Twisted skyrmions at domain boundaries and the method of image skyrmions

We predict a twisted skyrmion structure at the boundary of two antiparallel magnetic domains stabilized by a (synthetic or crystal) antiferromagnetic coupling on the border. Through this novel state, skyrmions with opposite polarities can be freely switched between each other by spin-polarized electric currents. We therefore propose the concept of double-track skyrmion racetrack memories and logic gates where the binary data are represented by skyrmions with different polarities. The dynamics of the skyrmion polarity reversal is theoretically studied. Using the method of image, we derive the analytical formula of the repulsive potential when a normal skyrmion approaches the domain boundary. A harmonic attractive potential is obtained for the twisted skyrmion across the border. Micromagnetic simulations compare well with analytical results. The method of image skyrmion proposed in this work can be used to deal with a large class of skyrmion-boundary interaction problems.

Figure 1

(a) Illustration of a Néel skyrmion and a twisted skyrmion at the domain and the domain boundary, respectively, in a coplanar [Co/Pt]/A/[Co/Pt] heterostructure with a synthetic antiferromagnetic coupling on the border (shown by the gray strip). (b) Top view of data bits consisting of four skyrmions in a double-track SRM. The binary data “1101” is rewritten to be “0101” by a local current $j_e$. (c) Position dependence of the skyrmion charge $Q$ and its polarity $p$.

Figure 2

Potential energy $U$ vs $X$ for $J_{\text{AFM}}/J_{\text{FM}}=1$ (yellow circles) and 3 (red squares), obtained numerically. The solid black curve is Eq. (5a) with $R=12\text{nm}$. The solid blue line is Eq. (5b) with a smaller $R=5\text{nm}$ due to the size shrinking of skyrmion when it is very close to the boundary. The solid brown curve is Eq. (7) with $R^{\prime }=20\text{nm},R^{\prime \prime }=4.5\text{nm}$, and $U_1=0.15\text{eV}$. Inset: Three representative skyrmion structures in different regions.

Figure 3

Schematic plot of the method of image skyrmion. For a straight domain boundary, the skyrmion and the image skyrmion is of mirror symmetry under a fixed BC.

Figure 4

STAR phase diagram plotted in the $J_{\text{AFM}}\text{−}{j}_e$ plane, with black squares for “S,” red circles for “T,” blue rhombus for “A,” and purple triangles for “R.” Arrows label the two critical currents $j_{c,1}$ and $j_{c,2}$ analytically obtained in the large $J_{\text{AFM}}$ limit.

Figure 5

Three steps toward the coplanar heterostructure. (a) A [Co/Pt] bilayer with a perpendicular magnetic anisotropy and a wedged edge is grown on the substrate. The magnetizations point down. (b) A Ru layer with thickness less than 1 nm (a few atomic layers) is sputtered on the wedged [Co/Pt] edge. (c) A [Co/Pt] bilayer is deposited as the right magnetic domain with magnetizations pointing up.

Figure 6

(a) Schematic plot of the antiferromagnetically coupled coplanar heterostructure with a wedged edge (black dashed line) with a tilted angle $\delta =\pi /4$. A current $j_e={10}^{13}\text{A}/{\text{m}}^2$ is applied along $-x$ direction. (b) Spin configurations of the coplanar [Co/Pt] domains at $t=0.09\text{ns}$ with both the top view in the region: $166\le x\le 234\text{nm},86\le y\le 114\text{nm},z=5\text{nm}$ and the sectional view in the region: $166\le x\le 234\text{nm},y=100\text{nm},1\le z\le 10\text{nm}$. (c) and (d) Same as (b) at $t=0.11\text{ns}$ (twisted skyrmion) and $t=0.13\text{ns}$ (skyrmion with flipped polarity), respectively.

Figure 7

Skyrmion velocity $v$ as a function of the driving current $j_e$ in [Co/Pt] domains.

Figure 8

Schematic plot of four states in MTJs (sectional view): (a) no skyrmion, (b) a twisted skyrmion, (c) a skyrmion with $p=1$, and (d) a skyrmion with $p=-1$.

Figure 9

(a) A NOT gate (top view). White dashed circle represents an origin skyrmion with $p=-1$ (encoding 1). Driven by a current, the origin skyrmion passes through the boundary and becomes a skyrmion with opposite polarity $p=1$ (encoding 0). Dashed line is the trajectory of the skyrmion. Arrows show the direction of movement. (b)–(d) Snapshots of the working process of logic NOT gates: (b) Skyrmion with $p=-1$ at $t=0$; (c) twisted skyrmion at $t=0.11\text{ns}$; (d) skyrmion with $p=1$ at $t=0.22\text{ns}$. An electric current of magnitude ${10}^{13}\text{A}/{\text{m}}^2$ is applied (green arrow).

Figure 10

(a) Schematic plot of a logic AND gate with two input skyrmions of $p=-1$ (encoding 1) (top view). The logic gate contains the following parts: a double-track cross with two inputs and two outputs, two MTJs located at the inputs, four VCMA gates on the outputs as well as the peripheral complementary metal-oxide semiconductor (CMOS) circuits. $V_1$ represents the high voltage. $V_2\left(V_3\right)$ depends on the resistance of the connected MTJ: high (low) voltage for high (low) resistance. The operation of logic gates is realized by applying electric currents generated by the electrodes at the terminals of the logic gates (not show here). (b) Truth table for the logic AND gate. (c) Schematic plot of a logic OR gate with two input skyrmions of $p=1$ (encoding 0) (top view). The logic OR gate can be implemented by changing the circuit connections in the structure of a logic AND gate: MTJs connect with VCMA21 and VCMA22. (d) Truth table for the logic OR gate.

Figure 11

Snapshots of the working process of skyrmions logic AND gate (top view). (a1)–(a5) Logic operation $1+1=1$. (b1)–(b5) $1+0=0$. (c1)–(c5) $0+0=0$. Green arrows label the directions of electric currents. The magnitude of the current is $j={10}^{12}\text{A}/{\text{m}}^2$. Here $\beta =0.3$.

Figure 12

Snapshots of the working process of skyrmions logic OR gate (top view). (a1)–(a5) Logic operation $1+1=1$. (b1)–(b5) $1+0=1$. (c1)–(c5) $0+0=0$. Green arrows label the directions of electric currents. The magnitude of the current is $j={10}^{12}\text{A}/{\text{m}}^2$. Here $\beta =0.3$.

Figure 13

(a) Schematic of the double-track SRM containing four skyrmions with opposite polarities. An electric current along $-y$ direction with $j_e={10}^{12}\text{A}/{\text{m}}^2$ and $\beta =0.45>\alpha =0.3$ is applied. The directions of the Magnus force and the repulsive force from the boundary are schematically shown by the white arrows in skyrmions 1 and 2. The two forces are in balance at steady states. (b) Time dependence of velocities for the four skyrmions, denoted as $v_{1x\left(1y\right)},v_{2x\left(2y\right)},v_{3x\left(3y\right)},v_{4x\left(4y\right)}$, respectively. At steady motion, $v_x=0$ and $v_y\simeq 100\text{m}/\text{s}$ for all skyrmions.

Figure 14

(a) Skyrmion trajectory driven by a transverse current with $j_e={10}^{12}\text{A}/{\text{m}}^2$. The dashed circle represents the origin skyrmion in the left domain. The dashed green line guides the eyes for the trajectory of the skyrmion. Arrows show the direction of skyrmion propagation. (b) $\mathrm{\Delta }y=Y\left(t\right)-Y\left(0\right)$ vs $X$ for the process in (a). $(X,Y)$ is the position of the skyrmion/twisted-skyrmion center. The dashed orange line labels the boundary $x=0$.

Figure 15

Radial distribution of the skyrmion polar angle by micromagnetic simulations (blue circles). Solid lines are different skyrmion profile models [Eq. (C17)]: ${\mathrm{\Theta }}_1$ (red) and ${\mathrm{\Theta }}_2$ (orange).

Figure 16

Schematic plot of a skyrmion and its image in a disk. For an origin Néel skyrmion at $(X,0)$ (inside the disk), its image (outside the disk) is a Bloch-type skyrmion with the same polarity at $(X^{\prime },0)=(R_0^2/X,0)$.

Figure 17

Same as Fig. 3 for antiskyrmions at (a) a semi-infinite system and (b) a ferromagnetic disk.

Figure 18

A stretched twisted skyrmion. An ellipse of semimajor and semiminor axes $R^{\prime }$ and $R^{\prime \prime }$ at $(X,0)$ is found to be a good approximation of its shape (see the Supplemental movie T.avi [36]). $L$ is the length of the boundary inside the twisted skyrmion.

Figure 19

(a) $U$ vs $X$ for $J_{\text{AFM}}/J_{\text{FM}}=1$ and $\beta =0.15<\alpha =0.3$. Note that $U$ is not symmetric with respect to $x=0$ in this case. (b) STAR phase diagram in the $J_{\text{AFM}}-j_e$ plane with black squares for “Success” (S phase), red circles for “Trapping” (T phase), blue rhombus for “Annihilation” (A phase), and purple triangles for “Rejection” (R phase). Arrows locate the critical currents $j_{c,1}$ and $j_{c,2}$ obtained analytically by Eq. (8).

Figure 20

(a) $U$ vs $X$ for antiskyrmions. Here $J_{\text{AFM}}/J_{\text{FM}}=1$ and $\alpha =\beta =0.3$. The solid black curve is Eq. (5a) with $R=10.8\text{nm}$, and the solid blue curve is Eq. (5b) with $R=4\text{nm}$. The solid brown curve is Eq. (7) with $R^{\prime }=28.5\text{nm}$ and $R^{\prime \prime }=22.5\text{nm}$. (b) STAR phase diagram in the $J_{\text{AFM}}\text{−}{j}_e$ plane with black squares for success (S phase), red circles for trapping (T phase), blue rhombus for annihilation (A phase), and purple triangles for rejection (R phase). Arrows locate the critical currents $j_{c,1}$ and $j_{c,2}$ obtained analytically by Eq. (8). (c) Same as (a) but for $\beta =0.15$. ${\kappa }_1\left[{\mathrm{\Theta }}_1\right]$ is plotted with $R=11.5\text{nm}$. ${\kappa }_2\left[{\mathrm{\Theta }}_1\right]$ and $U_0\left[{\mathrm{\Theta }}_1\right]({\kappa }_2^{\prime }\left[{\mathrm{\Theta }}_1\right]$ and $U_0^{\prime }\left[{\mathrm{\Theta }}_1\right])$ are obtained with $R=4.2\text{nm}\left(R=5.8\text{nm}\right)$. $k_0\left(k_0^{\prime }\right)$ is calculated by $R^{\prime }=20\text{nm}$ and $R^{\prime \prime }=6.5\text{nm}(R^{\prime }=23\text{nm}$ and $R^{\prime \prime }=4.8\text{nm})$. (d) Same as (b) but for $\beta =0.15$.

Figure 21

(a) $U$ vs $X$ for skyrmions in [Pd/Fe] domains. Here $J_{\text{AFM}}/J_{\text{FM}}=1$, and $\alpha =\beta =0.3$. The solid black curve is Eq. (5a) with $R=3.2\text{nm}$. The solid blue curve is Eq. (5b) with $R=1.5\text{nm}$. The solid brown curve is Eq. (7) with $R^{\prime }=3.4\text{nm}$ and $R^{\prime \prime }=3.2\text{nm}$. (b) STAR phase diagram in the $J_{\text{AFM}}\text{−}{j}_e$ plane with black squares for success (S phase), red circles for trapping (T phase), blue rhombus for annihilation (A phase), and purple triangles for rejection (R phase). Arrows locate the critical currents $j_{c,1}$ and $j_{c,2}$ obtained by Eq. (8).