Dynamics of chiral solitons driven by polarized currents in monoaxial helimagnets

Chiral solitons are one dimensional localized magnetic structures that are metastable in some ferromagnetic systems with Dzyaloshinskii–Moriya interactions and/or uniaxial magnetic anisotropy. Though topological textures in general provide a very interesting playground for new spintronics phenomena, how to properly create and control single chiral solitons is still unclear. We show here that chiral solitons in monoaxial helimagnets, characterized by a uniaxial Dzyaloshinskii–Moriya interaction, can be stabilized with external magnetic fields. Once created, the soliton moves steadily in response to a polarized electric current, provided the induced spin-transfer torque has a dissipative (nonadiabatic) component. The structure of the soliton depends on the applied current density in such a way that steady motion exists only if the applied current density is lower than a critical value, beyond which the soliton is no longer stable.


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The supplemental material contais details about computations of the chiral soliton stability domain, of the BVP that provides the steady solution, of the implementation of monoaxial DMI in the MuMax program and numerical simulations, and of the dynamics of the unfavored χ = −1 chiral soliton.

S1. CHIRAL SOLITON STABILITY
The chiral soliton is metastable if the differential operators K t and K z defined by Eqs.
(8) and (9) in the main text are (semi)positive definite. Let us write them here again for the reader convenience: where the prime stands for derivative respect to z. By Fourier transform in x and y, and changing the variable to w = χ h y q 0 z, the eigenvalues of K t and K z , denoted by µ t and µ z respectively, can be written as µ t = q 2 0 h y (λ t + 1) + k 2 x + k 2 y (3) where k x and k y are the components of the Fourier wave vector, and λ t and λ z are the eigenvalues ofK respectively. From now on the prime stands for the derivative with respect to w.
Let us consider firstK t , which is the Schrödinger operator with a Pöschl-Teller potential.
Its lowest lying eigenvalue is λ t = −1 [1]. Hence, µ t ≥ 0 and K t is always semidefinite positive. This result can be quickly obtained by noticing that ϕ 0 is an eigenstate ofK t with zero eigenvalue. This can be checked either by direct application ofK t to ϕ , and it is due to the fact that ϕ is the generator of infinitesimal translations of the soliton along z: FIG. 1: Lowest lying eigenvalue ofK z .
andn 0 (w + δw) andn 0 (w) have the same energy. Given that ϕ 0 has no nodes (it does not vanish at any point), Sturm theorem [2] guarantees that it corresponds to the lowest lying eigenvalue.
Hence, the stability of the soliton is solely determined by the lowest lying eigenvalue of Notice that the spectrum ofK z depends only on χ/ h y , or, equivalently, on χ h c /h y , where h c = π 2 /16 is the critical field for soliton proliferation. If we set χ = 0 inK z , it becomes a Pöschl-Teller potential whose lowest lying eigenvalue is λ Hence, in absence of DMI, the chiral soliton is stable for h y < −κ/3, and, in a simple ferromagnet, without DMI and anisotropy, it is always unstable, since h y ≥ 0 by definition. The lowest lying eigenvalue ofK z as a function of χ h c /h y is displayed in Fig. 1. The soliton (metas)tability boundary as a function of χ, κ, and h y can be determined from Eq. (8) and Fig. 1.

S2. SOLUTION OF THE BOUNDARY VALUE PROBLEM
The soliton steady motion driven by a polarized torque is determined by the Boundary Value Problem (BVP) set by equations (12) and (13) , which is zero to machine precision.
below for the reader convenience, and the boundary conditions (BCs) where θ and ϕ are functions of the variable w = q 0 (z − vt) and the prime stands for the derivative with respect to w.
This BVP has no solution in general. To obtain a solution it is necessary to impose some relation between Ω and Γ. To see why the BVP has no solution in general, let us analyze the general form of the solution in the asymptotic region w → ±∞.
Hence, two values of ν are positive and two negatives. Therefore, at least for small Ω and Γ, two values of ν will have positive real part and two negative real part. Let us call ν i , with i = 1, . . . , 4, the four solutions of (14), with i = 1, 2 having positive real part and i = 3, 4 negative real part. The general asymptotic solution as w → −∞ is where the vector ( To satisfy the BCs for z → −∞ it is necessary (and sufficient) that a For w → +∞ the functionsθ = π/2 − θ andφ = χ(2π − ϕ) are exponentially small and the linearized equations readθ Notice that these equations are obtained from Eqs. (12) and (13) just replacing Γ by −χΓ.
Hence, the general solution is where the ν i are the solutions of (14) and the vectors (u i , v i ) T are the corresponding solution of (16) in which Γ has to be replaced by −χΓ. The B.C. for w → +∞ require a In general, a BVP can have one, many, or no solution. An Initial Value Problem (IVP), however, has one and only one solution. We may try to solve the BVP problem as an IVP with initial conditions at w = 0 given by The asymptotic solutions of the IVP as w → ±∞ are given by Eqs.
have a solution, that can be obtained by solving the IVP and using ϕ ini and θ ini to enforce the BCs on each side (w < 0 or w > 0). Of course, the values of ϕ ini and θ ini will be different for each of the two BVPs.
We solved the BVP by splitting it into two pieces, one for w ≤ 0 and another one for w ≥ 0, with the BCs of Eqs. (22). We solved them numerically, using a relaxation method.
A solution of the complete BVP, for −∞ < w < ∞, is obtained from the two restricted BVP if the derivatives θ and ϕ are continuous at w = 0. To satisfy these two conditions, we have θ ini at our disposal as a variable, and, given that this is not enough, we need to tune also Ω and Γ. It turns out that ϕ is continuous at w = 0 if and only if Ω = 0, whatever θ ini or Γ. Therefore, we set Ω = 0 and use θ ini to enforce the continuity of θ at w = 0. The condition Ω = 0 determines the soliton velocity through Eq. (15) of the main text, which is Then, Γ = (β/α − 1)b j j/v 0 , which remains as a free parameter controlled by the current density.
Let us write θ ini = π/2 −θ 0 . For Ω = 0 and given Γ, we solved the two restricted BVP for a sufficiently dense mesh ofθ 0 from −π/2 to π/2 (since the polar angle θ takes values between 0 and π). Defining θ ± = lim w→0 ± θ (w), we obtain a solution of the complete BVP if θ + − θ − = 0. Fig. 2 shows θ + − θ − as a function ofθ 0 for κ = −5.0 and the values of h y and Γ displayed in the legends. When several zeros of θ + − θ − appear, the steady solution corresponding to θ 0 closest 0 is stable and the other unstable.  is presented in Fig. 4(b), with t * indicating the time beyond which the single soliton is destroyed. Finally, Fig. 4(c) shows how t * increases when j c is approached from above. 8