Skyrmion to ferromagnetic state transition: A description of the topological change as a finite-time singularity in the skyrmion dynamics

Alberto D. Verga

Phys. Rev. B 90, 174428 – Published 20 November 2014

Abstract

We investigate the topological change in a Belavin-Polyakov skyrmion under the action of a spin-polarized current. The dynamics is described by the Schrödinger equation for the electrons carrying the current coupled to the Landau-Lifshitz equation for the evolution of the magnetic texture in a square lattice. We show that the addition of an exchange dissipation term tends to smooth the transition from the skyrmion state to the ferromagnetic state. We demonstrate that this topological change in the continuum dissipationless limit can be described as a self-similar finite-time singularity by which the skyrmion core collapses.

Received 9 September 2014
Revised 29 October 2014

DOI:https://doi.org/10.1103/PhysRevB.90.174428

Authors & Affiliations

Alberto D. Verga^*

Aix-Marseille Universit'e, IM2NP, Campus Saint Jérôme, service 142, 13387 Marseille, France

^*Alberto.Verga@univ-amu.fr

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Issue

Vol. 90, Iss. 17 — 1 November 2014

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Images

Figure 1
Initial magnetization used in numerical computations in the form of a $Q = - 1$ skyrmion (left). Arrows give the magnetization in the plane, and color gives its $z$ component: At the center $S = (0, 0, - 1)$ . The three right panels show skyrmions with topological charge $Q = 1$ and size $λ$ defined by the stereographic projection on the complex plane $w = w (z)$ for (left) $w = z / λ$ , (center) $w = (1 + i) z / λ$ , and (right) $w = (1 - i) z / λ$ . The opposite charge is obtained by the transformation $w \to 1 / \bar{w}$ , which changes $S_{z} \to - S_{z}$ . A polarized current induces a precession motion that changes the arrow's orientation around the center in time. The lattice side is $L = 128 a$ and $λ = 20 a$ .
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Figure 2
Magnetization field near the skyrmion core at the topological change (at time $t \approx 5936)$ . The core collapse (two left panels $t = 5912, 5936 t_{0})$ is followed by the emission spin waves (two right panels $t = 5940, 5960 t_{0})$ . The magnetization is shown in full resolution, an arrow per lattice site, in a square of side $64 a$ . Parameters: $β = 0.001, E = 10^{- 3} E_{0}$ .
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Figure 3
Topological charge $Q$ as a function of time (in adimensional units) for different values of the exchange dissipation parameter: (left) $β = 0.001$ , (center) $β = 0.01$ , and (right) $β = 0.1$ . The initial skyrmion charge is $Q = - 1$ , and the current is polarized in the $- z$ direction. The change from $Q = - 1$ to $Q = 0$ corresponds to a transition from the skyrmion to the ferromagnetic sate. Increasing the dissipation strength results in a decrease in the time necessary to reach the transition: $t = 5936$ , 1748, and 1236 for $β = 0.001$ , 0.01, and 0.1, respectively. The electric field is $E = 10^{- 3}$ .
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Figure 4
Phenomenology of the topological transition. Contours of $S_{z}$ (fixed spins), arrows of $(s_{x}, s_{y})$ (itinerant spins), and color density of the topological $b$ field. The dissipation is (left) $β = 0.001$ , (center) $β = 0.01$ , and (right) $β = 0.1$ . The transition towards the ferromagnetic state is correlated with the appearance of intense $b$ -field structures possessing a topological charge opposite to the one of the original skyrmion.
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Figure 5
Magnetization components corresponding to the numerical computation $β = 0.001, t = 5936 t_{0}$ (left) and to the analytical self-similar solution (center) near the skyrmion core from Eqs. (24) and (11). Analytical magnetization texture (right) to be compared with Fig. 2. The size of the core shrinks to zero as the time approaches the critical singularity.
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Figure 6
Stochastic torque. The magnitude $τ (t)$ and polar angle $φ (t)$ of the electron torque $τ (x, t) = s (x, t) \times S (x, t)$ at the skyrmion core $x = x_{0} (t)$ as a function of time exhibit (a) a random dynamics with (b) a Poissonian distribution as shown by the exponential fit (red line). Parameters correspond to the intermediate dissipation case of $β = 0.01$ .
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