Small-angle neutron scattering of long-wavelength magnetic modulations in reduced sample dimensions

Abstract

Magnetic small-angle neutron scattering (SANS) is ideally suited to providing direct reciprocal-space information on long-wavelength magnetic modulations, such as helicoids, solitons, merons or skyrmions. SANS of such structures in thin films or micro-structured bulk materials is strongly limited by the tiny scattering volume vis a vis the prohibitively high background scattering by the substrate and support structures. Considering near-surface scattering just above the critical angle of reflection, where unwanted signal contributions due to substrate or support structures become very small, it is established that the scattering patterns of the helical, conical, skyrmion lattice and fluctuation-disordered phases in a polished bulk sample of MnSi are equivalent for conventional transmission and near-surface SANS geometries. This motivates the prediction of a complete repository of scattering patterns expected for thin films in the near-surface SANS geometry for each orientation of the magnetic order with respect to the scattering plane.

Keywords: MnSi; magnetism; near-surface SANS; non-collinear magnetism; skyrmions; small-angle neutron scattering; thin films.

Figure 1

Schematic depiction of the different SANS geometries. (a) Transmission SANS geometry where the neutron beam is oriented normal to the sample surface A located in the xy plane. The edge thickness t of the sample is positioned parallel to the incident neutron beam, which propagates along the z axis of the sample. (b) Reflection SANS geometry where the neutron beam is directed at a shallow incidence angle α_i with respect to the sample surface A located in the xy plane. The edge thickness t of the sample is positioned (almost) perpendicular to the incident neutron beam, which propagates along the x axis of the sample. For values of α_i greater than or equal to the material-specific critical angle of reflection α_c, the configuration is known as near-surface SANS. For values of α_i less than α_c an evanescent wave may form in the surface layers of the sample. This configuration is referred to as grazing-incidence SANS. For each geometry, rocking scans are obtained by rotating the sample about the vertical ω and horizontal ϕ instrument axes.

Figure 2

Relationship between the penetration depth D and the incidence angle α_i of the neutron beam. Results are calculated for MnSi at a neutron wavelength of 5.5 Å, resulting in a critical angle of α_c = 0.076°. For incidence angles less than the critical angle (α_i < α_c), surface-sensitive GI-SANS takes place. For incidence angles greater than or equal to the critical angle (α_i ≥ α_c) bulk-sensitive NS-SANS takes place.

Figure 3

Neutron scattering data. Scattering patterns obtained in the transmission SANS geometry (column 1) and in the NS-SANS geometry (column 2) in the (a) helical, (b) conical, (c) skyrmion and (d) fluctuation-disordered phases of MnSi. Equivalent scattering patterns obtained for the two SANS geometries implies that the same magnetic periodicities are probed.

Figure 4

Neutron scattering data. Scattering patterns obtained in the skyrmion phase of MnSi in the NS-SANS geometry. For data obtained (a1) at α_i = 0.3° without rocking and (a2) as a rocking sum about the ω and ϕ axes.

Figure 5

Fourier transformations of the real-space magnetization densities of helical, conical and skyrmion magnetic orders. Calculations were performed for a single-domain sample of MnSi exhibiting a bulk helical wavelength of 180 Å and a thickness of 1000 Å. The catalogue of results shown here serves as a point of reference for SANS studies of thin films in near-surface geometry.