Measuring the formation energy barrier of skyrmions in zinc-substituted ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$

We report small-angle neutron scattering (SANS) measurements of the skyrmion lattice in ${\left({\mathrm{Cu}}_{0.976}{\mathrm{Zn}}_{0.024}\right)}_2{\mathrm{OSeO}}_3$ under the application of electric and magnetic fields. These measurements show an expansion of the skyrmion lattice stability region with electric field. Furthermore, using time-resolved SANS, we observe the slow formation of skyrmions after an electric or magnetic field is applied, which has not been observed in pristine ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ crystals. The measured formation times are dramatically longer than the corresponding skyrmion annihilation times after the external field is removed, and increase exponentially from 100 s at 52.5 K to 10 000 s at 51.5 K. This thermally activated behavior indicates an energy barrier for skyrmion formation of 1.57(2) eV, the size of which demonstrates the huge cost for creating these complex chiral objects.

Figure 1

(a) SANS pattern measured at 54 K with $E=-2$ $\mathrm{V}/\mu \mathrm{m}$ and ${\mu }_{0}H=26\mathrm{mT}$. The gray ring shows the area that was summed on each pattern to characterize the skyrmion intensity. (b)–(d) Skyrmion phase diagrams of ${\left({\mathrm{Cu}}_{0.976}{\mathrm{Zn}}_{0.024}\right)}_2{\mathrm{OSeO}}_3$ measured by SANS with increasing magnetic field sweeps after zero magnetic field cooling in applied electric fields of (b) $+1.7\mathrm{V}/\mu \mathrm{m}$, (c) 0, and (d) $-2\mathrm{V}/\mu \mathrm{m}$. The lines show the conical–field-polarized boundary (black) and the helical-conical boundary (gray) measured by ac susceptibility at $E=0$.

Figure 2

Time dependence of skyrmion SANS intensity when applying and removing a $1.7\mathrm{V}/\mu \mathrm{m}$ electric field at 52.25 K and 34 mT.

Figure 3

Formation of skyrmions by applying an electric field of $1.7\mathrm{V}/\mu \mathrm{m}$ at 52.25 K in varied magnetic fields. (a) Skyrmion SANS intensity as a function of time. (b) Fit formation time (blue squares plotted on a log scale) and intensity scale factor (red triangles) from Eq. (1). Solid lines in (b) show fits to a Gaussian peak and $\tau =A{e}^{-\lambda {\mu }_{0}H}+{\tau }_a$. The dashed line in (b) shows the pure exponential behavior ($\tau =A_1{e}^{-{\lambda }_1{\mu }_{0}H}$) of the formation time below 30 mT.

Figure 4

(a) Skyrmion SANS intensity as a function of time after switching on $E=1.7\mathrm{V}/\mu \mathrm{m}$ at ${\mu }_{0}H=30\mathrm{mT}$ for temperatures between 51.75 and 52.62 K. (b) Formation times as a function of temperature. Circles show formation times after switching on $E=1.7\mathrm{V}/\mu \mathrm{m}$ in various magnetic fields. Purple triangles show the formation time after switching on ${\mu }_{0}H=26\mathrm{mT}$ in zero applied electric field. Lines show fits to an Arrhenius law. (c) Energy barriers from fits in panel (b) for $E=1.7\mathrm{V}/\mu \mathrm{m}$ (black circles) and $E=0$ (red triangle).

Figure 5

(a) Skyrmion SANS intensity as a function of time measured at $E=1.7\mathrm{V}/\mu \mathrm{m}$, ${\mu }_{0}H=30\mathrm{mT}$, and $T=52.25\mathrm{K}$. The red data show measurements when the electric field was turned on first, and then the magnetic field was turned on at $t=0\mathrm{s}$. These are fitted to $S=A(1-e^{-\frac{t}{\tau }})$. The blue data show measurements when the magnetic field was turned on first, and the electric field ramping begins at $t=0\mathrm{s}$. These are fitted to Eq. (1). (b) Skyrmion SANS intensity as a function of time measured at $E=0\mathrm{V}/\mu \mathrm{m}$, ${\mu }_{0}H=26\mathrm{mT}$, and $T=53\mathrm{K}$. The red data show measurements where the magnetic field was turned on at time = 0 in zero electric field. The blue data show measurements in a constant magnetic field where the electric field was changed from $-1.7\mathrm{V}/\mu \mathrm{m}$ to 0 at $t=0\mathrm{s}$. Both data sets were fitted to $S=A(1-e^{-\frac{t}{\tau }})$ as the magnetic field changes quickly and the electric field changes quickly when set to zero.

Figure 6

Gaussian widths and intensities for SANS rocking curves measured during skyrmion formation at 52.25 K in a 30 mT magnetic field after applying a $1.7\mathrm{V}/\mu \mathrm{m}$ electric field. The inset shows an example rocking curve constructed from data taken 20 minutes after applying the electric field.

Figure 7

Formation of skyrmions at 52.25 K in a magnetic field of 30 mT after applying various electric fields.

Figure 8

Intensity of skyrmions as a function of time measured at $T=52.25\mathrm{K}$ and $H=30\mathrm{mT}$ for two different ramp rates of the electric field. Blue is for ramping fast to $0.9\mathrm{V}/\mu \mathrm{m}$ and $0.009(\mathrm{V}/\mu \mathrm{m})/\mathrm{s}$ up to 1.7 $\mathrm{V}/\mu \mathrm{m}$, while red is ramping fast to $1.4\mathrm{V}/\mu \mathrm{m}$ and $0.009(\mathrm{V}/\mu \mathrm{m})/\mathrm{s}$ up to $1.7\mathrm{V}/\mu \mathrm{m}$. Solid lines show fits to Eq. (A6) using parameters for the two different ramp rates. Both are fitted to the same formation time of $133\pm 5\mathrm{s}$