- Open Access
Cubic magnetic anisotropy in magnets: Interplay of anisotropy and magnetic order in
Phys. Rev. B 113, 104444 – Published 25 March, 2026
DOI: https://doi.org/10.1103/h76w-8twm
Abstract
The metallic systems MnSi and are known to feature a generic magnetic phase diagram primarily determined by the isotropic exchange and Dzyaloshinskii-Moriya interactions. However, additional weaker anisotropies, lowest in the hierarchy of energy scales, play a crucial role: they determine the relative order of phases in the phase diagram and may even enable skyrmion stability far below the ordering temperature. Among cubic B20 helimagnets, the insulator is currently the only known example exhibiting a low-temperature, anisotropy-induced skyrmion pocket. In this manuscript, we present a systematic study of cubic magnetocrystalline anisotropy by means of angle-resolved SQUID magnetization measurements in MnSi and single crystals and provide quantitative values of the anisotropy constants. For , the cubic anisotropy is found to be strongly dependent on the Co concentration . From a theoretical point of view, for low Co concentrations , the anisotropy is expected to be sufficiently strong to stabilize a low-temperature skyrmion lattice. Therefore, may represent the first chiral metallic system to exhibit a low-temperature skyrmion phase controllably stabilized by cubic anisotropy for specific directions of the magnetic field.
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References (57)
- S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Böni, Skyrmion lattice in a chiral magnet, Science 323, 915 (2009).
- A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G. Niklowitz, and P. Böni, Topological Hall effect in the phase of MnSi, Phys. Rev. Lett. 102, 186602 (2009).
- Y. Ishikawa, K. Tajima, D. Bloch, and M. Roth, Helical spin structure in manganese silicide MnSi, Solid State Commun. 19, 525 (1976).
- P. Bak and M. H. Jensen, Theory of helical magnetic structures and phase transitions in MnSi and FeGe, J. Phys. C 13, L881 (1980).
- O. Nakanishi, A. Yanase, A. Hasegawa, and M. Kataoka, The origin of the helical spin density wave in MnSi, Solid State Commun. 35, 995 (1980).
- S. V. Grigoriev, S. V. Maleyev, A. I. Okorokov, Y. O. Chetverikov, P. Böni, R. Georgii, D. Lamago, H. Eckerlebe, and K. Pranzas, Magnetic structure of MnSi under an applied field probed by polarized small-angle neutron scattering, Phys. Rev. B 74, 214414 (2006).
- S. V. Grigoriev, S. V. Maleyev, A. I. Okorokov, Y. O. Chetverikov, and H. Eckerlebe, Field-induced reorientation of the spin helix in near , Phys. Rev. B 73, 224440 (2006).
- A. Bauer and C. Pfleiderer, Magnetic phase diagram of MnSi inferred from magnetization and ac susceptibility, Phys. Rev. B 85, 214418 (2012).
- A. Bauer, M. Garst, and C. Pfleiderer, Specific heat of the skyrmion lattice phase and field-induced tricritical point in MnSi, Phys. Rev. Lett. 110, 177207 (2013).
- M. Janoschek, M. Garst, A. Bauer, P. Krautscheid, R. Georgii, P. Böni, and C. Pfleiderer, Fluctuation-induced first-order phase transition in Dzyaloshinskii-Moriya helimagnets, Phys. Rev. B 87, 134407 (2013).
- J. Beille, J. Voiron, F. Towfiq, M. Roth, and Z. Y. Zhang, Helimagnetic structure of the alloys, J. Phys. F: Met. Phys. 11, 2153 (1981).
- J. Beille, J. Voiron, and M. Roth, Long period helimagnetism in the cubic B20 and alloys, Solid State Commun. 47, 399 (1983).
- M. Motokawa, S. Kawarazaki, H. Nojiri, and T. Inoue, Magnetization measurements of , J. Magn. Magn. Mater. 70, 245 (1987).
- S. V. Grigoriev, V. A. Dyadkin, D. Menzel, J. Schoenes, Y. O. Chetverikov, A. I. Okorokov, H. Eckerlebe, and S. V. Maleyev, Magnetic structure of in a magnetic field studied via small-angle polarized neutron diffraction, Phys. Rev. B 76, 224424 (2007).
- S. V. Grigoriev, S. V. Maleyev, V. A. Dyadkin, D. Menzel, J. Schoenes, and H. Eckerlebe, Principal interactions in the magnetic system : Magnetic structure and critical temperature by neutron diffraction and SQUID measurements, Phys. Rev. B 76, 092407 (2007).
- W. Münzer, A. Neubauer, T. Adams, S. Mühlbauer, C. Franz, F. Jonietz, R. Georgii, P. Böni, B. Pedersen, M. Schmidt, A. Rosch, and C. Pfleiderer, Skyrmion lattice in the doped semiconductor , Phys. Rev. B 81, 041203 (2010).
- A. Bauer, M. Garst, and C. Pfleiderer, History dependence of the magnetic properties of single-crystal , Phys. Rev. B 93, 235144 (2016).
- J. Kindervater, I. Stasinopoulos, A. Bauer, F. X. Haslbeck, F. Rucker, A. Chacon, S. Mühlbauer, C. Franz, M. Garst, D. Grundler, and C. Pfleiderer, Weak crystallization of fluctuating skyrmion textures in MnSi, Phys. Rev. X 9, 041059 (2019).
- N. Manyala, Y. Sidis, J. F. DiTusa, G. Aeppli, D. Young, and Z. Fisk, Magnetoresistance from quantum interference effects in ferromagnets, Nature (London) 404, 581 (2000).
- N. Manyala, Y. Sidis, J. F. DiTusa, G. Aeppli, D. P. Young, and Z. Fisk, Large anomalous Hall effect in a silicon-based magnetic semiconductor, Nat. Mater. 3, 255 (2004).
- S. V. Grigoriev, V. A. Dyadkin, S. V. Maleyev, D. Menzel, J. Schoenes, D. Lamago, E. V. Moskvin, and H. Eckerlebe, Noncentrosymmetric cubic helical ferromagnets and , Phys. Solid State 52, 907 (2010).
- J. Grefe, P. Herre, Y. Hilgers, F. Labbus, N. Lüer-Epping, N. Radomski, M. A. C. de Melo, F. J. Litterst, D. Menzel, and S. Süllow, Near coincidence of metal-insulator transition and quantum critical fluctuations: Electronic ground state and magnetic order in , Phys. Rev. B 109, 054414 (2024).
- D. Bloch, J. Voiron, V. Jaccarino, and J. Wernick, The high field-high pressure magnetic properties of MnSi, Phys. Lett. A 51, 259 (1975).
- C. Thessieu, K. Kamishima, T. Goto, and G. Lapertot, Magnetization under high pressure in MnSi, J. Phys. Soc. Jpn. 67, 3605 (1998).
- K. Koyama, T. Goto, T. Kanomata, and R. Note, Observation of an itinerant metamagnetic transition in MnSi under high pressure, Phys. Rev. B 62, 986 (2000).
- R. Ritz, M. Halder, C. Franz, A. Bauer, M. Wagner, R. Bamler, A. Rosch, and C. Pfleiderer, Giant generic topological Hall resistivity of MnSi under pressure, Phys. Rev. B 87, 134424 (2013).
- Y. Nishihara, S. Waki, and S. Ogawa, Mössbauer study of in external magnetic fields, Phys. Rev. B 30, 32 (1984).
- S. V. Grigoriev, V. A. Dyadkin, E. V. Moskvin, D. Lamago, T. Wolf, H. Eckerlebe, and S. V. Maleyev, Helical spin structure of under a magnetic field: Small angle neutron diffraction study, Phys. Rev. B 79, 144417 (2009).
- A. Bauer, A. Neubauer, C. Franz, W. Münzer, M. Garst, and C. Pfleiderer, Quantum phase transitions in single-crystal and : Crystal growth, magnetization, ac susceptibility, and specific heat, Phys. Rev. B 82, 064404 (2010).
- S. V. Grigoriev, E. V. Altynbaev, S.-A. Siegfried, K. A. Pschenichnyi, D. Menzel, A. Heinemann, and G. Chaboussant, Spin-wave stiffness in the Dzyaloshinskii-Moriya helimagnets , Phys. Rev. B 97, 024409 (2018).
- A. Petrova, S. Gavrilkin, G. Rybalchenko, D. Menzel, I. Zibrov, and S. Stishov, Physical properties of ()Si at : Quantum criticality, Phys. Rev. B 103, L180401 (2021).
- S. Stishov, A. Petrova, D. Menzel, and A. Belemuk, Quantum criticality features in the Co,Fe doped MnSi, Physica B 675, 415607 (2024).
- Y. Fang, S. Ran, W. Xie, S. Wang, Y. S. Meng, and M. B. Maple, Evidence for a conducting surface ground state in high-quality single crystalline FeSi, Proc. Natl. Acad. Sci. 115, 8558 (2018).
- K. E. Avers, Y. S. Eo, H. Yoon, J. A. Horn, S. R. Saha, A. Suarez, P. Zavalij, and J. Paglione, Disordered two-dimensional ferromagnetism at the surface of FeSi, Phys. Rev. B 110, 134416 (2024).
- S. Grigoriev, N. Chubova, L. Azarova, and O. Utesov, Transition from spiral to ferromagnetic structure in compounds: Small-angle neutron scattering study, Ann. Phys. 447, 169132 (2022).
- L. J. Bannenberg, F. Qian, R. M. Dalgliesh, N. Martin, G. Chaboussant, M. Schmidt, D. L. Schlagel, T. A. Lograsso, H. Wilhelm, and C. Pappas, Reorientations, relaxations, metastabilities, and multidomains of skyrmion lattices, Phys. Rev. B 96, 184416 (2017).
- Y. Luo, S.-Z. Lin, D. M. Fobes, Z. Liu, E. D. Bauer, J. B. Betts, A. Migliori, J. D. Thompson, M. Janoschek, and B. Maiorov, Anisotropic magnetocrystalline coupling of the skyrmion lattice in MnSi, Phys. Rev. B 97, 104423 (2018).
- T. Adams, M. Garst, A. Bauer, R. Georgii, and C. Pfleiderer, Response of the skyrmion lattice in MnSi to cubic magnetocrystalline anisotropies, Phys. Rev. Lett. 121, 187205 (2018).
- J. Kindervater, T. Adams, A. Bauer, F. X. Haslbeck, A. Chacon, S. Mühlbauer, F. Jonietz, A. Neubauer, U. Gasser, G. Nagy, N. Martin, W. Häußler, R. Georgii, M. Garst, and C. Pfleiderer, Evolution of magnetocrystalline anisotropies in and as inferred from small-angle neutron scattering and bulk properties, Phys. Rev. B 101, 104406 (2020).
- M. Preißinger, K. Karube, D. Ehlers, B. Szigeti, H.-A. Krug von Nidda, J. S. White, V. Ukleev, H. M. Ronnow, Y. Tokunaga, A. Kikkawa, Y. Tokura, Y. Taguchi, and I. Kezsmarki, Vital role of magnetocrystalline anisotropy in cubic chiral skyrmion hosts, npj Quantum Mater. 6, 65 (2021).
- N. Akulov, Zur Quantentheorie der Temperaturabhängigkeit der Magnetisierungskurve, Z. Phys. 100, 197 (1936).
- H. Callen and E. Callen, The present status of the temperature dependence of magnetocrystalline anisotropy, and the l(l+1)2 power law, J. Phys. Chem. Solids 27, 1271 (1966).
- A. Chacon, L. Heinen, M. Halder, A. Bauer, W. Simeth, S. Mühlbauer, H. Berger, M. Garst, A. Rosch, and C. Pfleiderer, Observation of two independent skyrmion phases in a chiral magnetic material, Nat. Phys. 14, 936 (2018).
- M. Halder, A. Chacon, A. Bauer, W. Simeth, S. Mühlbauer, H. Berger, L. Heinen, M. Garst, A. Rosch, and C. Pfleiderer, Thermodynamic evidence of a second skyrmion lattice phase and tilted conical phase in , Phys. Rev. B 98, 144429 (2018).
- L. J. Bannenberg, H. Wilhelm, R. Cubitt, A. Labh, M. P. Schmidt, E. Lelièvre-Berna, C. Pappas, M. Mostovoy, and A. O. Leonov, Multiple low-temperature skyrmionic states in a bulk chiral magnet, npj Quantum Mater. 4, 11 (2019).
- M. Crisanti, A. O. Leonov, R. Cubitt, A. Labh, H. Wilhelm, M. P. Schmidt, and C. Pappas, Tilted spirals and low-temperature skyrmions in , Phys. Rev. Res. 5, 033033 (2023).
- S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Observation of skyrmions in a multiferroic material, Science 336, 198 (2012).
- S. Seki, J.-H. Kim, D. S. Inosov, R. Georgii, B. Keimer, S. Ishiwata, and Y. Tokura, Formation and rotation of skyrmion crystal in the chiral-lattice insulator , Phys. Rev. B 85, 220406 (2012).
- A. Aharoni, Demagnetizing factors for rectangular ferromagnetic prisms, J. Appl. Phys. 83, 3432 (1998).
- J. H. van Vleck, On the anisotropy of cubic ferromagnetic crystals, Phys. Rev. 52, 1178 (1937).
- L. Ludgren, O. Beckman, V. Attia, S. P. Bhattacheriee, and M. Richardson, Helical spin arrangement in cubic FeGe, Phys. Scr. 1, 69 (1970).
- I. Stasinopoulos, S. Weichselbaumer, A. Bauer, J. Waizner, H. Berger, S. Maendl, M. Garst, C. Pfleiderer, and D. Grundler, Low spin wave damping in the insulating chiral magnet , Appl. Phys. Lett. 111, 032408 (2017).
- S. M. Sauther, De Haas–van Alphen effect and magnetic anisotropy in correlated electron systems studied by cantilever magnetometry, Ph.D. thesis, Technische Universität München (2021).
- R. Skomski, A. Kashyap, A. Solanki, A. Enders, and D. J. Sellmyer, Magnetic anisotropy in itinerant magnets, J. Appl. Phys. 107, 09A735 (2010).
- A. O. Leonov, G. Gödecke, J. Grefe, S. Süllow, and D. Menzel, Low-temperature skyrmions and spiral reorientation processes in chiral magnets with cubic anisotropy: Guidelines for bridging theory and experiment, Phys. Rev. Res. 8, 013130 (2026).
- S. V. Grigoriev, A. S. Sukhanov, and S. V. Maleyev, From spiral to ferromagnetic structure in B20 compounds: Role of cubic anisotropy, Phys. Rev. B 91, 224429 (2015).
- Y. Onose, N. Takeshita, C. Terakura, H. Takagi, and Y. Tokura, Doping dependence of transport properties in , Phys. Rev. B 72, 224431 (2005).