pith. machine review for the scientific record. sign in

arxiv: 2604.02824 · v1 · submitted 2026-04-03 · ❄️ cond-mat.mtrl-sci

Recognition: 2 theorem links

· Lean Theorem

Disorder-induced chirality in superconductor-ferromagnet heterostructures revealed by neutron scattering and multiscale modeling

Alicia Backs, Anders Bergman, Angela B. Klautau, Annika Stellhorn, Connie Bednarski-Meinke, Elizabeth Blackburn, Emmanuel Kentzinger, Helena M. Petrilli, Ivan P. Miranda, Juan G. C. Palma, Juri Barthel, Nina-Juliane Steinke, Steffen Tober

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:22 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords magnetic chiralityDzyaloshinskii-Moriya interactionFePd filmsneutron scatteringdisorder-induced effectssuperconductor-ferromagnet heterostructuresmultiscale modeling
0
0 comments X

The pith

Chemical disorder and compositional gradients in FePd generate net magnetic chirality through finite Dzyaloshinskii-Moriya interactions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines why nominally centrosymmetric FePd layers in superconductor-ferromagnet stacks show net chirality that affects triplet superconductivity. Structural probes and polarization-analyzed grazing-incidence small-angle neutron scattering reveal partial L1_{0} order, atomic intermixing, anti-phase boundaries, and a depth-dependent defect gradient together with room-temperature chiral magnetic modulations. First-principles calculations and multiscale modeling then trace the chirality to Dzyaloshinskii-Moriya terms that arise when chemical disorder is combined with the measured compositional gradient, producing mixed Bloch-Néel structures whose in-plane length scale matches experiment. A reader would care because the result relocates the source of chirality from interfaces alone to intrinsic bulk-like features of the ferromagnet.

Core claim

Chemical disorder in FePd, especially when paired with a compositional gradient, produces finite Dzyaloshinskii-Moriya interactions that stabilize chiral finite-q magnetic modulations of mixed Bloch-Néel character; the resulting in-plane modulation length approaches the value measured by PA-GISANS, demonstrating that the observed net chirality is an intrinsic property of the disordered FePd layer rather than solely an interface effect.

What carries the argument

Depth-dependent defect gradient combined with atomic intermixing that induces dominant Dzyaloshinskii-Moriya interactions and finite-q chiral modulations.

If this is right

  • Net chirality is generated inside the FePd volume and does not require an external interface.
  • The main chiral component lies in-plane while an out-of-plane component tracks the depth-dependent inhomogeneity.
  • The in-plane modulation length produced by the disorder-induced Dzyaloshinskii-Moriya terms falls in the experimentally observed range.
  • Similar disorder profiles in other centrosymmetric ferromagnets can be expected to produce comparable chiral modulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Growth protocols that control intermixing and defect gradients could be used to tune the strength of chirality in device stacks.
  • The same mechanism may operate in other Fe-based or Pd-based thin films where anti-phase boundaries and composition slopes are common.
  • Modeling of triplet superconductivity in these hybrids should now incorporate bulk disorder contributions to the magnetic texture rather than interface-only terms.

Load-bearing premise

The multiscale model correctly maps the measured defect gradient and intermixing profile onto the Dzyaloshinskii-Moriya terms that set the observed modulation length and mixed character.

What would settle it

Observation of zero net chirality and zero Dzyaloshinskii-Moriya strength in a perfectly ordered, compositionally uniform FePd film of the same thickness would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.02824 by Alicia Backs, Anders Bergman, Angela B. Klautau, Annika Stellhorn, Connie Bednarski-Meinke, Elizabeth Blackburn, Emmanuel Kentzinger, Helena M. Petrilli, Ivan P. Miranda, Juan G. C. Palma, Juri Barthel, Nina-Juliane Steinke, Steffen Tober.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a,b) shows high-resolution HAADF-STEM and HAADF-STEM EDX elemental maps of i.Nb/FePd, respectively. In a well-ordered L10 FePd structure, the alternating Fe and Pd layers produce a clear contrast in the STEM image due to their different scattering cross sections, whereas defect-richer regions appear with much weaker contrast. This becomes evident in the reduced contrast between atomic rows near the FePd/P… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: shows the sensitivity analysis results with respect to K. A finite-q solution exists and is stable within an anisotropy window, collapsing into the homo￾geneous Q = 0 state above a critical value Kc. The characteristic in-plane modulation period λ = 2π/|Q∥| = 2π/q Q2 x + Q2 y is found to be in the range 68-72.5 nm. This represents a clear improvement over the localized 0 1 2 3 4 5 6 K (MJ/m3 ) 1e 2 68 69 … view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
read the original abstract

Chirality in superconductor-ferromagnet hybrids strongly influences phenomena such as the observable signatures of long-range triplet superconductivity, but its microscopic origin in nominally centrosymmetric ferromagnets is still unclear. Here, we combine structural characterization, polarization-analyzed grazing-incidence small-angle neutron scattering (PA-GISANS), first-principles calculations, and deep-learning-assisted multiscale modeling to study FePd and Nb/FePd heterostructures. Experimentally, we observe partial L1$_0$ order, atomic intermixing, anti-phase boundaries, and a depth-dependent defect gradient across the FePd layer, together with a finite net magnetic chirality at room temperature. The GISANS asymmetry indicates that the main chiral contribution lies in-plane, with an additional out-of-plane component associated with depth-dependent magnetic inhomogeneity. Theoretically, we show that chemical disorder in FePd, especially when combined with a compositional gradient, produces finite Dzyaloshinskii-Moriya interactions and stabilizes chiral finite-$\mathbf{q}$ magnetic modulations with mixed Bloch-N\'eel character. In the mesoscopic model, the resulting in-plane modulation length approaches the experimentally observed range. These results identify disorder and compositional gradients as intrinsic microscopic sources of net chirality in FePd-based films, showing that the observed chirality does not arise only from interface effects.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript combines PA-GISANS neutron scattering, structural characterization, first-principles calculations, and deep-learning-assisted multiscale modeling on FePd and Nb/FePd heterostructures. It reports partial L1_{0} order, atomic intermixing, anti-phase boundaries, and a depth-dependent defect gradient in the FePd layer, together with finite net magnetic chirality at room temperature. The modeling shows that chemical disorder combined with a compositional gradient generates finite Dzyaloshinskii-Moriya interactions that stabilize chiral finite-q modulations of mixed Bloch-Néel character whose in-plane length approaches the experimental value, leading to the claim that disorder and gradients are intrinsic sources of net chirality rather than interface effects alone.

Significance. If the central claim is substantiated, the work would be significant for understanding chirality in nominally centrosymmetric ferromagnets and its role in triplet superconductivity. The multiscale approach linking measured defect gradients to DMI and modulation lengths provides a concrete microscopic mechanism that could guide materials design, and the combination of polarization-analyzed GISANS with first-principles-derived mesoscopic modeling is a methodological strength.

major comments (2)
  1. [Theoretical modeling / mesoscopic model] The abstract and modeling description state that disorder plus compositional gradient produces finite DMI and that the resulting in-plane modulation length approaches experiment, yet no control simulation with uniform composition (zero gradient) or with interface DMI terms explicitly set to zero is presented. This control is required to isolate the internal contributions and to support the claim that 'the observed chirality does not arise only from interface effects.'
  2. [Experimental results and data analysis] The GISANS asymmetry is used to assign the main chiral contribution as in-plane with an additional out-of-plane component tied to depth-dependent inhomogeneity. Quantitative details on how the measured defect gradient is converted into the dominant DMI terms (including any fitting or scaling choices) are needed to confirm that the model translation is not post-hoc.
minor comments (2)
  1. [Abstract] The abstract could quote the numerical range of the modeled modulation length alongside the experimental value for direct comparison.
  2. [Throughout] Notation for the mixed Bloch-Néel character and the definition of the modulation wavevector q should be made consistent between the abstract and the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We are pleased that the significance of the work is recognized. Below we address the major comments point by point and outline the revisions we will make.

read point-by-point responses

  1. Referee: [Theoretical modeling / mesoscopic model] The abstract and modeling description state that disorder plus compositional gradient produces finite DMI and that the resulting in-plane modulation length approaches experiment, yet no control simulation with uniform composition (zero gradient) or with interface DMI terms explicitly set to zero is presented. This control is required to isolate the internal contributions and to support the claim that 'the observed chirality does not arise only from interface effects.'

    Authors: We agree that control simulations are necessary to rigorously isolate the contributions from chemical disorder and compositional gradients versus potential interface effects. In the revised version, we will add two sets of control simulations: (1) with uniform composition (zero gradient) across the FePd layer, which we expect to yield vanishing net DMI and no stable finite-q modulations, and (2) with the interface DMI terms explicitly set to zero while retaining the internal disorder and gradient terms. These results will be presented in a new figure or supplementary section to directly support our claim that the observed chirality originates intrinsically from the disorder and gradients within the FePd film. revision: yes

  2. Referee: [Experimental results and data analysis] The GISANS asymmetry is used to assign the main chiral contribution as in-plane with an additional out-of-plane component tied to depth-dependent inhomogeneity. Quantitative details on how the measured defect gradient is converted into the dominant DMI terms (including any fitting or scaling choices) are needed to confirm that the model translation is not post-hoc.

    Authors: We will revise the manuscript to include more quantitative details on the conversion process. Specifically, we will describe how the depth-dependent defect gradient measured by structural characterization (e.g., from XRD and TEM) is mapped to the DMI coefficients in the mesoscopic model using parameters derived from first-principles calculations. This will include the explicit scaling factors, any averaging or fitting procedures used to match the experimental modulation length, and the sensitivity analysis to these choices. We believe this addition will demonstrate that the modeling is grounded in the experimental inputs rather than being post-hoc. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The paper's chain proceeds from measured structural defects and gradients (via PA-GISANS and characterization) to first-principles calculations on explicit disordered supercells that compute DMI terms, followed by a mesoscopic model whose output modulation length is compared to (rather than fitted to) the experimental range. No equation reduces a claimed prediction to a fitted parameter by construction, no self-citation is invoked as a uniqueness theorem that forbids alternatives, and no ansatz is smuggled via prior work. The central claim that disorder plus gradients produce net chirality is therefore an independent computational result rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard spin-lattice models plus the assumption that the measured defect profile can be mapped onto effective DMI parameters without additional fitted constants beyond those derived from first-principles.

axioms (2)
  • standard math Heisenberg spin Hamiltonian with Dzyaloshinskii-Moriya terms derived from first-principles on disordered configurations
    Invoked in the theoretical section to link atomic disorder to finite DMI.
  • domain assumption Mesoscopic model parameters can be upscaled from the atomistic DMI without loss of the dominant modulation length
    Required for the claim that the in-plane modulation length approaches the experimental value.

pith-pipeline@v0.9.0 · 5602 in / 1411 out tokens · 17233 ms · 2026-05-13T18:22:05.225141+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Reference graph

Works this paper leans on

101 extracted references · 101 canonical work pages · 2 internal anchors

  1. [1]

    Modeling via deep learning Figure 8(a-h) shows the parity plots for the obtained Jij,|D ij|, andµ i and the corresponding error distribu- tions (insets). The first aspect to note is that the net- work captures a much more complex scenario than the baseline model (see Methods), which produces, by con- struction, a series of discrete values (plateaus) for e...

    work page

  2. [2]

    Construction and analysis of the mesoscopic FePd model To construct the mesoscopic model of the FePd thick- ness in the FePd/Nb system, we employed the modified version of the gradient model (MGM), as described in detail in Supplementary Note 4, aiming to reproduce, as closely as possible, the observed characteristics of sam- ples i.Nb/FePd and ii.FePd. S...

    work page

  3. [3]

    Nature of the disorder-induced DMI: insights from the DFT dataset Apart from serving as a reference for training and com- parison withSAGNNpredictions on the validation set, the DFT results that constitute the database can also pro- vide some interesting and non-trivial insights into the nature of the DMI induced by chemical disorder, par- ticularly for F...

    work page 2022

  4. [4]

    The schematic scattering geometry is shown in Fig. 11(a). In this schematic, the domain walls are drawn as straight lines for simplicity, whereas the actual domain- wall network in i.Nb/FePd and ii.FePd forms a maze- like pattern. All experiments have been performed at room temperature, under a guide field of 10 G, with a wavelength of 8 ˚A, a sample-to-d...

    work page

  5. [5]

    All calculations were performed within the local spin density approximation (LSDA) as parametrized by von Barth and Hedin [79]

    computed in each variational step [78], whereLand Shere denote the orbital and spin angular momentum operators, respectively. All calculations were performed within the local spin density approximation (LSDA) as parametrized by von Barth and Hedin [79]. As we fo- cus on spin-related quantities, no orbital polarization [80] was included. Since the main ele...

    work page

  6. [6]

    The data are divided into shells (s) and pair types (p), so that the neural network training is fo- cused on such subspaces of the whole dataset

    Graph neural network architecture Supplementary Figure 7 shows the architecture of the Species-Aware Graph Neural Network (SAGNN), as used in this work. The data are divided into shells (s) and pair types (p), so that the neural network training is fo- cused on such subspaces of the whole dataset. A total of 9197 graphs were used for training, and 2300 fo...

    work page

  7. [7]

    Baseline models The baseline model predictedJ ij and|D ij|using the mean value computed over the training set for each shell sand pair typep. Formally, ¯Jbase ij (s, p) = 1 n(Etrains,p ) X (i,j)∈E trains,p Jij | ¯Dij|base(s, p) = 1 n(Etrains,p ) X (i,j)∈E trains,p |Dij| ,(20) whereE train s,p denotes the set of edges in the training data with shellsand pa...

    work page

  8. [8]

    Computational details The weighted atom-centered symmetry functions (ACSFs), here modified from the original formulation of Ref. [91] and as described in the Supplementary Note 5, were implemented in Python, employing the open-source libraries Numpy 2.0.2 (for dataset handling) and PyTorch 2.7.1/PyTorch Geometric [92] 2.6.1 (for GNN handling). To overcome...

    work page

  9. [9]

    The atoms were arranged in the ideal fcc crystal structure, with Pd and Fe randomly dis- tributed over the lattice sites to represent chemical disor- der

    Dataset filtering The dataset was constructed fromab-initiocalcula- tions of 100 4×2×2 supercells (32 atoms each), composed of 50% Pd and 50% Fe. The atoms were arranged in the ideal fcc crystal structure, with Pd and Fe randomly dis- tributed over the lattice sites to represent chemical disor- der. Magnetic interactions were evaluated by considering ever...

    work page

  10. [10]

    A. I. Buzdin, Rev. Mod. Phys.77, 935 (2005)

    work page 2005

  11. [11]

    Linder and J

    J. Linder and J. W. A. Robinson, Nat. Phys.11, 307 (2015)

    work page 2015

  12. [12]

    I. P. Kulik, Sov. Phys. JETP22, 841 (1966)

    work page 1966

  13. [13]

    Z. Yang, M. Lange, A. Volodin, R. Szymczak, and V. Moshchalkov, Nat. Mater.3, 793 (2004)

    work page 2004

  14. [14]

    A. F. Volkov, Y. V. Fominov, and K. B. Efetov, Phys. Rev. B72, 184504 (2005)

    work page 2005

  15. [15]

    A. F. Volkov, F. S. Bergeret, and K. B. Efetov, Phys. Rev. Lett.90, 117006 (2003)

    work page 2003

  16. [16]

    Di Bernardo, S

    A. Di Bernardo, S. Komori, G. Livanas, G. Divitini, P. Gentile, M. Cuoco, and J. W. A. Robinson, Nat. Mater.18, 1194 (2019)

    work page 2019

  17. [17]

    Nikoli´ c, N

    D. Nikoli´ c, N. L. Schulz, A. I. Buzdin, and M. Eschrig, Phys. Rev. B112, 224507 (2025)

    work page 2025

  18. [18]

    A. Y. Aladyshkin, A. V. Silhanek, W. Gillijns, and V. V. Moshchalkov, Supercond. Sci. Technol.22, 053001 (2009)

    work page 2009

  19. [19]

    L. R. Tagirov, Phys. Rev. Lett.83, 2058 (1999)

    work page 2058

  20. [20]

    Garnier, A

    M. Garnier, A. Mesaros, and P. Simon, Commun. Phys. 2, 126 (2019)

    work page 2019

  21. [21]

    K. L. Svalland, M. T. Mercaldo, and M. Cuoco, Phys. Rev. B111, 184501 (2025)

    work page 2025

  22. [22]

    Sardinero, R

    I. Sardinero, R. S. Souto, and P. Burset, Phys. Rev. B 110, L060505 (2024)

    work page 2024

  23. [23]

    Nothhelfer, S

    J. Nothhelfer, S. A. D´ ıaz, S. Kessler, T. Meng, M. Rizzi, K. M. D. Hals, and K. Everschor-Sitte, Phys. Rev. B105, 224509 (2022)

    work page 2022

  24. [24]

    S. Rex, I. V. Gornyi, and A. D. Mirlin, Phys. Rev. B100, 064504 (2019)

    work page 2019

  25. [25]

    S. M. Dahir, A. F. Volkov, and I. M. Eremin, Phys. Rev. Lett.122, 097001 (2019)

    work page 2019

  26. [26]

    S. S. Apostoloff, E. S. Andriyakhina, and I. S. Bur- mistrov, Phys. Usp.68, 1092 (2025)

    work page 2025

  27. [27]

    A. P. Petrovi´ c, M. Raju, X. Y. Tee, A. Louat, I. Maggio- Aprile, R. M. Menezes, M. J. Wyszy´ nski, N. K. Duong, M. Reznikov, C. Renner, M. V. Miloˇ sevi´ c, and C. Panagopoulos, Phys. Rev. Lett.126, 117205 (2021)

    work page 2021

  28. [28]

    Y.-J. Xie, A. Qian, B. He, Y.-B. Wu, S. Wang, B. Xu, G. Yu, X. Han, and X. G. Qiu, Phys. Rev. Lett.133, 166706 (2024)

    work page 2024

  29. [29]

    S.-H. Yang, R. Naaman, Y. Paltiel, and S. S. P. Parkin, Nat. Rev. Phys.3, 328 (2021)

    work page 2021

  30. [30]

    Maggiora, X

    J. Maggiora, X. Wang, and R. Zheng, Phys. Rep.1076, 1 (2024)

    work page 2024

  31. [31]

    Moriya, Phys

    T. Moriya, Phys. Rev.120, 91 (1960)

    work page 1960

  32. [32]

    R. E. Camley and K. L. Livesey, Surf. Sci. Rep.78, 100605 (2023)

    work page 2023

  33. [33]

    Allenspach, A

    R. Allenspach, A. Bischof, B. Boehm, U. Drechsler, O. Reich, M. Sousa, S. Tacchi, and G. Carlotti, Phys. Rev. B110, 014402 (2024)

    work page 2024

  34. [34]

    E. Chen, A. Tamm, T. Wang, M. E. Epler, M. Asta, and T. Frolov, npj Comput Mater.8(2022)

    work page 2022

  35. [35]

    Carvalho, I

    P. Carvalho, I. Miranda, J. Brand˜ ao, A. Bergman, J. Cezar, A. Klautau, and H. Petrilli, Nano Lett.23, 10.1021/acs.nanolett.3c00428 (2023)

    work page doi:10.1021/acs.nanolett.3c00428 2023

  36. [36]

    Michels, D

    A. Michels, D. Mettus, I. Titov, A. Malyeyev, M. Ber- sweiler, P. Bender, I. Peral, R. Birringer, Y. Quan, P. H¨ autle, J. Kohlbrecher, D. Honecker, J. R. Fern´ andez, L. F. Barqu´ ın, and K. L. Metlov, Phys. Rev. B99, 014416 (2019)

    work page 2019

  37. [37]

    Liang, M

    J. Liang, M. Chshiev, A. Fert, and H. Yang, Nano Lett. 22, 10128 (2022)

    work page 2022

  38. [38]

    I. P. Miranda, A. B. Klautau, A. Bergman, and H. M. Petrilli, Phys. Rev. B105, 224413 (2022)

    work page 2022

  39. [39]

    Heinze, K

    S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Ku- betzka, R. Wiesendanger, G. Bihlmayer, and S. Bl”ugel, Nat. Phys.7, 713 (2011)

    work page 2011

  40. [40]

    Pajda, J

    M. Pajda, J. Kudrnovsk´ y, I. Turek, V. Drchal, and P. Bruno, Phys. Rev. B64, 174402 (2001)

    work page 2001

  41. [41]

    Fert and P

    A. Fert and P. M. Levy, Phys. Rev. Lett.44, 1538 (1980)

    work page 1980

  42. [42]

    Kentzinger, H

    E. Kentzinger, H. Frielinghaus, U. R”ucker, A. Ioffe, D. Richter, and T. Br”uckel, Phys. B Condens. Matter 397, 43 (2007)

    work page 2007

  43. [44]

    B. E. Warren,X-ray Diffraction(Addison-Wesley, 1969)

    work page 1969

  44. [45]

    Gehanno, R

    V. Gehanno, R. Hoffmann, Y. Samson, A. Marty, and S. Auffret, Eur. Phys. J. B10, 457 (1999)

    work page 1999

  45. [46]

    Gehanno,Perpendicular magnetic anisotropy of epi- taxial thin films of FePd ordered alloys, Ph.D

    V. Gehanno,Perpendicular magnetic anisotropy of epi- taxial thin films of FePd ordered alloys, Ph.D. thesis, Uni- versit´ e Joseph-Fourier – Grenoble I (1997). 21

    work page 1997

  46. [47]

    Stellhorn,Interplay of proximity effects in super- conductor/ferromagnet heterostructures, Ph.D

    A. Stellhorn,Interplay of proximity effects in super- conductor/ferromagnet heterostructures, Ph.D. thesis, RWTH Aachen (2021)

    work page 2021

  47. [48]

    Nieves, S

    P. Nieves, S. Arapan, T. Schrefl, and S. Cuesta-Lopez, Phys. Rev. B96, 224411 (2017)

    work page 2017

  48. [49]

    Kentzinger, U

    E. Kentzinger, U. R”ucker, B. Toperverg, F. Ott, and T. Br”uckel, Phys. Rev. B77, 104435 (2008)

    work page 2008

  49. [50]

    Fermon, F

    C. Fermon, F. Ott, B. Gilles, A. Marty, A. Menelle, Y. Samson, G. Legoff, and G. Francinet, Physica B Con- dens. Matter.267-268, 162 (1999)

    work page 1999

  50. [51]

    Tasset, P

    F. Tasset, P. J. Brown, E. Leli` evre-Berna, T. Roberts, S. Pujol, J. Allibon, and E. Bourgeat-Lami, Phys. B Con- dens. Matter267–268, 69 (1999)

    work page 1999

  51. [52]

    Schweika, J

    W. Schweika, J. Phys.: Conf. Ser.211, 012026 (2010)

    work page 2010

  52. [53]

    Hubert and R

    A. Hubert and R. Sch”afer,Magnetic Domains(Springer, 2008)

    work page 2008

  53. [54]

    Pathak, O

    R. Pathak, O. A. Golovnia, E. G. Gerasimov, A. G. Popov, N. I. Vlasova, R. Skomski, and A. Kashyap, J. Magn. Magn. Mater.499, 166266 (2020)

    work page 2020

  54. [55]

    G. Yu, T. Cheng, X. Zhang, and W. Gong, J. Magn. Magn. Mater.510, 166904 (2020)

    work page 2020

  55. [56]

    Ghosh, Intermetallics17, 708 (2009)

    S. Ghosh, Intermetallics17, 708 (2009)

    work page 2009

  56. [57]

    G. Yu, T. Cheng, Z. Cheng, and X. Zhang, Comput. Mater. Sci.188, 110168 (2021)

    work page 2021

  57. [58]

    Shima, K

    H. Shima, K. Oikawa, A. Fujita, K. Fukamichi, and K. Ishida, J. Magn. Magn. Mater.272–276, 2173 (2004)

    work page 2004

  58. [59]

    S. V. Meschel, J. Pavlu, and P. Nash, J. Alloys Compd. 509, 5256 (2011)

    work page 2011

  59. [60]

    M. A. Ruderman and C. Kittel, Phys. Rev.96, 99 (1954)

    work page 1954

  60. [61]

    Y. O. Kvashnin, R. Cardias, A. Szilva, I. Di Marco, M. I. Katsnelson, A. I. Lichtenstein, L. Nordstr”om, A. B. Klautau, and O. Eriksson, Phys. Rev. Lett.116, 217202 (2016)

    work page 2016

  61. [62]

    J. A. Mydosh, Phys. Rev. Lett.33, 1562 (1974)

    work page 1974

  62. [63]

    Cheong and X

    S.-W. Cheong and X. Xu, npj Quantum Mater.7, 10.1038/s41535-022-00447-5 (2022)

    work page doi:10.1038/s41535-022-00447-5 2022

  63. [64]

    EvaluatingC(Eq

    This can be seen by considering a flat-spiralansatz m(r) =m 0[ˆe1 cos(q0 ·r) + ˆe2 sin(q0 ·r)], with ˆe1 ⊥ ˆe2, ˆe1,2 ⊥ ˆq0. EvaluatingC(Eq. 2) and theα-component of the chirality tensorL α = (∂ αm)×m(see Supplemen- tary Note 3), one finds, up to a proportionality constant, C(q0)∝K α(q= 0), whereK α is the Fourier transform ofL α. Explicitly,K α(q= 0) =−2...

    work page

  64. [65]

    Zunger, S.-H

    A. Zunger, S.-H. Wei, L. G. Ferreira, and J. E. Bernard, Phys. Rev. Lett.65, 353 (1990)

    work page 1990

  65. [66]

    The specific heat at constant volume was calculated as the change in specific internal energy (e) with respect to temperature at constant volumec v = ∂e dT v

    work page

  66. [67]

    Frota-Pessˆ oa, R

    S. Frota-Pessˆ oa, R. B. Muniz, and J. Kudrnovsk´ y, Phys. Rev. B62, 5293 (2000)

    work page 2000

  67. [68]

    W. L. Bragg and E. J. Williams, Proc. R. Soc. London, Ser. A145, 699 (1934)

    work page 1934

  68. [69]

    D. E. Laughlin, K. Srinivasan, M. Tanase, and L. Wang, Scr. Mater.53, 383 (2005)

    work page 2005

  69. [70]

    P. Kamp, A. Marty, B. Gilles, R. Hoffmann, S. March- esini, M. Belakhovsky, C. Boeglin, H. A. D”urr, S. S. Dhesi, G. van der Laan, and A. Rogalev, Phys. Rev. B 59, 1105 (1999)

    work page 1999

  70. [71]

    J. Cui, T. W. Shield, and R. D. James, Acta Mater.52, 35 (2004)

    work page 2004

  71. [72]

    Shima, K

    H. Shima, K. Oikawa, A. Fujita, K. Fukamichi, K. Ishida, and A. Sakuma, Phys. Rev. B70, 224408 (2004)

    work page 2004

  72. [73]

    Kota and A

    Y. Kota and A. Sakuma, J. Phys. Soc. Jpn.81, 084705 (2012)

    work page 2012

  73. [74]

    R”ussmann and S

    P. R”ussmann and S. Bl”ugel, Phys. Rev. B105, 125143 (2022)

    work page 2022

  74. [75]

    R. Reho, N. Wittemeier, A. H. Kole, P. Ordej´ on, and Z. Zanolli, Phys. Rev. B110, 134505 (2024)

    work page 2024

  75. [76]

    Stellhorn, A

    A. Stellhorn, A. Sarkar, E. Kentzinger, M. Waschk, P. Sch”offmann, S. Schr”oder, G. Abuladze, Z. Fu, V. Pipich, and T. Br”uckel, J. Magn. Magn. Mater.476, 483 (2019)

    work page 2019

  76. [77]

    Kov´ acs, R

    A. Kov´ acs, R. Schierholz, and K. Tillmann, JLSRF2, 10.17815/jlsrf-2-68 (2016)

    work page doi:10.17815/jlsrf-2-68 2016

  77. [78]

    A. R. Wildes, Rev. Sci. Instrum.70, 4241 (1999)

    work page 1999

  78. [79]

    W. T. Lee, J. Hagman, D. Martin Rodriguez, A. Stell- horn, A. Backs, T. Arnold, E. Blackburn, P. Deen, C. Durniak, M. Feygenson, A. T. Holmes, J. Hous- ton, S. Jaksch, O. Kirstein, D. Mannix, M. M˚ ansson, M. Morgano, G. Nilsen, D. Noferini, T. Nylan- der, D. Orlov, V. Santoro, S. Schmidt, M. Schulz, W. Schweika, M. Strobl, A. Tartaglione, R. Toft- Peters...

    work page 2023

  79. [80]

    Frota-Pessoa, Phys

    S. Frota-Pessoa, Phys. Rev. B46, 14570 (1992)

    work page 1992

  80. [82]

    O. K. Andersen, Phys. Rev. B12, 3060 (1975)

    work page 1975

Showing first 80 references.