Quantum Depinning of a Magnetic Skyrmion

We investigate the quantum depinning of a weakly driven skyrmion out of an impurity potential in a mesoscopic magnetic insulator. For small barrier height, the Magnus force dynamics dominates over the inertial term, and the problem is reduced to a massless charged particle in a strong magnetic field. The universal form of the WKB exponent, the rate of tunneling, and the crossover temperature between thermal and quantum tunneling are provided, independently of the detailed form of the pinning potential. The results are discussed in terms of macroscopic parameters of the insulator ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ and various skyrmion radii. We demonstrate that small enough magnetic skyrmions, with a radius of $\sim 10$ lattice sites, consisting of some thousands of spins, can behave as quantum objects at low temperatures in the millikelvin regime.

Figure 1

Parameters $V_0,a$ of the pinning potential $V_p(r_0)$ of Eq. (4), as a function of the defect size ${\lambda }_d$, with $\kappa =0.463$ and $h=0.25$, for a skyrmion of radius $\lambda =1.86\alpha d$, with $\alpha$ the lattice constant. Vertical dashed line indicates the value ${\lambda }_d=\alpha d$.

Figure 2

Schematic illustration of the pinning potential $U$ as function of position, for two values of the external field, $F_{\text{ext}}=0$ and $0<F_{\text{ext}}<F_c$. $U_0$ denotes the height of the potential barrier and $X_d$ the position of the turning point.

Figure 3

(a) Equipotential lines of the potential landscape $U(X,iY)$, for a skyrmion of radius $\lambda =1.86\alpha d$, trapped by a pinning center of radius ${\lambda }_d=2.5\lambda$, and $\epsilon =0.096$. The instanton trajectory describing the tunneling of the skyrmion is depicted by the black solid line, while the red dashed line signals the position of the turning point $X_d$. (b) $X_d$ as a function of $\epsilon$, calculated using the full potential $U$ as well as the expanded potential $\overline{U}$ of Eq. (9), for $\lambda =1.86\alpha d$ and ${\lambda }_d=\alpha d$. The expansion is valid for $\epsilon \lesssim 0.05$. The inset illustrates the linear dependence of $X_d$ on the defect size, for a choice of $\epsilon =0.01$. (c) The tunneling exponent ${\mathcal{S}}_0$ as a function of $\epsilon$ for $\lambda =1.86\alpha d$ and ${\lambda }_d=\alpha d$, while the inset depicts the dependence of ${\mathcal{S}}_0$ on the skyrmion size.