Mesoscale Dzyaloshinskii-Moriya interaction: geometrical tailoring of the magnetochirality

Crystals with broken inversion symmetry can host fundamentally appealing and technologically relevant periodical or localized chiral magnetic textures. The type of the texture as well as its magnetochiral properties are determined by the intrinsic Dzyaloshinskii-Moriya interaction (DMI), which is a material property and can hardly be changed. Here we put forth a method to create new artificial chiral nanoscale objects with tunable magnetochiral properties from standard magnetic materials by using geometrical manipulations. We introduce a mesoscale Dzyaloshinskii-Moriya interaction that combines the intrinsic spin-orbit and extrinsic curvature-driven DMI terms and depends both on the material and geometrical parameters. The vector of the mesoscale DMI determines magnetochiral properties of any curved magnetic system with broken inversion symmetry. The strength and orientation of this vector can be changed by properly choosing the geometry. For a specific example of nanosized magnetic helix, the same material system with different geometrical parameters can acquire one of three zero-temperature magnetic phases, namely, phase with a quasitangential magnetization state, phase with a periodical state and one intermediate phase with a periodical domain wall state. Our approach paves the way towards the realization of a new class of nanoscale spintronic and spinorbitronic devices with the geometrically tunable magnetochirality.

∂ s e e e α = F αβ e e e β , whereas Greek indices α, β enumerate curvilinear coordinates (TNB) and components of vector field, κ and τ are the curvature and torsion of the wire, respectively. Our study is based on the Landau-Lifshitz phenomenological equation: 1,2 where m m m = M M M/M s is the normalized magnetization vector, M s is the saturation magnetization, ∂ t is derivative with respect to the time, γ 0 is the gyromagnetic ratio, η is a Gilbert damping constant and E is the total magnetic energy. We consider here a narrow magnetic wire, the total energy E = E an + E ex + E DMI of which consists of three parts: exchange, anisotropic and intrinsic Dzyaloshinskii-Moriya interaction (DMI) contributions. The uniaxial anisotropy term reads E an = −K S ds (m m m ·e e e an ) 2 , where S is the wire cross-section area, s is wire arc length, K = K 0 + πM 2 s is the effective anisotropy constant, with K 0 > 0 being the magnetocrystalline anisotropy constant of easy-tangential type, the term πM 2 s comes from the magnetostatic contribution. 3 The vector e e e an in (S3) is the unit vector along the anisotropy axis, which is assumed to be oriented along the tangential direction of the curvilinear wire e e e an = e e e T .
The exchange energy has the form where A is an exchange constant and w = A/K is the characteristic magnetic length. General form for the iDMI energy in the case of curvilinear magnetic system can be obtained from the atomic description of the DMI Hamiltonian: 4 d d d n n n,n n n+δ δ δ · [S S S n n n ×S S S n n n+δ δ δ ] , where S S S n n n ≡ (S x , S y , S z ) is a classical spin vector with fixed length S in units of action on the site n n n = (n x , n y , n z ) of a threedimensional cubic lattice with integers n x , n y , n z . The vector δ δ δ connects nearest neighbors and d d d n n n,n n n+δ δ δ is the DMI vector for the bond n n n,n n n +δ δ δ . The direction of d d d n n n,n n n+δ δ δ depends on the crystallographic symmetry of the non-centrosymmetric materials: In the case of cubic crystals with T and O classes of symmetry, the vector d d d n n n,n n n+δ δ δ is aligned in the same direction as the vector ξ ξ ξ n n n,n n n+δ δ δ , 4 which is the unit vector between sites n n n an n n n +δ δ δ ; in the case of other classes of crystal symmetry, the vector d d d n n n,n n n+δ δ δ is tilted from the vector ξ ξ ξ n n n,n n n+δ δ δ by some angle, which can even reach the value π/2 for ultrathin magnetic layers on heavy metal of the adjacent layer. 5,6 The continuum dynamics of the spin system can be described in terms of magnetization unit vector m m m and due to the assumption that spin direction evolves slowly at the atomic scale, one can derive the DMI energy term for the arbitrary one-dimensional curvilinear magnetic system: where D D D I is the DMI vector with the continuous DMI constant D I . The link between D I and d depends on the lattice type, but it scales with d S 2 /a 2 with a being the lattice constant. In (S6) prime denotes the derivative with respect to the dimensionless coordinate ξ = s/w, the D D D I = D D D/ √ A K is the reduced vector of the iDMI. Using the TNB-parametrization of the magnetization m m m = m T e e e T + m N e e e N + m B e e e B , one can represent the exchange energy as follows: 7 where the Einstein notation is used for summation. The first term is the isotropic part of the exchange expression and it has formally the same form as for a straight wire. The second term in (S7) is the chiral term, which has the form of the set of the Lifshitz invariants and describe the geometry symmetry braking. This set of the Lifshitz invariants in the curvilinear frame of reference can be referred to a curvature-induced DMI, which is linear with respect to the reduced curvature κ = κ w and torsion σ = τ w. The tensor F αβ = w F αβ is the dimensionless Frenet-Serret tensor The third term in (S7) describes an effective curvature-induced anisotropy, where the components of the tensor K αβ = F αν F β ν , the components of which are bilinear with respect to κ and σ : It is also possible to rewrite the exchange energy expression (S7) in the following form: due to the use of the Frenet-Serret formulas and the property |m m m| = 1. In the (S8) D D D E = −2 σ e e e T − 2 κ e e e B denotes the reduced vector of the extrinsic curvature-driven DMI.
Using the Frenet-Serret formulas one can write the density of the total energy in the form: where explicit forms of the effective anisotropy and DMI tensors, respectively, D eff αβ and K eff αβ , read due to the use of the property |m m m| = 1.

Rotated frame of references
In the following we consider that the vector of the intrinsic DMI D D D is directed along the tangential direction of a curvilinear wire. As it was shown in (2), the tensor of the effective anisotropy coefficients K meso αβ has a non-diagonal components, which is typical for the biaxial magnets and means that the homogeneous magnetization state is not oriented along one fixed axis of the TNB basis. (S10) By choosing the rotation angle ψ as follows one can easily reduce the anisotropy energy E A eff to the form (S12) In the same way one can derive the effective DMI energy terms in the new frame of reference: (S13) The coefficient K 1 characterizes the strength of the effective easy-axis anisotropy while K 2 gives the strength of the effective easy-surface anisotropy. The parameters D 1 and D 2 are the effective mDMI constants, which are responsible for two types of magnetization rotation: around the direction e e e 1 and e e e 3 , respectively. The directions of the rotated ψ-frame {e e e 1 ,e e e 2 ,e e e 3 } have the following form: e e e 1 = e e e T cos ψ +e e e B sin ψ, e e e 2 = e e e N , e e e 3 = −e e e T sin ψ +e e e B cos ψ. (S14)

Cycloidal state
As a specific case of the general elliptical helicoidal state in a helix wire, we treat analytically cycloidal state in the rotated frame of references. It is realized under the condition σ = σ 0 = D I T /2 and corresponds to the situation when the vector of mDMI is aligned along the hard axis e e e 3 . The effective anisotropy and DMI constants read In order to derive the exact solution of the cycloidal state in the rotated ψ-frame, it is convenient to use an angular parametrization m m m = {sin θ cos φ , sin θ sin φ , cos θ }. The minimization of the energy density (5) with (S15) results in θ cy = π/2 and the azimuthal angle φ cy = π/2 − Φ, where Φ satisfies the pendulum equation: 7 The inhomogeneous solution of this equation reads where am(ζ , k) is Jacobi's amplitude 8 and the modulus k cy is determined by the condition with λ cy being a period of cycloidal modulations and K(k) being the complete elliptic integral of the first kind. 8 The energy density of this state reads where dn(ζ , k) is the Jacobi's elliptic function. 8 Integrating the energy density over one period of the modulations and minimizing with respect to the period λ cy leads to the exact form of the energy of cycloid modulations E cy . The equality of energies E qt (κ, D I T ) = E cy (κ, D I T ) determines the boundary curve, which separates the quasitangential and the cycloidal states, Fig. 2 (g1) (σ = D I T /2 line) and solid magenta line on the phase diagram Fig. S1 (a).

Spin-lattice simulations
We verify our predictions about the mDMI and its influence on the magnetic system by performing three-dimensional spin-lattice simulations using in-house developed program SLaSi. 9 We model the ferromagnetic helix by considering a classical chain of magnetic moments m m m i , with i = 1, N, situated on a helix. The system is described by the set of N discrete Landau-Lifshitz-Gilbert equations,

4/6
where ω 0 = 4π γ 0 M s and E = E ex + E an + E dmi is a dimensionless magnetic energy of the helix wire, normalized by 4π M 2 s a 2 with a being the lattice constant. In our simulation we consider the exchange exchange energy of the magnetic system, where = A/(4πM 2 s ) is an exchange length. The easy-tangential anisotropy term, where Q = K/(2πM 2 s ) is a quality factor and u u u i, j is the unit vector which connects i-th and j-th sites. And the iDMI term, where it was estimated that the DMI vector is aligned parallel to u u u i,i+1 .
To study the equilibrium magnetization states we consider a chain of N = 1000 sites. In all simulations the magnetic length w = 15a. The curvature and torsion of the helix wire are varied in the wide range of parameters, respectively, κ ∈ (0, 1) and σ ∈ (−1, 3), with the step ∆κ = ∆σ = 0.1. The reduced value of the iDMI constant D I T is varied in the range D I T ∈ (0, 3) with the step 0.3.
In order to verify our analytical predictions, we perform our simulations starting from different initial magnetization distribution, namely: uniform magnetization state in the Cartesian frame of reference, which aligns along theẑ z z direction; periodical magnetization states in the Cartesian frame of reference, which align along the e e e T and e e e N directions in the TNB frame; and quasitangential magnetization state, which align along the e e e 1 directions in the rotated curvilinear ψ-frame. We simulate numerically the set of N discrete Landau-Lifshitz-Gilbert equations (S20) in the overdamped regime with damping coefficient η = 0.5 during the long-time interval ∆t (η ω 0 ) −1 . The final state with the lowest energy is considered to be the equilibrium magnetization state. Simulations data are presented on the Fig.2 (g) by filled symbols together with theoretical results. One can see a good correspondence between simulations and our analytical results.