#
Skyrmion Crystals and Phase Transitions in Magneto-Ferroelectric Superlattices: Dzyaloshinskii–Moriya Interaction in a Frustrated J_{1} − J_{2} Model

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Model and Skyrmion Crystal

#### 2.1. Model

#### 2.2. Ground State

## 3. Skyrmion Phase Transition

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Bogdanov, A.N.; Yablonskii, D. Thermodynamically stable vortices in magnetically ordered crystals. The mixed state of magnets. Z. Eksp. Teor. Fiz
**1989**, 95, 178. [Google Scholar] - Yu, X.; Onose, Y.; Kanazawa, N.; Park, J.; Han, J.; Matsui, Y.; Nagaosa, N.; Tokura, Y. Real-space observation of a two-dimensional skyrmion crystal. Nature
**2010**, 465, 901–904. [Google Scholar] [CrossRef] [PubMed] - Yu, X.; Kanazawa, N.; Onose, Y.; Kimoto, K.; Zhang, W.; Ishiwata, S.; Matsui, Y.; Tokura, Y. Near room-temperature formation of a skyrmion crystal in thin-films of the helimagnet FeGe. Nat. Mater.
**2011**, 10, 106–109. [Google Scholar] [CrossRef] [PubMed] - Heinze, S.; Von Bergmann, K.; Menzel, M.; Brede, J.; Kubetzka, A.; Wiesendanger, R.; Bihlmayer, G.; Blügel, S. Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions. Nat. Phys.
**2011**, 7, 713–718. [Google Scholar] [CrossRef] - Romming, N.; Hanneken, C.; Menzel, M.; Bickel, J.E.; Wolter, B.; von Bergmann, K.; Kubetzka, A.; Wiesendanger, R. Writing and deleting single magnetic skyrmions. Science
**2013**, 341, 636–639. [Google Scholar] [CrossRef] [Green Version] - Rosch, A. Skyrmions: Moving with the current. Nat. Nanotechnol.
**2013**, 8, 160. [Google Scholar] [CrossRef] - Leonov, A.; Togawa, Y.; Monchesky, T.; Bogdanov, A.; Kishine, J.; Kousaka, Y.; Miyagawa, M.; Koyama, T.; Akimitsu, J.; Koyama, T.; et al. Chiral surface twists and skyrmion stability in nanolayers of cubic helimagnets. Phys. Rev. Lett.
**2016**, 117, 087202. [Google Scholar] [CrossRef] - Moreau-Luchaire, C.; Moutafis, C.; Reyren, N.; Sampaio, J.; Vaz, C.; Van Horne, N.; Bouzehouane, K.; Garcia, K.; Deranlot, C.; Warnicke, P.; et al. Additive interfacial chiral interaction in multilayers for stabilization of small individual skyrmions at room temperature. Nat. Nanotechnol.
**2016**, 11, 444–448. [Google Scholar] [CrossRef] - Soumyanarayanan, A.; Raju, M.; Oyarce, A.G.; Tan, A.K.; Im, M.Y.; Petrović, A.P.; Ho, P.; Khoo, K.; Tran, M.; Gan, C.; et al. Tunable room-temperature magnetic skyrmions in Ir/Fe/Co/Pt multilayers. Nat. Mater.
**2017**, 16, 898–904. [Google Scholar] [CrossRef] [Green Version] - Dupé, B.; Bihlmayer, G.; Böttcher, M.; Blügel, S.; Heinze, S. Engineering skyrmions in transition-metal multilayers for spintronics. Nat. Commun.
**2016**, 7, 11779. [Google Scholar] [CrossRef] [Green Version] - Müller, J.; Rosch, A.; Garst, M. Edge instabilities and skyrmion creation in magnetic layers. New J. Phys.
**2016**, 18, 065006. [Google Scholar] [CrossRef] [Green Version] - Rosch, A. Spintronics: Electric control of skyrmions. Nat. Nanotechnol.
**2017**, 12, 103–104. [Google Scholar] [CrossRef] [PubMed] - Shen, L.; Xia, J.; Zhao, G.; Zhang, X.; Ezawa, M.; Tretiakov, O.A.; Liu, X.; Zhou, Y. Spin torque nano-oscillators based on antiferromagnetic skyrmions. Appl. Phys. Lett.
**2019**, 114, 042402. [Google Scholar] [CrossRef] [Green Version] - Fert, A.; Cros, V.; Sampaio, J. Skyrmions on the track. Nat. Nanotechnol.
**2013**, 8, 152–156. [Google Scholar] [CrossRef] - Bessarab, P.; Yudin, D.; Gulevich, D.; Wadley, P.; Titov, M.; Tretiakov, O.A. Stability and lifetime of antiferromagnetic skyrmions. Phys. Rev. B
**2019**, 99, 140411. [Google Scholar] [CrossRef] [Green Version] - Tomasello, R.; Martinez, E.; Zivieri, R.; Torres, L.; Carpentieri, M.; Finocchio, G. A strategy for the design of skyrmion racetrack memories. Sci. Rep.
**2014**, 4, 6784. [Google Scholar] [CrossRef] [Green Version] - Koshibae, W.; Kaneko, Y.; Iwasaki, J.; Kawasaki, M.; Tokura, Y.; Nagaosa, N. Memory functions of magnetic skyrmions. Jpn. J. Appl. Phys.
**2015**, 54, 053001. [Google Scholar] [CrossRef] [Green Version] - Kang, W.; Huang, Y.; Zheng, C.; Lv, W.; Lei, N.; Zhang, Y.; Zhang, X.; Zhou, Y.; Zhao, W. Voltage controlled magnetic skyrmion motion for racetrack memory. Sci. Rep.
**2016**, 6, 23164. [Google Scholar] [CrossRef] [Green Version] - Zhang, X.; Xia, J.; Zhou, Y.; Liu, X.; Zhang, H.; Ezawa, M. Skyrmion dynamics in a frustrated ferromagnetic film and current-induced helicity locking-unlocking transition. Nat. Commun.
**2017**, 8, 1717. [Google Scholar] [CrossRef] [Green Version] - Mühlbauer, S.; Binz, B.; Jonietz, F.; Pfleiderer, C.; Rosch, A.; Neubauer, A.; Georgii, R.; Böni, P. Skyrmion lattice in a chiral magnet. Science
**2009**, 323, 915–919. [Google Scholar] [CrossRef] [Green Version] - Du, H.; Che, R.; Kong, L.; Zhao, X.; Jin, C.; Wang, C.; Yang, J.; Ning, W.; Li, R.; Jin, C.; et al. Edge-mediated skyrmion chain and its collective dynamics in a confined geometry. Nat. Commun.
**2015**, 6, 8504. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jiang, W.; Upadhyaya, P.; Zhang, W.; Yu, G.; Jungfleisch, M.B.; Fradin, F.Y.; Pearson, J.E.; Tserkovnyak, Y.; Wang, K.L.; Heinonen, O.; et al. Blowing magnetic skyrmion bubbles. Science
**2015**, 349, 283–286. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Leonov, A.; Monchesky, T.; Romming, N.; Kubetzka, A.; Bogdanov, A.; Wiesendanger, R. The properties of isolated chiral skyrmions in thin magnetic films. New J. Phys.
**2016**, 18, 065003. [Google Scholar] [CrossRef] - Woo, S.; Litzius, K.; Krüger, B.; Im, M.Y.; Caretta, L.; Richter, K.; Mann, M.; Krone, A.; Reeve, R.M.; Weigand, M.; et al. Observation of room-temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic ferromagnets. Nat. Mater.
**2016**, 15, 501–506. [Google Scholar] [CrossRef] - Jiang, W.; Zhang, X.; Yu, G.; Zhang, W.; Wang, X.; Jungfleisch, M.B.; Pearson, J.E.; Cheng, X.; Heinonen, O.; Wang, K.L.; et al. Direct observation of the skyrmion Hall effect. Nat. Phys.
**2017**, 13, 162–169. [Google Scholar] [CrossRef] - Litzius, K.; Lemesh, I.; Krüger, B.; Bassirian, P.; Caretta, L.; Richter, K.; Büttner, F.; Sato, K.; Tretiakov, O.A.; Förster, J.; et al. Skyrmion Hall effect revealed by direct time-resolved X-ray microscopy. Nat. Phys.
**2017**, 13, 170–175. [Google Scholar] [CrossRef] - Woo, S.; Song, K.M.; Han, H.S.; Jung, M.S.; Im, M.Y.; Lee, K.S.; Song, K.S.; Fischer, P.; Hong, J.I.; Choi, J.W.; et al. Spin-orbit torque-driven skyrmion dynamics revealed by time-resolved X-ray microscopy. Nat. Commun.
**2017**, 8, 15573. [Google Scholar] [CrossRef] - Seki, S.; Yu, X.; Ishiwata, S.; Tokura, Y. Observation of skyrmions in a multiferroic material. Science
**2012**, 336, 198–201. [Google Scholar] [CrossRef] [Green Version] - Nahas, Y.; Prokhorenko, S.; Louis, L.; Gui, Z.; Kornev, I.; Bellaiche, L. Discovery of stable skyrmionic state in ferroelectric nanocomposites. Nat. Commun.
**2015**, 6, 8542. [Google Scholar] [CrossRef] - Kézsmárki, I.; Bordács, S.; Milde, P.; Neuber, E.; Eng, L.; White, J.; Rønnow, H.M.; Dewhurst, C.; Mochizuki, M.; Yanai, K.; et al. Néel-type skyrmion lattice with confined orientation in the polar magnetic semiconductor GaV 4 S 8. Nat. Mater.
**2015**, 14, 1116–1122. [Google Scholar] [CrossRef] [Green Version] - El Hog, S.; Bailly-Reyre, A.; Diep, H.T. Stability and phase transition of skyrmion crystals generated by Dzyaloshinskii-Moriya interaction. J. Magn. Magn. Mater.
**2018**, 455, 32–38. [Google Scholar] [CrossRef] [Green Version] - Butenko, A.; Leonov, A.; Rößler, U.; Bogdanov, A. Stabilization of skyrmion textures by uniaxial distortions in noncentrosymmetric cubic helimagnets. Phys. Rev. B
**2010**, 82, 052403. [Google Scholar] [CrossRef] [Green Version] - Rößler, U.K.; Leonov, A.A.; Bogdanov, A.N. Chiral skyrmionic matter in non-centrosymmetric magnets. J. Phys. Conf. Ser.
**2011**, 303, 012105. [Google Scholar] [CrossRef] [Green Version] - Zverev, V.; Tishin, A.; Chernyshov, A.; Mudryk, Y.; Gschneidner, K.A., Jr.; Pecharsky, V.K. Magnetic and magnetothermal properties and the magnetic phase diagram of high purity single crystalline terbium along the easy magnetization direction. J. Phys. Condens. Matter
**2014**, 26, 066001. [Google Scholar] [CrossRef] [PubMed] - Zverev, V.; Tishin, A.; Min, Z.; Mudryk, Y.; Gschneidner, K., Jr.; Pecharsky, V. Magnetic and magnetothermal properties, and the magnetic phase diagram of single-crystal holmium along the easy magnetization direction. J. Phys. Condens. Matter
**2015**, 27, 146002. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Stishov, S.M.; Petrova, A.E.; Khasanov, S.; Panova, G.K.; Shikov, A.A.; Lashley, J.C.; Wu, D.; Lograsso, T.A. Magnetic phase transition in the itinerant helimagnet MnSi: Thermodynamic and transport properties. Phys. Rev. B
**2007**, 76, 052405. [Google Scholar] [CrossRef] [Green Version] - Leonov, A.; Mostovoy, M. Edge states and skyrmion dynamics in nanostripes of frustrated magnets. Nat. Commun.
**2017**, 8, 14394. [Google Scholar] [CrossRef] - Lin, S.Z.; Hayami, S. Ginzburg-Landau theory for skyrmions in inversion-symmetric magnets with competing interactions. Phys. Rev. B
**2016**, 93, 064430. [Google Scholar] [CrossRef] [Green Version] - Hayami, S.; Lin, S.Z.; Batista, C.D. Bubble and skyrmion crystals in frustrated magnets with easy-axis anisotropy. Phys. Rev. B
**2016**, 93, 184413. [Google Scholar] [CrossRef] [Green Version] - Hayami, S.; Lin, S.Z.; Kamiya, Y.; Batista, C.D. Vortices, skyrmions, and chirality waves in frustrated Mott insulators with a quenched periodic array of impurities. Phys. Rev. B
**2016**, 94, 174420. [Google Scholar] [CrossRef] [Green Version] - Lin, S.Z.; Hayami, S.; Batista, C.D. Magnetic vortex induced by nonmagnetic impurity in frustrated magnets. Phys. Rev. Lett.
**2016**, 116, 187202. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Batista, C.D.; Lin, S.Z.; Hayami, S.; Kamiya, Y. Frustration and chiral orderings in correlated electron systems. Rep. Prog. Phys.
**2016**, 79, 084504. [Google Scholar] [CrossRef] [PubMed] - Yuan, H.; Gomonay, O.; Kläui, M. Skyrmions and multisublattice helical states in a frustrated chiral magnet. Phys. Rev. B
**2017**, 96, 134415. [Google Scholar] [CrossRef] [Green Version] - Rózsa, L.; Deák, A.; Simon, E.; Yanes, R.; Udvardi, L.; Szunyogh, L.; Nowak, U. Skyrmions with attractive interactions in an ultrathin magnetic film. Phys. Rev. Lett.
**2016**, 117, 157205. [Google Scholar] [CrossRef] [Green Version] - Rózsa, L.; Palotás, K.; Deák, A.; Simon, E.; Yanes, R.; Udvardi, L.; Szunyogh, L.; Nowak, U. Formation and stability of metastable skyrmionic spin structures with various topologies in an ultrathin film. Phys. Rev. B
**2017**, 95, 094423. [Google Scholar] [CrossRef] [Green Version] - Sutcliffe, P. Skyrmion knots in frustrated magnets. Phys. Rev. Lett.
**2017**, 118, 247203. [Google Scholar] [CrossRef] [Green Version] - Sharafullin, I.F.; Kharrasov, M.K.; Diep, H.T. Dzyaloshinskii-Moriya interaction in magnetoferroelectric superlattices: Spin waves and skyrmions. Phys. Rev. B
**2019**, 99, 214420. [Google Scholar] [CrossRef] [Green Version] - Zheng, H.; Wang, J.; Lofland, S.; Ma, Z.; Mohaddes-Ardabili, L.; Zhao, T.; Salamanca-Riba, L.; Shinde, S.; Ogale, S.; Bai, F.; et al. Multiferroic batio3-cofe2o4 nanostructures. Science
**2004**, 303, 661–663. [Google Scholar] [CrossRef] [Green Version] - Bibes, M.; Barthélémy, A. Multiferroics: Towards a magnetoelectric memory. Nat. Mater.
**2008**, 7, 425–426. [Google Scholar] [CrossRef] - Mathur, N. Materials science: A desirable wind up. Nature
**2008**, 454, 591–592. [Google Scholar] [CrossRef] - Nan, C.W. Magnetoelectric effect in composites of piezoelectric and piezomagnetic phases. Phys. Rev. B
**1994**, 50, 6082–6088. [Google Scholar] [CrossRef] [PubMed] - Sergienko, I.A.; Dagotto, E. Role of the Dzyaloshinskii-Moriya interaction in multiferroic perovskites. Phys. Rev. B
**2006**, 73, 094434. [Google Scholar] [CrossRef] [Green Version] - Udalov, O.; Beloborodov, I. The Coulomb based magneto-electric coupling in multiferroic tunnel junctions and granular multiferroics. AIP Adv.
**2018**, 8, 055810. [Google Scholar] [CrossRef] [Green Version] - Ortiz-Álvarez, H.; Bedoya-Hincapié, C.; Restrepo-Parra, E. Monte Carlo simulation of charge mediated magnetoelectricity in multiferroic bilayers. Phys. B Condens. Matter
**2014**, 454, 235–239. [Google Scholar] [CrossRef] - Janssen, T. Dynamics of (anti) ferromagnetic/electric domain walls. Ferroelectrics
**1994**, 162, 265–273. [Google Scholar] [CrossRef] - Janssen, T.; Tjon, J. Microscopic model for incommensurate crystal phases. Phys. Rev. B
**1982**, 25, 3767–3785. [Google Scholar] [CrossRef] - Li, Q.; Chen, X.; Gao, X.; Liu, J.M.; Liu, Z. Monte-carlo study on phase transitions of ferroelectromagnets. Ferroelectrics
**2002**, 279, 67–81. [Google Scholar] [CrossRef] - Pyatakov, A. Magnetoelectricity goes local: From bulk multiferroic crystals to ferroelectricity localized on magnetic topological textures. Phys. B Condens. Matter
**2018**, 542, 59–62. [Google Scholar] [CrossRef] - Maruyama, T.; Shiota, Y.; Nozaki, T.; Ohta, K.; Toda, N.; Mizuguchi, M.; Tulapurkar, A.; Shinjo, T.; Shiraishi, M.; Mizukami, S.; et al. Large voltage-induced magnetic anisotropy change in a few atomic layers of iron. Nat. Nanotechnol.
**2009**, 4, 158–161. [Google Scholar] [CrossRef] - Alberca, A.; Munuera, C.; Azpeitia, J.; Kirby, B.; Nemes, N.; Perez-Muñoz, A.; Tornos, J.; Mompean, F.; Leon, C.; Santamaria, J.; et al. Phase separation enhanced magneto-electric coupling in La 0.7 Ca 0.3 MnO 3/BaTiO 3 ultra-thin films. Sci. Rep.
**2015**, 5, 17926. [Google Scholar] [CrossRef] [Green Version] - Karthik, T.; Rao, T.D.; Srinivas, A.; Asthana, S. A-Site Cation disorder and Size variance effects on the physical properties of multiferroic Bi0. 9RE0. 1FeO3 Ceramics (RE = Gd3+, Tb3+, Dy3+). arXiv
**2012**, arXiv:1206.5606. [Google Scholar] - Garcia-Castro, A.C.; Spaldin, N.A.; Romero, A.; Bousquet, E. Geometric ferroelectricity in fluoroperovskites. Phys. Rev. B
**2014**, 89, 104107. [Google Scholar] [CrossRef] [Green Version] - Xiang, H.; Kan, E.; Zhang, Y.; Whangbo, M.H.; Gong, X. General theory for the ferroelectric polarization induced by spin-spiral order. Phys. Rev. Lett.
**2011**, 107, 157202. [Google Scholar] [CrossRef] [PubMed] - Balents, L. Spin liquids in frustrated magnets. Nature
**2010**, 464, 199–208. [Google Scholar] [CrossRef] [PubMed] - Pei, H.; Guo, S.; Ren, L.; Chen, C.; Luo, B.; Dong, X.; Jin, K.; Ren, R.; Zeeshan, H.M. The Frustration-induced Ferroelectricity of a Manganite Tricolor Superlattice with Artificially Broken Symmetry. Sci. Rep.
**2017**, 7, 6201. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Göbel, B.; Mook, A.; Henk, J.; Mertig, I. Antiferromagnetic skyrmion crystals: Generation, topological Hall, and topological spin Hall effect. Phys. Rev. B
**2017**, 96, 060406. [Google Scholar] [CrossRef] [Green Version] - Yadav, A.; Nelson, C.; Hsu, S.; Hong, Z.; Clarkson, J.; Schlepütz, C.; Damodaran, A.; Shafer, P.; Arenholz, E.; Dedon, L.; et al. Observation of polar vortices in oxide superlattices. Nature
**2016**, 530, 198–201. [Google Scholar] [CrossRef] - Leonov, A.; Mostovoy, M. Multiply periodic states and isolated skyrmions in an anisotropic frustrated magnet. Nat. Commun.
**2015**, 6, 8275. [Google Scholar] [CrossRef] [Green Version] - Koshibae, W.; Nagaosa, N. Theory of skyrmions in bilayer systems. Sci. Rep.
**2017**, 7, 42645. [Google Scholar] [CrossRef] [Green Version] - Martinez, J.; Jalil, M. Topological dynamics and current-induced motion in a skyrmion lattice. New J. Phys.
**2016**, 18, 033008. [Google Scholar] [CrossRef] [Green Version] - Lin, S.Z.; Reichhardt, C.; Batista, C.D.; Saxena, A. Driven Skyrmions and Dynamical Transitions in Chiral Magnets. Phys. Rev. Lett.
**2013**, 110. [Google Scholar] [CrossRef] [PubMed] - Iwasaki, J.; Mochizuki, M.; Nagaosa, N. Universal current-velocity relation of skyrmion motion in chiral magnets. Nat. Commun.
**2013**, 4, 1463. [Google Scholar] [CrossRef] [PubMed] - Diep, H.T. Frustrated Spin Systems; World Scientific: Singapore, 2013; 648p. [Google Scholar]
- Pinettes, C.; Diep, H.T. Phase transition and phase diagram of the J1-J2 Heisenberg model on a simple cubic lattice. J. Appl. Phys.
**1998**, 83, 6317. [Google Scholar] [CrossRef] - Hoang, D.T.; Magnin, Y.; Diep, H.T. Spin Resistivity in the Frustrated J1-J2 Model. Mod. Phys. Lett.
**2011**, 25, 937–945. [Google Scholar] [CrossRef] [Green Version] - Yang, H.; Chen, G.; Cotta, A.A.C.; N’ Diaye, A.T.; Nikolaev, S.A.; Soares, E.A.; Macedo, W.A.A.; Liu, K.; Schmid, A.K.; Fert, A.; et al. Significant Dzyaloshinskii-Moriya interaction at graphene-ferromagnet interfaces due to the Rashba effect. Nat. Mater.
**2018**, 17, 605–609. [Google Scholar] [CrossRef] - Manchon, A.; Koo, H.C.; Nitta, J.; Frolov, S.; Duine, R. New perspectives for rashba spin-orbit coupling. Nat. Mater.
**2015**, 14, 871–882. [Google Scholar] [CrossRef] - Landau, D.P.; Binder, K. A Guide to Monte Carlo Simulations in Statistical Physics; Cambridge University Press: London, UK, 2009. [Google Scholar]
- Brooks, S.; Gelman, A.; Jones, S.L.; Meng, X.L. Handbook of Markov Chain Monte Carlo; CRC Press: Boca Raton, FL, USA, 2011. [Google Scholar]
- Mézard, M.; Parisi, M.; Virasoro, M. Spin Glass Theory and Beyond An Introduction to the Replica Method and Its Applications; World Scientific: Singapore, 1986. [Google Scholar]
- El Hog, S.; Kato, F.; Koibuchi, H.; Diep, H.T. Skyrmions on 2D Elastic Surfaces with Fixed Boundary Frame. J. Mag. Mag. Mat.
**2019**, in press. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**(

**a**) Magneto-ferroelectric superlattice; (

**b**) positions of the spins in the $xy$ plane and the position of the non-magnetic ion Oxygen, defining the Dzyaloshinskii-Moriya (DM) vector (see text); (

**c**) interfacial coupling between a polarization P with five spins in a DM interaction.

**Figure 2.**(

**a**) Two-dimensional (2D) view of the ground state (GS) configuration of the interface for $H=0$ with ${J}^{m}={J}^{f}=1$, ${J}^{2m}={J}^{2f}=-0.3$, and ${J}^{mf}=-1.25$; (

**b**) Three-dimensional (3D) view.

**Figure 3.**(

**a**) 3D view of the GS configuration of the interface for moderate frustration ${J}^{2m}={J}^{2f}=-0.2$. (

**b**) 3D view of the GS structure of the interior magnetic layers; (

**c**) zoom of a skyrmion on the interface layer: Red denotes up spin, the four spins with clear blue color are down spin, and other colors correspond to spin orientations between the two. The skyrmion is of the Bloch type; (

**d**) z-components of spins across the skyrmion shown in (

**c**). Other parameters: ${J}^{m}={J}^{f}=1$, ${J}^{mf}=-1.25$, and $H=0.25$.

**Figure 4.**3D view of the GS configuration of (

**a**) the interface and (

**b**) the second layer, for stronger frustration ${J}^{2m}={J}^{2f}=-0.3$. ${J}^{m}={J}^{f}=1$, ${J}^{2m}={J}^{2f}=-0.3$, ${J}^{mf}=-1.25$, and $H=0.25$.

**Figure 5.**Strongest frustration ${J}^{2m}={J}^{2f}=-0.4$ with (

**a**) 3D view of the GS configuration of the interface and (

**b**) 3D view of the GS configuration of the second magnetic layers. Other parameters: ${J}^{m}={J}^{f}=1$, ${J}^{mf}=-1.25$, and $H=0.25$.

**Figure 6.**(

**a**) 3D view of the GS configuration of the interface for ${J}^{m}={J}^{f}=1$, ${J}^{2m}=-0.3,{J}^{2f}=-0.1$, ${J}^{mf}=-1.25$, and $H=0.25$ and (

**b**) 3D view of the GS configuration of the second magnetic layers, for ${J}^{2m}=-0.3$ and ${J}^{2f}=-0.1$. Other parameters: ${J}^{m}={J}^{f}=1$, ${J}^{mf}=-1.25$, and $H=0.25$.

**Figure 7.**(

**a**) 3D view of the GS configuration of the interface ${J}^{2m}=-0.4$ and ${J}^{2f}=-0.1$; (

**b**) 3D view of the GS configuration of the interface for ${J}^{2m}=-0.4$ and ${J}^{2f}=0$. Other parameters: ${J}^{m}={J}^{f}=1$, ${J}^{mf}=-1.25$, and $H=0.25$.

**Figure 8.**(

**a**) 3D view of the GS configuration of the interface for ${J}^{2m}=-0.1$ and ${J}^{2f}=-0.3$; (

**b**) 3D view of the GS configuration of the interface for ${J}^{2m}=-0.1$ and ${J}^{2f}=-0.4$. Other parameters: ${J}^{m}={J}^{f}=1$, ${J}^{mf}=-1.25$, and $H=0.25$.

**Figure 9.**$H=1$: 3D view of the GS configuration of (

**a**) the magnetic interface and (

**b**) the second magnetic layers. Other parameters: ${J}^{m}={J}^{f}=1$, ${J}^{2m}=-0.4$, ${J}^{2f}=-0.4$, and ${J}^{mf}=-1.25$.

**Figure 10.**(

**a**) Phase diagram in the ${J}^{2m}-{J}^{mf}$ plane for the case ${J}^{m}={J}^{f}=1$, ${J}^{2f}={J}^{2m}$, and $H=0.25$. The skyrmion phase is indicated by S (the yellow region). See the text for comments; (

**b**) dependence of the ratio of the number of skyrmions on the interior layer ${N}_{2}$ to that on the interface layer ${N}_{1}$.

**Figure 11.**(

**a**) Energy of the magnetic films versus temperature T for $({J}^{2m}={J}^{2f}=-0.4)$ (red), coinciding with the curve for $({J}^{2m}=-0.4,{J}^{2f}=0)$ (black, hidden behind the red curve). The blue curve is for $({J}^{2m}=0,{J}^{2f}=-0.4)$. (

**b**) Order parameter of the magnetic films versus temperature T for $({J}^{2m}={J}^{2f}=-0.4)$ (red), $({J}^{2m}=-0.4,{J}^{2f}=0)$ (black), and $({J}^{2m}=0,{J}^{2f}=-0.4)$ (blue). Other used parameters: ${J}^{mf}=-1.25$, $H=0.25$.

**Figure 12.**(

**a**) Energy of the ferroelectric films versus temperature T for $({J}^{2m}={J}^{2f}=-0.4)$ (red), $({J}^{2m}=-0.4,{J}^{2f}=0)$ (black), (${J}^{2m}=0,{J}^{2f}=-0.4$) (blue), (

**b**) Order parameter of the ferroelectric films versus temperature T for $({J}^{2m}={J}^{2f}=-0.4)$ (red), $({J}^{2m}=-0.4,{J}^{2f}=0)$ (black), $({J}^{2m}=0,{J}^{2f}=-0.4)$ (blue). Other used parameters: ${J}^{mf}=-1.25$, $H=0.25$.

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**MDPI and ACS Style**

Sharafullin, I.F.; Diep, H.T.
Skyrmion Crystals and Phase Transitions in Magneto-Ferroelectric Superlattices: Dzyaloshinskii–Moriya Interaction in a Frustrated *J*_{1} − *J*_{2} Model. *Symmetry* **2020**, *12*, 26.
https://doi.org/10.3390/sym12010026

**AMA Style**

Sharafullin IF, Diep HT.
Skyrmion Crystals and Phase Transitions in Magneto-Ferroelectric Superlattices: Dzyaloshinskii–Moriya Interaction in a Frustrated *J*_{1} − *J*_{2} Model. *Symmetry*. 2020; 12(1):26.
https://doi.org/10.3390/sym12010026

**Chicago/Turabian Style**

Sharafullin, Ildus F., and Hung T. Diep.
2020. "Skyrmion Crystals and Phase Transitions in Magneto-Ferroelectric Superlattices: Dzyaloshinskii–Moriya Interaction in a Frustrated *J*_{1} − *J*_{2} Model" *Symmetry* 12, no. 1: 26.
https://doi.org/10.3390/sym12010026