Framework for Polarized Magnetic Neutron Scattering from Nanoparticle Assemblies with Vortex-Type Spin Textures
Michael P. Adams Evelyn P. Sinaga Stefan Liscak Andreas Michels




[Electronic address: ]michael.adams@uni.lu








[Electronic address: ]andreas.michels@uni.lu





Within the framework of the recently introduced multi-nanoparticle power-series expansion method for the polarized small-angle neutron scattering (SANS) cross section, we present analytical expressions for the polarized SANS observables arising from dilute nanoparticle assemblies with antisymmetric vortex-type spin structures. We establish connections between the magnetic correlation coefficients and the magnetic field-dependent vortex-axes distribution function, which is related to the random orientations of the magnetocrystalline anisotropy axes of the nanoparticles. Our analytical results are validated through a comparative analysis with micromagnetic simulations. This framework contributes to a comprehensive understanding of polarized magnetic neutron scattering from spherical nanoparticle systems exhibiting vortex-type spin structures.




Framework for Polarized Magnetic Neutron Scattering from Nanoparticle Assemblies with Vortex-Type Spin Textures
    Andreas Michels
    April 29, 2024
===============================================================================================================





Introduction. Magnetic nanoparticles, which are in the scope of immense interdisciplinary research, offer versatile applications, e.g., in materials science, nanotechnology, and biomedicine <cit.>. They open up new possibilities in the nanoscopic realm and drive technological advances and breakthrough discoveries. But still, at the current stage of research, it is an immense challenge to characterize their internal spin structure, which is generally to be expected as nonuniform (e.g., <cit.>).

Magnetic small-angle neutron scattering (SANS) is possibly the only technique to probe the spatial variation of spin structures on a scale of ∼1-100  nm and in the bulk of the material <cit.>. Recent advances in the understanding of magnetic SANS from complex nanoparticle systems have been achieved by the marriage of micromagnetic theory and magnetic neutron scattering formalism, through both computer simulations and analytical calculations <cit.>. Although computer simulations offer considerable potency in predicting neutron scattering observables for intricate nanoparticle assemblies, their drawback lies in their time-intensive nature, vast parameter space, and the inherent challenge of interpreting results. This complexity hinders the derivation of overarching conclusions and poses a substantial obstacle in formulating generalized statements.

To address these challenges, Adams et al. <cit.> introduced the multi-nanoparticle power-series expansion (MNPSE) method to study the neutron scattering signatures from spherical nanoparticle assemblies featuring diverse types of magnetic surface anisotropy. Here, we use the MNPSE approach to predict the main features of nanoparticle assemblies with inherent vortex-type spin textures as to be seen in the neutron scattering observables. Vortex-type structures are ubiquitous in magnetism research and are encountered in many systems, such as in type-II superconductors <cit.>, GdCo_2 micropillars <cit.>, Nd-Fe-B magnets <cit.>, iron oxide nanoparticles <cit.> and nanoflowers <cit.>, or the very recently discovered topological vortex rings in a chiral magnetic nanocylinder <cit.>. Our results, which replace the conventional analytical formulation for the superspin model, enable the straightforward prediction of the spin-flip SANS cross section I_sf(q) and the corresponding spin-flip pair-distance distribution function p_sf(r) arising from spatially antisymmetric spin structures, such as nanovortices, through easily-applicable analytical expressions.

The Letter is organized as follows: We start out by analyzing the main features of the first-order MNPSE method for the spin-flip SANS cross section and the pair-distance distribution function. This approach is valid for an arbitrary linear magnetization distribution. Subsequently, for the particular case of a linear vortex, we derive analytical expressions for the two- and one-dimensional SANS observables. The analytical expressions are compared to the results of micromagnetic computations. We refer to the Supplemental Material <cit.> for details regarding the analytical derivations and the micromagnetic SANS simulations.



Linear MNPSE method. This approach is based on the following expansion for the magnetization vector field,

    𝐌'(𝐫') = 
        [ m_0^x; m_0^y; m_0^z ]
        +
        [ m_1^xx m_1^xy m_1^xz; m_1^yx m_1^yy m_1^yz; m_1^zx m_1^zy m_1^zz ]·[ x'; y'; z' ] ,

where 𝐫' = [x', y', z'] denotes the position vector in the local particle frame. The model consists of 12 expansion coefficients per particle, i.e., 3 zero-order coefficients m_0^i and 9 first-order coefficients m_1^jk. For a dilute assembly of spherical nanoparticles, the MNPSE formalism yields the following expression for the azimuthally-averaged spin-flip SANS cross section <cit.>:

    I_sf(q)    = I_sf^0 [f(qR)]^2 +
        I_sf^1 [ f'(qR) ]^2,

where R is the particle radius, and the field-dependent coefficients I_sf^0 and I_sf^1 represent complicated averages of the magnetization coefficients m_0^i and m_1^jk over the particle assembly and over the detector plane. The corresponding basis functions are given by (u=qR):

    f(u)    = sin u - u cos u/u^3 ,
        
    
        f'(u)    = d f/du = (u^2 - 3) sin u + 3 u cos u/u^4 .

Here, f(u) is the form-factor function of the unit sphere <cit.> and f'(u) is the related first-order derivative. By inverse Fourier transformation we find from Eq. (<ref>) the related pair-distance distribution function <cit.>: 

    p_sf(r)    = I_sf^0π r^2/6 R^3[ 1 - 3r/4R + r^3/16 R^3] 
       +  I_sf^1π r^2/10 R^3[ 1 - 5r/4R + 5r^3/16R^3 - r^5/32R^5]  .

While the zero-order contribution (I_sf^0) arises from symmetric (parallel, positive) correlations only, the first-order contribution (I_sf^1) contains antisymmetric (antiparallel, negative) correlations. By comparison to micromagnetic simulations using Mumax3 <cit.>—including isotropic exchange, a random cubic anisotropy [In the numerical micromagnetic simulations we use the materials parameters for (cubic) iron, and we also consider the case of a random uniaxial anisotropy (see <cit.>).], the Zeeman interaction, and the demagnetizing field—we find that this linear approach [Eqs. (<ref>) and (<ref>)] already captures the main features of vortex-type spin textures seen in the SANS observables.

The results that are embodied by Eqs. (<ref>) and (<ref>) are summarized in Fig. <ref>. Prominent features regarding vortex-type spin structures are the decreased spin-flip scattering intensity I_sf(q) at momentum transfer q=0 [see Fig. <ref>(a,c,e,g)] and the damped oscillatory behavior of the pair-distance distribution function p_sf(r) exhibiting negative (antiparallel) correlations related to a vortex (see Fig. <ref>(b,d,f,h), compare to <cit.>). In the limiting case of ι = I_sf^1/I_sf^0→∞ (modeling the remanent state) our linear theory predicts a maximum of I_sf(q) at q_max≅ 2.50/R [maximum of (f'(u))^2]. This prediction is in excellent agreement with the result from our micromagnetic simulations, where we find q_max(B_0 = 0 T) ≅ 2.50/R [see Fig. <ref>(g)]. Furthermore, the relevant zero of p_sf(r) is predicted as the result of the following cubic equation that is derived from Eq. (<ref>):

    ν^3 + 4 ν^2 + ( 2 - 10/3ι) ν - ( 8 + 40/3ι) = 0 ,

where ν = r/R. For ι→∞, Eq. (<ref>) predicts the zero at r_z≅ 1.07 R, whereas in our micromagnetic simulations we find r_z(B_0 = 0 T) ≅ 1.03 R [see Fig. <ref>(h)].

Beyond these limits for the momentum transfer q_max and the “zero” correlation length r_z we find two specific transition points for ι in the two-dimensional (2D) maps shown in Fig. <ref>(a) and (b). In Fig. <ref>(b) we observe for ι > 1 the occurrence of negative correlations [negative values of p_sf(r)], and in Fig. <ref>(a) we see that the scattering intensity at the origin of reciprocal space, I_sf(q=0)/I_sf^max, is constant for ι < 5 and decreases for ι > 5. The micromagnetic simulation results reveal an analogous behavior [compare Fig. <ref>(e) and (f)].
 
 

MNPSE method: The case of a linear vortex. In the formulation of the linear MNPSE method the parameters I_sf^0 and I_sf^1 [in Eqs. (<ref>) and (<ref>)] are arbitrary functions of the 12 magnetization expansion coefficients in Eq. (<ref>). In the following, we aim to adapt the linear MNPSE method to include physically motivated parameters (replacing I_sf^0 and I_sf^1). This approach allows us to obtain a scattering model that is closer related to the underlying micromagnetic Hamiltonian in the sense that it contains information on the vortex helicity, on the orientation distribution of the vortex axes, and on the transformation behavior of the energies in the Hamiltonian under space inversion.

We consider a dilute assembly of noninteracting spherical nanoparticles that are rigidly embedded in a homogeneous and nonmagnetic matrix. Each particle is assumed to have a random orientation of its (cubic or uniaxial) magnetic anisotropy axis with respect to the externally applied magnetic field 𝐇_0 ∥𝐞_z, which defines the laboratory frame of reference. In addition to magnetic anisotropy and the Zeeman interaction, we consider an isotropic exchange energy and, most importantly, the magnetodipolar interaction (see the Supplemental Material <cit.>). When the spin structure of such a spherical nanoparticle is computed starting from saturation, we always find—using the materials parameters of iron—a vortex-type texture at low fields and for particle sizes larger than about 20  nm <cit.>. It is the dipolar interaction that is responsible for the vortex formation.

Based on these simulation results, and with the aim to obtain an approximate expression for the spin-flip SANS cross section of an ensemble of vortex-carrying randomly-oriented nanoparticles, we introduce a magnetization model with a uniform (constant) part of magnitude m_0 and a linear vortex term of magnitude m_1. More specifically, the basic magnetization vector field is written as:

    𝐌'(𝐫') = m_0 𝐞_z' + m_1 𝐯(𝐫') ,

where 𝐞_z'= [0, 0, 1] is the unit vector in z' direction, 𝐯(𝐫') = [-y', x', 0] is the linear vortex field, and 𝐫' = [x',y',z'] is the position vector with reference to the local vortex frame. Compared to Eq. (<ref>) the number of expansion coefficients in Eq. (<ref>) has been reduced to two. A positive m_1 indicates a counterclockwise (CCW) or right-handed sense of rotation, while a negative m_1 corresponds to a clockwise (CW) or left-handed sense of rotation. We note that for a micromagnetic Hamiltonian that contains the isotropic exchange interaction, magnetic anisotropy, the Zeeman, and magnetodipolar interaction, there exists no preference for CCW or CW vortex rotation senses in the particles. CCW and CW vortices appear with equal probability so that the chiral function averages to zero (see below). However, by including the Dzyaloshinskii-Moriya interaction (DMI), which breaks space-inversion symmetry, chirality selection takes place and leads to a nonzero chiral function <cit.>.

Equation (<ref>) models a linear vortex in the local vortex frame. We introduce a zy rotation matrix 𝐑(α,β) that transforms the local magnetization 𝐌' into the laboratory frame of reference, where α and β denote the (global) polar and azimuthal angles, respectively. The resulting global magnetization vector field is then obtained as:

    𝐌(𝐫; α, β) = 𝐑(α, β) ·𝐌'(𝐑^T(α, β) ·𝐫) .

Using Eq. (<ref>) in the MNPSE method <cit.>, we define the ensemble-averaged (dilute) SANS cross sections as:

    ⟨dΣ_sf,χ/dΩ⟩   = 1/2∫_0^4π[ dΣ_sf,χ^CCW/dΩ +dΣ_sf,χ^CW/dΩ] ψ(α, β) dΥ

where dΥ = sinα dα dβ is the solid-angle differential, and dΣ_sf,χ^CCW/dΩ(𝐪; α,β) and dΣ_sf,χ^CW/dΩ(𝐪; α,β) are the SANS cross sections referring to two nanoparticles with the same orientation (α, β), but opposite senses of vortex rotation (m_1^CCW = - m_1^CW). The function ψ(α, β) is a field-dependent probability distribution that models the orientation of both the CCW and CW vortex rotation axes (no distinction between the different polarities); its origin is related to the distribution of the net magnetization vectors of the nanoparticles. For simplicity, we assume a uniform distribution ψ_u on the spherical surface, which is limited by a field-dependent conical opening angle 0^∘≤α_c≤ 90^∘. The azimuthally-symmetric distribution is then given by (see <cit.> for details): 

    ψ_u(α, β) = Θ(1 - α/α_c)/2π (1 - cosα_c) ,

where Θ(ξ) is the Heaviside function. In the fully saturated case (B_0→∞) it follows that α_c→ 0, and α_c increases with decreasing applied magnetic field. By inserting Eqs. (<ref>)-(<ref>) into the formalism of the MNPSE method, we obtain the following final expressions for the randomly-averaged 2D spin-flip SANS cross section and chiral function <cit.>:


    ⟨dΣ_sf/dΩ⟩(q, θ)    = W/8
    [m_0 f(qR)]^2 ×[ 12 - (cos^2α_c +  cosα_c)(3cos^2(2θ) + 2cos(2θ) + 3) + 4 cos(2θ) ] 
        + W/2 [R m_1 f'(qR)]^2 ×[ 3 -  (2 cos^2α_c + 2 cosα_c - 1)cos(2θ) ] ,
    ⟨dΣ_χ^CCW, CW/dΩ⟩(q, θ)    = ± W  
     [ R m_0 |m_1| f(qR) f'(qR) cosθ]
       ×[4 + cos^2α_c + cosα_c - 3 (cos^2α_c + cosα_c) cos^2θ] ,


where W is a scaling constant. In Eq. (<ref>) we have separated the chiral function into CCW (“+” sign) and CW (“-” sign) contributions. For the here-considered micromagnetic energy contributions <cit.> (with no chirality selection taking place), it then follows that

    ⟨dΣ_χ/dΩ⟩ = 1/2[⟨dΣ_χ^CCW/dΩ⟩ + ⟨dΣ_χ^CW/dΩ⟩] = 0 .

Figure <ref> displays Eqs. (<ref>) and (<ref>). At saturation [Fig. <ref>(a,c) and (g,i)], with α_c = 0^∘ and m_1/m_0 = 0, the spin-flip SANS cross section exhibits the well-known sin^2θcos^2θ angular anisotropy. At remanence [Fig. <ref>(b,d) and (h,j)], with α_c = 27^∘ and m_1/m_0 = 0.7, we observe for the spin-flip signal an anisotropy that strongly differs from the saturated case, with maxima for θ = 90^∘. This observation strongly suggests that the magnetization Fourier components are anisotropic, i.e., M_x,y,z = M_x,y,z(q,θ) (compare to Eq. (4) in <cit.>). Consequently, to not lose this information, the experimental data analysis should be concentrated on 2D spin-flip data rather than on the azimuthally-averaged 1D data. In micromagnetic simulations of spherical nanoparticles, very similar scattering patterns were observed <cit.>.

Averaging Eq. (<ref>) over the angle θ, i.e., (2π)^-1∫_0^2π (...) dθ, yields the 1D quantity <cit.>

    ⟨ I_sf⟩ (q)    = 3W/16 [m_0 f(qR)]^2 (8 - 3 cos^2α_c - 3 cosα_c)
       + 
    8 3W/16 [m_1 R f'(qR)]^2   .

By comparison to Eq. (<ref>) we note that the new parameters m_0, m_1, and α_c in Eqs. (<ref>) and (<ref>) are related to the coefficient ratio I_sf^1/I_sf^0 as follows:

    ι   = I_sf^1/I_sf^0 = 8 m_1^2 R^2/m_0^2 (8 - 3cos^2α_c - 3cosα_c) ,

which emphasizes the importance of the vortex-axes distribution function. The angle α_c may be obtained from the analysis of (preferentially 2D) experimental spin-flip SANS data (compare to Fig. <ref>(d) and (j) and the video clip in <cit.>).

Conclusion. In this Letter we have demonstrated that the linear MNPSE approach captures the main effects in the spin-flip SANS cross section and pair-distance distribution function stemming from dilute assemblies of spherical nanoparticles exhibiting vortex-type spin textures. A crucial insight is that the linear functionality represents the most important contribution to the magnetic neutron scattering cross section. Based on the specific case of a linear vortex model, we have derived analytical expressions for the 2D and 1D spin-flip and chiral cross sections of an ensemble of randomly oriented vortex-carrying nanoparticles. The maximum of the spin-flip scattering intensity and the zero of the pair-distance distribution function appear, respectively, at momentum transfer q_max≅ 2.50/R and position r_z≅ 1.07 R, where R denotes the radius of the spherical nanoparticles. The analytical predictions, which enable e.g. the determination of the field-dependent conical opening angle α_c of the vortex-axes distribution from experimental data, are in very good agreement with the results of micromagnetic simulations. The chiral SANS cross section is sensitive to the vortex rotation sense, but in a many-particle system with no chirality selection, it averages to zero (as expected). A candidate for chirality selection is the DMI interaction that breaks space-inversion symmetry.

We acknowledge financial support from the National Research Fund of Luxembourg (AFR Grant No. 15639149, CORE Grant DeQuSky, and PRIDE MASSENA Grant).











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  author A. Michels, and author K. L. Metlov, @noop 
  journal journal J. Appl. Cryst. volume 55, pages 592 (year
  2022)NoStop
[Kons et al.(2023)Kons,
  Krycka, Robles, Ntallis,
  Pereiro, Phan, Srikanth,
  Borchers, and Arena]arena2023
  author author C. Kons, author K. L. Krycka,
  author J. Robles, author N. Ntallis, author
  M. Pereiro, author M.-H. Phan, author H. Srikanth, author J. A.  Borchers, and author D. A.  Arena, @noop  journal journal ACS Appl. Nano Mater. volume 6,  pages 10986 (year 2023)NoStop
[Ukleev et al.(2024)Ukleev,
  Ajejas, Devishvili, Vorobiev,
  Steinke, Cubitt, Luo,
  Abrudan, Radu, Cros,
  Reyren, and White]ukleev2024
  author author V. Ukleev, author F. Ajejas,
  author A. Devishvili, author A. Vorobiev, author
  N.-J. Steinke, author
  R. Cubitt, author C. Luo, author R.-M.  Abrudan, author F. Radu, author V. Cros, author N. Reyren, and author
  J. S. White, @noop  journal journal STAM Methods volume
  25, pages 2315015 (year 2024)NoStop
[Sinaga et al.(2024)Sinaga,
  Adams, Hasdeo, and Michels]evelynprb2024
  author author E. P. Sinaga, author M. P. Adams,
  author E. H. Hasdeo, and  author A. Michels, @noop
    (year 2024), https://arxiv.org/abs/2402.00558
  arXiv:2402.00558 [cond-mat.mes-hall] NoStop
[Adams et al.(2024)Adams,
  Sinaga, Kachkachi, and Michels]adamsprb2024
  author author M. P. Adams, author E. P. Sinaga,
  author H. Kachkachi, and  author A. Michels, @noop
   journal journal Phys. Rev. B volume 109, pages 024429 (year
  2024)NoStop
[Eskildsen et al.(2011)Eskildsen, Forgan, and Kawano-Furukawa]forgan2011
  author author M. R. Eskildsen, author E. M. Forgan, and author H. Kawano-Furukawa, @noop  journal journal Rep. Prog. Phys. volume 74, pages 124504 (year 2011)NoStop
[Donnelly et al.(2021)Donnelly, Metlov, Scagnoli, Guizar-Sicairos, Holler, Bingham,
  Raabe, Heyderman, Cooper,  and Gliga]metlovnp2021
  author author C. Donnelly, author K. L. Metlov, author V. Scagnoli,
  author M. Guizar-Sicairos,
  author M. Holler, author N. S. Bingham, author
  J. Raabe, author L. J. Heyderman, author N. R.  Cooper, and author S. Gliga, @noop  journal journal Nat. Phys. volume 17, pages
  316 (year 2021)NoStop
[Bersweiler et al.(2023)Bersweiler, Oba, Sinaga, Peral, Titov, Adams, Rai,
  Metlov, and Michels]bersweilerprb2023
  author author M. Bersweiler, author Y. Oba,
  author E. P. Sinaga, author I. Peral, author
  I. Titov, author M. P. Adams, author V. Rai, author K. L.  Metlov, and author A. Michels, @noop  journal journal Phys. Rev. B volume 108, pages 094434 (year 2023)NoStop
[Samardak et al.(2016)Samardak, Davydenko, Ognev, Jeon, Choi, and Kim]samardak2016
  author author A. S. Samardak, author A. V. Davydenko, author A. V. Ognev, author Y. S. Jeon,
  author Y. S. Choi, and  author Y. K. Kim, @noop
   journal journal Jap. J. Appl. Phys. volume 55, pages 100303 (year
  2016)NoStop
[Usov et al.(2018)Usov,
  Nesmeyanov, and Tarasov]usov2018
  author author N. A. Usov, author M. S. Nesmeyanov, and author V. P. Tarasov, @noop  journal journal Sci.
  Rep. volume 8, pages 1224 (year 2018)NoStop
[Niraula et al.(2023)Niraula, Toneto, Goya, Zoppellaro, Coaquira, Muraca, Denardin, Almeida, Knobel, Ayesh, and Sharma]knobel2023
  author author G. Niraula, author D. Toneto,
  author G. F. Goya, author G. Zoppellaro, author
  J. A. H. Coaquira, author
  D. Muraca, author J. C. Denardin, author T. P.  Almeida, author M. Knobel, author A. I. Ayesh,  and author S. K. Sharma,  @noop  journal journal Nanoscale Adv.  volume 5, pages 5015 (year
  2023)NoStop
[Moya et al.(2024)Moya,
  Escoda-Torroella, Rodríguez-Álvarez,
  Figueroa, García, Ferrer-Vidal, Gallo-Cordova, Puerto Morales, Aballe, Fraile Rodríguez, Labarta, and Batlle]BATLLE2024
  author author C. Moya, author M. Escoda-Torroella, author J. Rodríguez-Álvarez, author A. I.  Figueroa, author Í. García, author I. B. Ferrer-Vidal, author A. Gallo-Cordova, author M. Puerto Morales, author L. Aballe, author A. Fraile Rodríguez, author A. Labarta, and author X. Batlle, @noop  journal journal Nanoscale volume 16, pages
  1942 (year 2024)NoStop
[Liu and Nagaosa(2024)]nagaosaprl2024
  author author Y. Liu and author N. Nagaosa,  @noop  journal journal Phys. Rev. Lett.  volume 132, pages 126701 (year 2024)NoStop
[mic()]michael2024sm
  @noop  note See the Supplemental Material [URL] for
  further details on the numerical micromagnetic calculations and on the MNPSE
  method used for the analytical calculation of the spin-flip SANS cross
  section and chiral function of vortex-carrying nanoparticles. We also provide
  a video clip that features the field evolution of the spin structure, the 2D
  and 1D spin-flip SANS cross section, as well as the pair-distance
  distribution function along the hysteresis loop.Stop
[Vansteenkiste et al.(2014)Vansteenkiste, Leliaert, Dvornik,
  Helsen, Garcia-Sanchez, and Van Waeyenberge]mumax3new
  author author A. Vansteenkiste, author J. Leliaert, author M. Dvornik,
  author M. Helsen, author F. Garcia-Sanchez, and author B. Van Waeyenberge, @noop 
  journal journal AIP Advances volume 4, pages 107133 (year
  2014)NoStop
[Leliaert et al.(2018)Leliaert, Dvornik, Mulkers, De Clercq, Milošević, and Van Waeyenberge]Leliaert_2018
  author author J. Leliaert, author M. Dvornik,
  author J. Mulkers, author J. De Clercq, author
  M. V. Milošević, and  author B. Van Waeyenberge,  @noop  journal journal J. Phys. D: Appl.
  Phys. volume 51, pages 123002
  (year 2018)NoStop
[Note1()]Note1
  note In the numerical micromagnetic simulations we use the
  materials parameters for (cubic) iron, and we also consider the case of a
  random uniaxial anisotropy (see <cit.>).Stop
[Sinaga et al.(2023)Sinaga,
  Adams, Bersweiler, Vivas,
  Hasdeo, Leliaert, Bender,
  Honecker, and Michels]evelynprb2023
  author author E. P. Sinaga, author M. P. Adams,
  author M. Bersweiler, author L. G. Vivas, author
  E. H. Hasdeo, author
  J. Leliaert, author
  P. Bender, author D. Honecker, and author A. Michels, @noop  journal journal Phys. Rev. B volume 107, pages 014416 (year 2023)NoStop
[Adams et al.(2023)Adams,
  Sinaga, and Michels]michaeliucrj2023
  author author M. P. Adams, author E. P. Sinaga,  and author A. Michels,  @noop  journal journal IUCrJ volume 10, pages 261 (year
  2023)NoStop


