Entropy-limited topological protection of skyrmions

Abstract

Magnetic skyrmions are topologically protected whirls that decay through singular magnetic configurations known as Bloch points. We used Lorentz transmission electron microscopy to infer the energetics associated with the topological decay of magnetic skyrmions far from equilibrium in the chiral magnet Fe_1-x Co _x Si. We observed that the lifetime τ of the skyrmions depends exponentially on temperature, [Formula: see text]. The prefactor τ₀ of this Arrhenius law changes by more than 30 orders of magnitude for small changes of the magnetic field, reflecting a substantial reduction of the lifetime of skyrmions by entropic effects and, thus, an extreme case of enthalpy-entropy compensation. Such compensation effects, being well known across many different scientific disciplines, affect topological transitions and, thus, topological protection on an unprecedented level.

Fig. 1. Magnetic phases and skyrmion decay in Fe_1-xCo_xSi.

(A) Magnetic phase diagram of Fe_1−xCo_xSi with x = 0.5 obtained by first cooling the system at a fixed magnetic field, B ≈ 23 mT, and then raising or lowering the field at fixed temperature T. The decrease in the applied field triggers the decay into a helical configuration, whereas either a conical or ferromagnetic state is reached for an increase of the field. (B) Typical LTEM images of helical and skyrmion lattice order, respectively. (C) Schematic image of an early state of the decay of the skyrmion lattice toward a ferromagnetic state. The skyrmion splits by the formation of a pair of Bloch points located at the end of the skyrmion strings, which move toward the surface. (D) Decay of skyrmion lattice order toward the helical state. Neighboring skyrmions merge, and a Bloch point at the merging points moves toward the surface.

Fig. 2. Key characteristics of the decay of skyrmions into a helical order.

The sample was field-cooled (FC) from above the helical transition temperature (T_c ≈ 38 K) under an applied magnetic field B = 23 mT down to T_m, where the field was reduced to B_m and data recorded as a function of time t. (A to C) Typical LTEM patterns at T_m = 16.7 K after reaching B_m = −2.6 mT for t = 0.1 , 4.8, and 20.2 s, respectively. (D) Evolution of the intensity across the white box marked in (A), (B), and (C) as a function of time (vertical axis). (E) Typical time dependence of the number of skyrmions for T_m = 20.4 K and B_m = −2.6 mT. The blue curve represents an exponential fit.

Fig. 3. Key characteristics of the decay of skyrmions for increasing magnetic fields.

The sample was field-cooled (FC) from above the helical transition temperature (T_c ≈ 38 K) under an applied magnetic field B = 23 mT down to T_m, where the field was increased to B_m and the data were recorded as a function of time t. (A to C) Typical LTEM patterns at T_m = 18.5 K after reaching B_m = 57 mT, for t = 6.8, 23.6, and 56.0 s, respectively. (D) Evolution of the intensity across the white box marked in (A), (B), and (C) as a function time (vertical axis). (E) Typical time dependence of the number of skyrmions for T_m = 20.4 K and B_m = 57 mT, fitted by an exponential (green line). (F) Time dependence of the intensity within the red dashed circle in (A), (B), and (C). For a small number of skyrmions, a two-step decay via an intermediate state with lower intensity is observed. (G) Statistics of the intermediate-state intensities.

Fig. 4. Key characteristics of the decay rates of supercooled skyrmions in Fe_1−xCo_xSi (x = 0.5).

(A) Typical decay times τ after field cooling at B = 23 mT, followed by a decrease/increase to B_m. (B and C) Decay time τ as a function of thermal energy for increasing and decreasing magnetic fields, respectively, and various values of B_m. (D) Attempt time τ₀ as a function of energy barrier ΔE inferred from the exponential time dependence of the skyrmion decay. The variation of more than 30 orders of magnitude of τ₀ reflects the extreme enthalpy-entropy compensation.