arXiv:1001.1292v3 [cond-mat.stat-mech] 17 Dec 2010 Confinement of Skyrmion states in noncentrosymmetric magnets A. A. Leonov, A. N. Bogdanov, U. K. R¨ oßler IFW Dresden, Postfach 270116, D-01171 Dresden, Germany (Dated: November 3, 2018) Skyrmionic states in noncentrosymmetric magnets with Lifshitz invariants are investigated within the phenomenological Dzyaloshinskii model. The interaction between the chiral Skyrmions, being repulsive in a broad temperature range, changes into attraction at high temperatures. This leads to a remarkable confinement effect: near the ordering temperature Skyrmions exist only as bound states, and Skyrmion lattices are formed by an unusual instability-type nucleation transition. Nu- merical investigations on two-dimensional models demonstrate the confinement and the occurrence of different Skyrmion lattice precursor states near the ordering transition that can become thermo- dynamically stable by anisotropy or longitudinal softness in cubic helimagnets. We introduce a new fundamental parameter, confinement temperature T$_{p}$ separating the peculiar region with ”confined” chiral modulations from the main part of the phase diagram with regular helical and Skyrmion states. PACS numbers: 75.10.-b 75.30.Kz 61.30.Mp 03.75.Lm sition where the amplitude of one fundamental mode acts as small parameter of the transition [12]. Here, we show for the standard model of chiral isotropic ferromagnets [1, 7, 13] that Skyrmions are con- fined very close to the ordering temperature. In that part of the phase diagram, the creation of Skyrmions as stable units and their condensation into extended tex- tures occurs simultaneously through a rare case of an instability-type nucleation transition [14], but the con- fined Skyrmions as discernible units may arrange in dif- ferent mesophases. This is a consequence of the coupling between the magnitude and the angular part of the order parameter. Thus, near the ordering transitions, the local magnetization is not only multiply twisted but also lon- gitudinally modulated. From numerical investigations on 2D models of isotropic chiral ferromagnets, a staggered half-Skyrmion square lattice at zero and low fields and a hexagonal Skyrmion lattice at larger fields are found in overlapping regions of the phase diagram near the tran- sition temperature. Furthermore, the thermodynamic stability of Skyrmionic states can be favoured with re- spect to one-dimensional modulations by supplementing the model with cubic exchange anisotropy or in a modi- fied model for metallic chiral magnets [7]. Within the standard phenomenological (Dzyaloshin- skii) theory [1] the magnetic energy density of a non- centrosymmetric ferromagnet can be written in the di- mensionless form [7, 13] Here, reduced values of the spatial variable x = r /L$_{D}$ , the magnetization m = M /M$_{0}$ , and the applied magnetic field h n ( h = | H | /H$_{0}$ , n is a unity vector along H ), are expressed via the parameters of the energy density [13] w ( M ) = A ( gradM ) $^{2}$+ Dw$_{D}$ ( M ) − H · M + f ( M ) (where f ( M ) = a$_{1}$M 2 + a$_{2}$M $^{4}$, and M = | M | ): L$_{D}$ = A/D , H$_{0}$ = κM$_{0}$ , M$_{0}$ = ( κ/a$_{2}$ ) 1 / $^{2}$, a = a$_{1}$/κ , κ = D $^{2}$/ ( A ). Skyrmionic textures usually form through nucleation, following a classification introduced by DeGennes [12] for (continuous) transitions into incommensurate mod- ulated phases. As demonstrated in Refs. [5, 6], for the low-temperature micromagnetic model of chiral magnets, at the transition from the field-driven Skyrmion lattices into the polarized homogeneous state the lattice period diverges and the Skyrmions are set free as localized exci- tations. As the Skyrmions retain their size and axisym- metric shape, there is a full spectrum of lattice modes up to the transition, in contrary to an instability type tran- In certain noncentrosymmetric magnetic systems, the asymmetric Dzyaloshinskii-Moriya (DM) exchange re- sults in chiral couplings that can stabilize long-period, non-collinear modulations of the magnetization with a fixed sense of rotation [1, 2]. These chiral couplings are phenomenologically described by Lifshitz invariants that destroy the homogeneity of ordered phases [1]. In highly symmetric systems with Lifshitz invariants multi- ple modulations occur as textures with localized twists in two or more spatial directions [3, 4]. In noncentrosym- metric magnetic systems with Lifshitz invariants these double-twist motifs exist as smooth (baby)-Skyrmions [5–7], static solitonic textures localized in two spatial directions, which can be extended into the third direc- tion as Skyrmion strings or Hopfions. These magnetic Skyrmions are stabilized solely by the chiral DM cou- plings [6, 7], which prevent a spontaneous collapse into topological singularities. Skyrmionic matter is created by the condensation of these solitonic units, similar to vortex matter in type-II superconductors [5]. Just such chiral Skyrmions have been recently observed in thin layers of noncentrosymmetric ferromagnet (Fe,Co)Si [8]. Skyrmionic states stabilized by Lifshitz-type invariants may exist and form extended mesophases in various con- densed matter systems, as chiral liquid crystals, ferro- electrics, multiferroics, and in confined achiral systems (e.g., thin magnetic layers) [4, 9–11]. Φ = ( gradm ) 2 − w$_{D}$ ( m ) − h ( n · m ) + am 2 + m $^{4}$. (1)