Artificial Double-Helix for Geometrical Control of Magnetic Chirality

Chirality plays a major role in nature, from particle physics to DNA, and its control is much sought-after due to the scientific and technological opportunities it unlocks. For magnetic materials, chiral interactions between spins promote the formation of sophisticated swirling magnetic states such as skyrmions, with rich topological properties and great potential for future technologies. Currently, chiral magnetism requires either a restricted group of natural materials or synthetic thin-film systems that exploit interfacial effects. Here, using state-of-the-art nanofabrication and magnetic X-ray microscopy, we demonstrate the imprinting of complex chiral spin states via three-dimensional geometric effects at the nanoscale. By balancing dipolar and exchange interactions in an artificial ferromagnetic double-helix nanostructure, we create magnetic domains and domain walls with a well-defined spin chirality, determined solely by the chiral geometry. We further demonstrate the ability to create confined 3D spin textures and topological defects by locally interfacing geometries of opposite chirality. The ability to create chiral spin textures via 3D nanopatterning alone enables exquisite control over the properties and location of complex topological magnetic states, of great importance for the development of future metamaterials and devices in which chirality provides enhanced functionality.

Selection of X-ray polarisations using a slit, between UP and MID positions within the cone of X-rays emitted by a bending magnet. Reproduced with permission from the author 1 .

Protocol 1: Beamline calibration
Due to the large changes in thickness intrinsic to complex 3D nanostructures, a comprehensive beamline calibration and image correction protocol is necessary, as described here.
In ideal conditions, the intensity transmitted through a magnetic sample under X-ray illumination can be expressed as 2 : where I 0 is the incident intensity, the non-magnetic linear absorption coefficient, = ℎ⁄ is the XMCD coefficient (ℎ) normalized against the non-magnetic absorption ( ), ⃗⃗ • ⃗ is the projection of the magnetisation vector onto the photon beam wave vector, and the infinitesimal distance element through the beam path. The ℎ coefficient within models the polarization-dependent magnetic absorption due to XMCD. For Co at the L3 edge, it changes sign when photon helicity is changed, and becomes zero for linearly polarised light.
In a magnetic microscopy experiment, (⃗⃗ • ⃗ ) is probed by comparing two images using different photon polarisation (slit placement in Fig. M5b), thus changing the XMCD coefficient and , and then comparing the two transmitted intensities: Note that should be zero as it is associated to linearly polarised light, however we are considering it in all the expressions for generality and clarity of the analysis. From these two intensities, (⃗⃗ • ⃗ ) can be extracted by evaluating: In real experimental conditions, however, the intensity measured at the CCD camera has been found to deviate from the above case in at least two ways: I 0 varies across the image and an uneven background , formed by stray scattered photons, is found to hit the detector even when the field of view is blocked by a thick sample holder. The measured intensity (I ) at the CCD is then given by: Correcting for the presence of is particularly important when either I 0 exp(− ∫ ) becomes of the same order of magnitude as , or considerable changes in occur between images. If one tries to evaluate the resulting XMCD contrast by comparing images taken with the polarisation slit placed UP and MID, the following is obtained: which is not proportional to ⃗⃗ • ⃗ , and will display an artificial contrast that tends to ln( ⁄ ) as the absorption from the material increases. Differences between I 0 UP and I 0 MID also need to be corrected.
To correct for these effects, we have carried out an experimental procedure consisting of three steps: First, an image (Image 1) is acquired without sample, containing ( 0 + ). Then an image with the sample is taken (Image 2), which contains the full information described in Equation M5. Finally, a third image (Image 3) is taken using the sample holder to block the field of view (this image contains ).
The following correction is then applied to the data: This correction yields: All images in this work have been corrected according to equation M7.
In addition, the exact level of circular polarisation and beam energy are generally affected by small changes in beamline alignment. Since X-ray polarisation is chosen by moving vertically a slit inside the optical system (see Fig. 2a), we perform a precise beamline calibration which accounts for differences in absorption ( ) and magnetic dichroism ( ) between the two polarisation settings.
If we measure in two conditions (UP and MID), we expect for a general case: To extract (⃗⃗ • ⃗ ), we then measure and calculate In 2D samples (constant thickness), an absorption spectrum can be usually obtained for each illumination condition, and then is calculated. In very small 3D nanostructures as those studied here, however, this is challenging due to the lack of clear magnetic domains and the absence of regions with constant thickness.
As a calibration sample, we have employed a pair of long nanowires which are uniformly magnetized (due to their thin diameter) and present an antiparallel state due to the dipolar coupling during growth (see Fig. M6). The nanowires are grown by FEBID under the same conditions as the double-helix system. The top part of these long nanowires became bent towards the X-ray beam shortly after starting imaging them (not shown here, confirmed by in-situ rotation). We associate this effect to strain changes due to surface dehydration 3 under photon irradiation. This bending provides a net magnetic moment along the X-ray beam (⃗⃗ • ⃗ ) ≠ 0 at the top of the wires, while the base remains vertical, i.e. (⃗⃗ • ⃗ ) = 0 (see schematics in Fig. M6d-e). To explain the protocol used to calibrate and , we first analyse two images of the nanowire pair for both polarisations (see Fig. M6a). Fig. M6b and Fig. M6c correspond to X-ray absorption profiles across two different heights of the wires: At their base (Fig. M6c), the two slit positions show no change in absorption between the two wires. However, the MID profile shows an overall higher absorption. This contrasts with the one at the upper part of the wires (Fig. M6b), where an asymmetric absorption between the nanowires is observed for the UP case, which corresponds to elliptically-polarised light. This asymmetric feature is exploited to systematically evaluate the non-magnetic absorption (where is the diameter of the nanowires) and the dichroic change to the absorption coefficient ( ⃗⃗ • ⃗ ) , using the definitions shown in Fig. M7a, based on equation M1. The two parameters are extracted using horizontal absorption lines for the two slit positions as explained above, for a set of equidistant Z-pixel values in the images (see Fig. M7b and Fig. M7c, respectively). As expected, no significant magnetic contrast is observed for linearly-polarised X-rays (MID slit position) at any Z-pixel value (Fig. M7b). On the contrary, a clear signal is found for elliptically-polarised X-rays (UP slit position), for the top (low Z-pixel values) of the nanowires. This signal tends to zero at their base (high Z-pixel values), where no component of the magnetisation is parallel to the X-ray beam. The tilting of the nanowires at their upper part not only results in an increase in XMCD signal, but it also leads to an increased effective thickness, as the X-ray absorption evolution with Z-pixel shows (see Fig. M7c). Based on the data of Fig. M7c, a histogram of the ratio can be created (Fig. M7d), from which we conclude that there exists a mismatch ratio in nonmagnetic absorption between the two polarisations of 0.9 ± 0.03 in our experiments. We therefore calculate the real XMCD contrast in our system as: which is proportional to the projection of ⃗⃗ along the direction of X-ray propagation. As illustrated by Fig.  M7f and Fig. M7g, the application of this correction enables to remove an overall artificial dark contrast in the XMCD images (Fig. M7f), resulting in a well-centred dichroic contrast (Fig. M7g). Reproduced with permission from the author 1 .
To ensure that the geometrical and magnetic chirality correspondences described in the main text are correct, all sign conventions for the different experimental setups must coincide. Two checks have been performed in this regard: First, XMCD was confirmed to be negative (blue colour in the scales of the main text) when ⃗⃗ points towards the synchrotron ring (+x in all figures). Secondly, we cross-checked that the X-ray images saved by the instrument's control software correspond to the viewpoint after transmission, i.e. as seen by the CCD, which reverses the sign of the y-axis with respect to SEM images. To match X-ray and SEM images, we chose to flip the y-axis of the X-ray data, since SEM data provide the true geometry (and geometrical chirality) of the structures.    In (d), when a RH double helix is added, a Néel defect naturally forms to enable the transition from the bottom LH Bloch wall to the RH Bloch wall above the tendril perversion. A dotted arrow represents the magnetic circulation observed in the real 3D structure. This circulation does not constitute a Néel defect because the orientation of the vortex core and the Bloch wall below the core is the same, and therefore the circulating magnetic texture can be continuously deformed into a Bloch wall.
(e-f) The same process is exemplified for the case in which a LH double-helix is fabricated first, leading to an opposite chirality of the asymmetric vortex.