New Boundary-Driven Twist States in Systems with Broken Spatial Inversion Symmetry

Abstract

A full description of a magnetic sample includes a correct treatment of the boundary conditions (BCs). This is in particular important in thin film systems, where even bulk properties might be modified by the properties of the boundary of the sample. We study generic ferromagnets with broken spatial inversion symmetry and derive the general micromagnetic BCs of a system with Dzyaloshinskii-Moriya interaction (DMI). We demonstrate that the BCs require the full tensorial structure of the third-rank DMI tensor and not just the antisymmetric part, which is usually taken into account. Specifically, we study systems with C∞v symmetry and explore the consequences of the DMI. Interestingly, we find that the DMI already in the simplest case of a ferromagnetic thinfilm leads to a purely boundary-driven magnetic twist state at the edges of the sample. The twist state represents a new type of DMI-induced spin structure, which is completely independent of the internal DMI field. We estimate the size of the texture-induced magnetoresistance effect being in the range of that of domain walls.Over the past few years, there has been an increasing interest in magnets where interface-induced phenomena play a major role [1][2][3]. This includes the topics of magnetic heterostructures as well as thin films, where the main effects arise from the sample's boundary. Therefore, a rigorous understanding of the physical boundary conditions (BCs) is needed.The strong spin-orbit coupling (SOC) and broken spatial inversion symmetry of these nanostructures lead to an intricate interplay between spin, charge, and orbital degrees of freedom, which affect the magnetic equilibrium state as well as the current-driven spin phenomena. Important examples of the SOC effects include current-driven spin-orbit torques [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21], charge-pumping via magnetization precession [8,[22][23][24][25][26], and the formation of topologically nontrivial skyrmion textures [27][28][29][30][31][32][33][34][35][36][37][38] and chiral domain walls [15,16,[39][40][41], as well as the multiferroic behaviour of chiral magnets [42,43] and the ferroelectricity of magnetic textures [41,44].The underlying mechanism being responsible for chiral skyrmions and domain walls is the Dzyaloshinskii-Moriya interaction (DMI) [45,46]. The DMI is a relativistic magnetic exchange interaction that originates from broken spatial inversion symmetry. Phenomenologically, the DMI is modeled by a free-energy density term, which is linear in the spatial variations of the magnetization. In its most general form, the term can be written asas discussed for example explicitly in Landau & Lifshitz, Ref.[47].Here, m is a unit vector pointing along the magnetization M = M s m, and D ijk is the DMI tensor, which is linear in the relativistic interactions. The particular form of the DMI tensor is determined by the point group of the system. Here, and in what follows, we use the convention of a summation of repeated indices. To avoid confusion with the...

Chiral ferromagnetism beyond Lifshitz invariants

“…In Refs. [14,17], it was specified that, for a 2D system of the C ∞v class, D LLG should, in fact, coincide with D as given by Eq. (17).…”

Effect of boundary-induced chirality on magnetic textures in thin films

“…In this case, there are only four independent tensor coefficients in the DMI tensor: D x xz = D y yz , D z xx = D z yy , D x zx = D y zy , and D z zz . The remaining tensor coefficients vanish because of the symmetry 18 . The DMI energy density terms given in Eq.…”

“…The anisotropy constant K e is assumed to be positive, K e > 0, so that the magnetization is collinear with the z-axis in absence of H ext . As we consider a laterally infinite film, the equilibrium state close to the uniform state will not vary along the x or y direction, but a surface twist in the z direction is to be expected due to the boundary-induced DMI 18 . Under these considerations, the use of an effective anisotropy captures the effects of the demagnetization exactly [see Appendix A].…”

“…It has been shown in Ref. 18, by means of symmetry analysis, that this boundary condition with generalized DMI requires the full tensorial structure of the third-rank DMI tensor and not just the antisymmetric part. Already for the case of a simple ferromagnetic thin film, a new type of DMI-induced spin structure, i.e.…”

Boundary twists, instabilities, and creation of skyrmions and antiskyrmions

“…In this section, we study the static magnetization profile induced by DMI, generalizing results of Refs. [38,40] to anisotropic DMI. The non-zero elements of DMI tensor are determined by relations:…”