Reversible vector ratchets for skyrmion systems

We show that ac driven skyrmions interacting with an asymmetric substrate provide a realization of a class of ratchet system which we call a vector ratchet that arises due to the effect of the Magnus term on the skyrmion dynamics. In a vector ratchet, the dc motion induced by the ac drive can be described as a vector that can be rotated clockwise or counterclockwise relative to the substrate asymmetry direction. Up to a full ${360}^{\circ }$ rotation is possible for varied ac amplitudes or skyrmion densities. In contrast to overdamped systems, in which ratchet motion is always parallel to the substrate asymmetry direction, vector ratchets allow the ratchet motion to be in any direction relative to the substrate asymmetry. It is also possible to obtain a reversal in the direction of rotation of the vector ratchet, permitting the creation of a reversible vector ratchet. We examine vector ratchets for ac drives applied parallel or perpendicular to the substrate asymmetry direction, and show that reverse ratchet motion can be produced by collective effects. No reversals occur for an isolated skyrmion on an asymmetric substrate. Since a vector ratchet can produce motion in any direction, it could represent a method for controlling skyrmion motion for spintronic applications.

Figure 1

Circles: Pinning site locations. (a) Conformal gradient array. (b) Square gradient array. (c) Random gradient array. Green arrow: direction of longitudinal drive $F_x^{\text{ac}}$. Red arrow: direction of transverse drive $F_y^{\text{ac}}$.

Figure 2

The velocity-force curves $|\langle V_x\rangle |$ (green circles) and $|\langle V_y\rangle |$ (red squares) vs dc drive $F_{\text{dc}}$ for the conformal pinning array in Fig. 1 with a skyrmion density of $n_s=0.3$ and ${\alpha }_m/{\alpha }_d=9.962$. (a) For dc driving in the positive $x$ direction, the critical depinning threshold is $F_c\approx 0.015$. Inset: The skyrmion Hall angle ${\theta }_{\text{sk}}={tan}^{-1}\left(\right|\langle V_{\perp }\rangle |/|\langle V_{\left|\right|}\rangle \left|\right)$ vs dc drive amplitude $F_{\text{dc}}$, where $V_{\perp }=V_y$ and $V_{\left|\right|}=V_x$. The dashed line indicates the pin free limit of ${\theta }_{\text{sk}}=84.{267}^{\circ }$. (b) For dc driving in the positive $y$ direction, close to the depinning threshold the skyrmion motion is strongly guided along the $y$ direction by the pinning sites. Inset: ${\theta }_{\text{sk}}$ vs $F_{\text{dc}}$, where $V_{\perp }=V_x$ and $V_{\left|\right|}=V_y$, shows that ${\theta }_{\text{sk}}$ is nearly zero at low drives and increases with increasing $F_{\text{dc}}$. The dashed line indicates the pin free limit of ${\theta }_{\text{sk}}=84.{267}^{\circ }$.

Figure 3

A diagram showing the eight possible types of vector ratchet motion for the conformal pinning array in Fig. 1. The net dc drift in the $(x,y)$ direction for each type is I, $(+x,0)$ (light blue); II, $(+x,+y)$ (dark blue); III, $(0,+y)$ (light green); IV, $(-x,+y)$ (dark green); V, $(-x,0)$ (pink); VI, $(-x,-y)$ (red); VII, $(0,-y)$ (light orange); and VIII, $(+x,-y)$ (dark orange). In addition, we define type IX with $(0,0)$ (purple) to be a state with no ratcheting motion.

Figure 4

(a,b) The average cumulative displacement per skyrmion $\langle \mathrm{\Delta }X\rangle$ (a) and $\langle \mathrm{\Delta }Y\rangle$ (b) vs time in ac cycles for the conformal pinning array under longitudinal ac driving with $n_s=0.3$ and $F_x^{\text{ac}}=0.04$ at ${\alpha }_m/{\alpha }_d=0$ (dark green), 1.36 (dark brown), 4.0 (burgundy), 8.0 (dark pink), 10 (yellow), and 20 (orange). There is no ratchet motion when ${\alpha }_m/{\alpha }_d=0$, but for ${\alpha }_m/{\alpha }_d\ne 0$ we observe ratchet reversals in both the $x$ and $y$ directions. (c,d) $\langle \mathrm{\Delta }X\rangle$ (c) and $\langle \mathrm{\Delta }Y\rangle$ (d) vs time in ac cycles for the same system for transverse ac driving at $F_y^{\text{ac}}=0.04$ and ${\alpha }_m/{\alpha }_d=0$ (dark green), 1.2 (dark brown), 1.6 (burgundy), 2.6 (dark pink), 10 (yellow), and 20 (orange). In this case the ratchet motion for ${\alpha }_m/{\alpha }_d\ne 0$ is always in the negative $x$ direction and shows a reversal in the $y$ direction.

Figure 5

(a,b) $\langle \mathrm{\Delta }X\rangle$ (a) and $\langle \mathrm{\Delta }Y\rangle$ (b) vs time in ac cycles for the conformal pinning array for longitudinal ac driving with ${\alpha }_m/{\alpha }_d=10$ at $F_x^{\text{ac}}=0$ (gray), 0.025 (brown), 0.04 (burgundy), 0.06 (pink), and 0.08 (orange). There is no ratchet motion at $F_x^{\text{ac}}=0$, but for $F_x^{\text{ac}}>0$ ratchet reversals occur in both the $x$ and $y$ directions. (c,d) $\langle \mathrm{\Delta }X\rangle$ (c) and $\langle \mathrm{\Delta }Y\rangle$ (d) for the same system for transverse ac driving at ${\alpha }_m/{\alpha }_d=10$ and $F_y^{\text{ac}}=0$ (gray), 0.007 (brown), 0.015 (burgundy), 0.021 (pink), and 0.06 (orange). The ratchet effect is always in the negative $x$ direction and shows a reversal in the $y$ direction.

Figure 6

The value of $\langle \mathrm{\Delta }X\rangle$ (green circles) and $\langle \mathrm{\Delta }Y\rangle$ (red squares) after 400 ac cycles vs ${\alpha }_m/{\alpha }_d$ for the system in Fig. 4. (a) Driving in the $x$ direction with $F_x^{\text{ac}}=0.04$. The ratchet sequence is IX-IV-III-II-I-VIII, so that the flow rotates clockwise by ${180}^{\circ }$ as indicated in the inset, which is based on the schematic in Fig. 3. (b) Driving in the $y$ direction with $F_y^{\text{ac}}=0.04$. The ratchet sequence is IX-VI-V-IV giving a counterclockwise rotation of ${90}^{\circ }$ as shown in the inset.

Figure 7

$\langle \mathrm{\Delta }X\rangle$ (green circles) and $\langle \mathrm{\Delta }Y\rangle$ (red squares) after 400 ac cycles vs $F^{\text{ac}}$ for the system in Fig. 4 at ${\alpha }_m/{\alpha }_d=10.0$. (a) Driving in the $x$ direction with $F_x^{\text{ac}}$. The ratchet sequence is IX-VI-V-IV-III-II-I-VIII-VII-VI, giving a clockwise rotation of ${360}^{\circ }$ as indicated in the inset. (b) Driving in the $y$ direction with $F_y^{\text{ac}}$. The ratchet sequence is IX-VI-V-IV, giving a clockwise rotation of ${90}^{\circ }$, as shown in the inset.

Figure 8

Heat maps of (a) $\langle \mathrm{\Delta }X\rangle$ and (b) $\langle \mathrm{\Delta }Y\rangle$ as a function of $F_x^{\text{ac}}$ vs ${\alpha }_m/{\alpha }_d$ for the conformal array. Here there are ratchet reversals in both the $x$ and $y$ directions. (c) Heat map of $\langle \mathrm{\Delta }X\rangle$ for driving in the $y$ direction where the drift is always in the negative $x$ direction. (d) The corresponding $\langle \mathrm{\Delta }Y\rangle$ as a function of $F_y^{\text{ac}}$ vs ${\alpha }_m/{\alpha }_d$ showing a reversal.

Figure 9

$\langle \mathrm{\Delta }X\rangle$ (green circles) and $\langle \mathrm{\Delta }Y\rangle$ (red squares) after 400 ac cycles vs skyrmion density $n_s/n_p$ for $F_p=0.1,F^{\text{ac}}=0.05$, and ${\alpha }_m/{\alpha }_d=9.962$. (a) Driving in the $x$ direction, $F_x^{\text{ac}}$. (b) Driving in the $y$ direction, $F_y^{\text{ac}}$.

Figure 10

$\langle \mathrm{\Delta }X\rangle$ (green circles) and $\langle \mathrm{\Delta }Y\rangle$ (red squares) after 400 ac cycles vs skyrmion density $n_s/n_p$ for $F_p=0.5,F^{\text{ac}}=0.25$, and ${\alpha }_m/{\alpha }_d=9.962$. (a) Driving in the $x$ direction, $F_x^{\text{ac}}$. The ratchet flow direction initially rotates counterclockwise, followed by a clockwise rotation for $n_s/n_p>1.3$. (c) Driving in the $y$ direction, $F_y^{\text{ac}}$.

Figure 11

$\langle \mathrm{\Delta }X\rangle$ (green circles) and $\langle \mathrm{\Delta }Y\rangle$ (red squares) for a single skyrmion, $N_s=1$, after 400 ac cycles vs $F^{\text{ac}}$ for $F_p=0.1$ and ${\alpha }_m/{\alpha }_d=9.962$. (a) Driving in the $x$ direction, where there is no ratchet effect. (b) Driving in the $y$ direction, where there is a very weak ratchet effect for $F^{\text{ac}}/F_p>1.0$.

Figure 12

(a) $\langle \mathrm{\Delta }X\rangle$ (green circles) and (b) $\langle \mathrm{\Delta }Y\rangle$ (red squares) after 400 ac cycles vs ac frequency $\omega$ in samples with ${\alpha }_m/{\alpha }_d=9.962$ and $F_x^{\text{ac}}=0.05$. In both cases, the ratchet flux decreases with increasing ac drive frequency. The insets show normalized values (a) $\langle \overline{\mathrm{\Delta }X}\rangle$ and (b) $\langle \overline{\mathrm{\Delta }Y}\rangle$ vs $\omega$. Normalization is achieved by dividing by the total time required to perform 400 ac drive cycles at each frequency, and then dividing by the value at $\omega =0.04$.

Figure 13

Skyrmion positions (filled dots), pinning site locations (open circles), and trajectories (lines) for $y$ direction ac driving $F_y^{\text{ac}}$ in a sample with $n_s/n_p=1.0$. (a) The positive ac drive cycle for $F_y^{\text{ac}}=0.013$. (b) The negative ac drive cycle for $F_y^{\text{ac}}=0.013$. At this drive, a type-VI ratchet with motion in the negative $y$ and negative $x$ directions occurs. (a) The positive ac drive cycle for $F_y^{\text{ac}}=0.03$. (a) The negative ac drive cycle for $F_y^{\text{ac}}=0.03$. Here, there is a type-IV ratchet with motion in the negative $x$ and positive $y$ directions.

Figure 14

Ratchet motion in the different arrays illustrated in Fig. 1: conformal pinning array (red triangles), square gradient array (blue squares), and random gradient array (brown circles), in samples with $n_s/n_p=0.3$. (a) $\langle \mathrm{\Delta }X\rangle$ and (b) $\langle \mathrm{\Delta }Y\rangle$ after 400 ac cycles vs ${\alpha }_m/{\alpha }_d$ for $x$ direction driving $F_x^{\text{ac}}$. (c) $\langle \mathrm{\Delta }X\rangle$ and (d) $\langle \mathrm{\Delta }Y\rangle$ after 400 ac cycles vs ${\alpha }_m/{\alpha }_d$ for $y$ direction driving $F_y^{\text{ac}}$.