Abstract
We consider multiskyrmion configurations in 2D ferromagnets with Dzyaloshinskii-Moriya interaction and the magnetic field, using the stereographic projection method. In the absence of Dzyaloshinskii-Moriya interaction, D, and the field, B, the skyrmions do not interact and the exact multiskyrmion solution is a sum of individual projections. In certain range of B, D ≠ 0, skyrmions become stable and form a hexagonal lattice. The shape of one skyrmion on the plane is fully determined by D and B. We describe multiskyrmion configurations by simple sums of individual skyrmion projections, of the same shape and adjusted scale. This procedure reveals pairwise and triple interactions between skyrmions, and the energy of proposed hexagonal structure is found in a good agreement with previous studies. It allows an effective theory of skyrmion structures in terms of variables, referring to individual skyrmions, i.e., their position, size and phase, elliptic distortions, etc.
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References
N. Nagaosa and Y. Tokura, Nat. Nanotech. 8, 899 (2013).
M. Garst, J. Waizner, and D. Grundler, J. Phys. D: Appl. Phys. 50, 293002 (2017).
N. Bogdanov and D. A. Yablonskii, Sov. Phys. JETP 68, 101 (1989).
N. Bogdanov and A. Hubert, J. Magn. Magn. Mater. 138, 255 (1994).
U. K. Roßler, N. Bogdanov, and C. Pfleiderer, Nature (London, U.K.) 442, 797 (2006).
B. Binz, A. Vishwanath, and V. Aji, Phys. Rev. Lett. 96, 207202 (2006).
T. Okubo, S. Chung, and H. Kawamura, Phys. Rev. Lett. 108, 017206 (2012).
S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Kubetzka, R. Wiesendanger, G. Bihlmayer, and S. Brügel, Nat. Phys. 7, 713 (2011).
S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Boni, Science (Washington, DC, U. S.) 323, 915 (2009).
X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, Nature (London, U.K.) 465, 901 (2010).
C. Pfleiderer, Nat. Phys. 7, 673 (2011).
P. Bak and M. H. Jensen, J. Phys. C: Solid State Phys. 13, L881 (1980).
O. Nakanishi, A. Yanase, A. Hasegawa, and M. Kataoka, Solid State Commun. 35, 995 (1980).
A. A. Belavin and A. M. Polyakov, JETP Lett. 22, 245 (1975).
A. O. Sorokin, J. Exp. Theor. Phys. 118, 417 (2014).
J. Garaud, K. A. H. Sellin, J. Jäykkä, and E. Babaev, Phys. Rev. B 89, 104508 (2014).
D. F. Agterberg, E. Babaev, and J. Garaud, Phys. Rev. B 90, 064509 (2014).
M. S. Scheurer and J. Schmalian, Nat. Commun. 6, 6005 (2015).
D. N. Aristov and A. Luther, Phys. Rev. B 65, 165412 (2002).
A. B. Borisov, J. Kishine, I. G. Bostrem, and A. S. Ovchinnikov, Phys. Rev. B 79, 134436 (2009).
V. V. Kiselev and A. A. Raskovalov, Theor. Math. Phys. 173, 1565 (2012).
Y. Togawa, T. Koyama, K. Takayanagi, S. Mori, Y. Kousaka, J. Akimitsu, S. Nishihara, K. Inoue, A. S. Ovchinnikov, and J. Kishine, Phys. Rev. Lett. 108, 107202 (2012).
A. O. Leonov, T. L. Monchesky, N. Romming, A. Kubetzka, A. N. Bogdanov, and R. Wiesendanger, New. J. Phys. 18, 065003 (2016).
V. P. Kravchuk, D. D. Sheka, U. K. Roßler, J. van den Brink, and Yu. Gaididei, Phys. Rev. B 97, 064403 (2018).
D. N. Aristov, S. S. Kravchenko, and A. O. Sorokin, JETP Lett. 102, 455 (2015).
N. Bogdanov and A. Hubert, Phys. Status Solidi B 186, 527 (1994).
D. D. Sheka, B. A. Ivanov, and F. G. Mertens, Phys. Rev. B 64, 024432 (2001).
K. L. Metlov, Phys. Rev. B 88, 014427 (2013).
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Timofeev, V.E., Sorokin, A.O. & Aristov, D.N. Towards an Effective Theory of Skyrmion Crystals. Jetp Lett. 109, 207–212 (2019). https://doi.org/10.1134/S0021364019030056
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DOI: https://doi.org/10.1134/S0021364019030056