curvature of helix A, the effective thickness of the nanowire probed by the X-rays is higher, leading to a lower signal to noise ratio in the image compared with helix B under otherwise equal imaging conditions. For each orientation, XMCD images were measured by measuring STXM images with C+, C− and linear (horizontal) polarization. To obtain a quantitative measure of the projection of the magnetization, so-called ‘dark-field’ signal originating from leakage of the centre stop of the zone plate and from the higher-order light diffracted by the monochromator was removed from the projections by applying the following normalization procedure:
where T is the normalized transmitted intensity, I is the nominal transmitted intensity, I0 is the intensity incident on the sample obtained from an empty region of the image and DF is the dark-field signal, which is estimated from regions of the image where the incident beam is blocked. The linear light projections were used to cross-check the removal of the unfocused and higher-energy light. In this way, quantitative projections of the magnetization were obtained, which were used to obtain a correct reconstruction of the three-dimensional magnetization.
For the two-dimensional imaging of the helices shown in Fig. 1 before and after the application of a transverse magnetic field, the samples were mounted in the laminography stage, and aligned such that the X-rays were aligned with the direction of the long axis of the helix, resulting in an incident angle of the X-rays and long axis of the helix of 45°, providing sensitivity to the component of the magnetization parallel to the long axis of the helix. The images in Fig. 1f,h,g,i were measured with eight, two, nine and four averages, respectively, leading to slight changes in the noise level of the individual images.
For the three-dimensional imaging, the X-ray laminography set-up consisted of a rotation stage whose rotation axis is oriented at 45° to the X-ray beam. Projections were measured with an angular separation of 10°. In laminography, the number N of projections measured over 360° required to achieve a spatial resolution Δr is defined as44.
\(N = \uppi \frac{t}{{{\Delta}r}}\tan \theta _{\mathrm{L}}\), where t is the thickness of the sample and θL is the laminography angle, which defines the angle between the X-ray beam and the rotation axis—in this case 45°.
In this measurement, projections of the structure with a field of view of 3 × 3 μm2 and a pixel size of 25 nm were measured around 360° with an angular separation of 10°. For each angle, an average of two XMCD projections was measured to increase the signal to noise ratio. The angular separation of 10° corresponds to a nominal Δr of less than 20 nm. In reality, the spatial resolution of the final reconstruction was limited by the signal to noise ratio of the XMCD projections. Of the 36 projections, it was only possible to measure 27 because of shadowing of the X-ray beam due to the sample holder. Simulations of magnetic laminography revealed that, although this led to an asymmetry in the reconstructed magnetization, it does not prevent the identification of the locked domain wall state.
The three-dimensional magnetic configuration was then reconstructed using a graphics processing unit implementation of an arbitrary projection reconstruction algorithm developed in ref. 29 and used in refs. 12,13.
The three-dimensional magnetization structure was visualized with ParaView 5.5.0.
Magnetic laminography reconstruction
The reconstructed mx, my and mz components of the magnetic configuration are presented in Extended Data Fig. 1a–c, respectively, and are directly compared with the micromagnetic simulation of the locked domain wall state in Extended Data Fig. 1d–f. The multidomain structure is confirmed by the mx component, which reveals positive and negative domains on the upper and lower halves of the helix, respectively. However, it is only with the transverse components of the magnetization, my and mz, that the type of domain wall pair state can be identified: both the my and mz components exhibit an alternating contrast (Extended Data Fig. 1b,c), consistent with the locked domain wall state (Extended Data Fig. 1e,f), and not with the more standard unlocked domain wall pair configuration previously observed in planar systems.
Micromagnetic simulations
Finite-element meshes were created by first creating STL files corresponding to three-dimensional double helices of different parameters using the open-source program FreeCAD. For all simulations, the nanowire diameter was kept constant at 50 nm, while the helix radii and pitches were varied. The meshes were created with a mesh size of 5 nm using the program Gmsh45.
To map the phase diagram of the stable state after transverse saturation of the magnetization, micromagnetic simulations were performed using the program magnum.fe, which employs finite-element micromagnetics with a hybrid finite-element/boundary-element method for the magnetostatic field computation and a preconditioned implicit time integration scheme for the Landau–Lifshitz–Gilbert equation46. The spontaneous magnetization of the material was fixed to Ms = 8 × 105 A m−1. The magnetization was initialized in the transverse direction, assuming full saturation, and then the relaxation calculated using the Landau–Lifshitz–Gilbert equation with a damping parameter α = 1.
To calculate the stray field around the final magnetization state, the double-helix mesh was embedded within a boxed mesh using the program Gmsh, and the stray field calculated at each position using the magnum.fe solver.
To compare the helix configuration with that of a straight nanowire, a coordinate transformation was applied to the relaxed locked domain wall configuration result using ParaView, and the magnetization transformed accordingly. Following the unwinding of the magnetic state, the configuration was once more allowed to relax using magnum.fe to reach a stable configuration. The micromagnetic simulations presented in Figs. 3 and 4 of the main text correspond to a helix pitch of 500 nm, a helix radius of 30 nm and a nanowire diameter of 50 nm.