Ultrafast generation of skyrmionic defects with vortex beams: Printing laser profiles on magnets

Controlling electric and magnetic properties of matter by laser beams is actively explored in the broad region of condensed matter physics, including spintronics and magneto-optics. Here we theoretically propose an application of optical and electron vortex beams carrying intrinsic orbital angular momentum to chiral ferro- and antiferromagnets. We analyze the time evolution of spins in chiral magnets under irradiation of vortex beams by using the stochastic Landau-Lifshitz-Gilbert equation. We show that beam-driven nonuniform temperature leads to a class of ring-shaped magnetic defects, what we call skyrmion multiplex, as well as conventional skyrmions. We discuss the proper beam parameters and the optimal way of applying the beams for the creation of these topological defects. Our findings provide an ultrafast scheme of generating topological magnetic defects in a way applicable to both metallic and insulating chiral (anti-) ferromagnets.

Figure 1

(a) Schematics of our setup. The vortex beam is irradiated to a chiral magnet and induces nonuniform temperature proportional to the local intensity of the beam. Arrows represent spins in chiral magnets [see Eq. (8)] and the colors represent their $z$ component (red for $+1$, blue for $-1$, and green for 0). (b) A topological defect created by the vortex beam, skyrmionium. (c, d) Spin texture of skyrmions in ferromagnetic backgrounds with up and down spins. (e) Spin texture of an antiskyrmion.

Figure 2

Intensity of Laguerre-Gaussian modes at the focal plane $|u^{\mathrm{LG}}{(\rho ,\varphi ,z=0)|}^2$ for several choices of $p$ and $m$. The brightness is proportional to the intensity. For nonvanishing values of $m$, the intensity at the center is exactly zero and there appear $(p+1)$-fold rings in the intensity distribution. We also see that for larger $m$, the size of rings becomes larger with increasing $m$.

Figure 3

Ground-state phase diagram of the canonical model of chiral ferromagnets (8) (reproduced from Ref. [90]). There appear three distinct phases depending on the magnitude of the external magnetic field applied in the $z$ direction. For the helical order phase and skyrmion crystal phase, we give their spin texture obtained by numerical simulations based on the LLG equation, using arrows for the $x,y$ components and colors for the $z$ components.

Figure 4

Skyrmion duplex and quadplex obtained by numerical simulation of the model (8) for $J=1,D=0.15$, and $B_{\mathrm{z}}=0.014$. (a) Cumulative skyrmion number $N_{\mathrm{SK}}\left(R\right)$ of the skyrmion duplex as a function of $R/a$, where $a$ is the lattice constant. The skyrmion duplex, or skyrmionium, is a skyrmion with positive $N_{\mathrm{SK}}$ [Fig. 1] surrounded by that with negative $N_{\mathrm{SK}}$ [Fig. 1]. (b) Spin texture of the skyrmion duplex with diameter of about 50 sites. (c) Spin texture of the skyrmion quadplex whose whole size is approximately 160 sites in diameter. The arrows represent the in-plane components of spins and the color indicates their $z$ component.

Figure 5

Time evolution of the $z$ component of spins for a particular trial with parameters $B_z=0.01,D=0.15,w=12.5a,t_0=500$, and $T_0=2$. We start from the metastable ferromagnetic state at $t=0$. At $t=0$ we suddenly raise the temperature in accord with the intensity of the vortex beam with $p=0$ and $m=5$. Then the temperature is lowered gradually, and finally, a skyrmion duplex is formed. The time is measured in the unit of $\hslash /J$. When $J=1\mathrm{meV}$ the time unit corresponds to 0.7 ps. The system consists of $150\times 150$ sites and a periodic boundary condition is imposed.

Figure 6

Time evolution of the $z$ component of spins for a particular trial with parameters $B_z=0.011,D=0.15,\alpha =0.1,w=100a/3,T_0=4$, and $t_0=800$. For $p=1$ and $m=3$, the intensity of the beam takes the form of a double ring and a skyrmion quadplex is formed after the irradiation. The system consists of $200\times 200$ sites and a periodic boundary condition is imposed.

Figure 7

Time evolution of the $z$ component of spins for a particular trial with parameters $B_z=0.01,D=0.15,t_0=500$, and $T_0=1.5$. We still use a vortex beam with $p=0$ and $m=5$, but the beam waist is set to be small: $w=6a$. Due to the small beam waist, the topological singularity in the temperature profile is smeared and as a result, an ordinary skyrmion is created. The system consists of $150\times 150$ sites and a periodic boundary condition is imposed.

Figure 8

Success probability of creating skyrmion duplexes by vortex beams with $p=0$ and $m=5$, for $J=1,D=0.15$, and $\alpha =0.1$. (a) Probability for $B_z=0.01,\left(\text{b}\right)B_z=0.0125$, and $\left(\text{c}\right)B_z=0.015$. The local temperature $T(t,\vec{r})$ is set to be proportional to the intensity of the beam: $T(t,\vec{r})=T\left(t\right)\left[|u^{\mathrm{LG}}{(\rho ,\varphi ,0)|}^2/\max ({\left|u^{\mathrm{LG}}(\rho ,\varphi ,0)\right|}^2)\right]$. Here the time dependence of the temperatures is given as panel (d): $T\left(t\right)=T_0\left(1-\frac{t}{t_0}\right)\mathrm{\Theta }\left(t\right)\mathrm{\Theta }(t_0-t)$ with $t_0=500$. The highest probability is achieved when the magnetic field is small and the beam waist satisfies $w\approx 11.5a$, for which the wavelength of the beam is comparable with the size of skyrmions. We also show where the chosen magnetic fields $B_z$ locate in the phase diagram.

Figure 9

Schematics of a Néel-type antiferromagnetic (a) skyrmion and (b) skyrmion duplex. There are two magnetic sublattices in Néel ordered states in the square lattice, and a skyrmion and skyrmion duplex can be seen as bound states of their ferromagnetic counterparts living in different magnetic sublattices. Just as Fig. 1, the colors of arrows represent the $z$ component of spins.

Figure 10

Phase diagram of the canonical model of chiral AFMs Eq. (13) for $A/J=0.055$. When DM interaction is very weak, we have an antiferromagnetic (AFM) region where we cannot have skyrmions. As we increase DM interaction $D$, antiferromagnetic skyrmions become energetically stable at $D/J\approx 0.16$ as isolated defects. For larger DM interaction $\left(D/J\ge 0.22\right)$, skyrmions deform to lower their energy (d-AFM state) and eventually warm domains are formed (WD). The phase boundary between d-AFM and WD is unclear from our calculations. We visualize the spin texture of typical states in each phase obtained from LLG calculations by using staggered spins $m_{\vec{r}}^z\times {(-1)}^{|\vec{r}|}\equiv m_{\vec{r}=(i,j)}^z\times {(-1)}^{i+j}$.

Figure 11

Time evolution of staggered spins $m_{\vec{r}}^z\times {(-1)}^{|\vec{r}|}\equiv m_{\vec{r}=(i,j)}^z\times {(-1)}^{i+j}$ for a particular trial with parameters $D/J=0.205,A/J=0.055,T_0/J=1,w=12.5a$, and $\alpha =0.1$. The ring-shaped heating caused by vortex beams with $p=0$ and $m=5$ creates magnetic domains different from the background Néel order. These domains merge to form an antiferromagnetic skyrmion duplex. A time step corresponds to 0.7 ps for $J=1$ meV.

Figure 12

Time evolution of staggered spins $m_{\vec{r}}^z\times {(-1)}^{|\vec{r}|}\equiv m_{\vec{r}=(i,j)}^z\times {(-1)}^{i+j}$ for a particular trial for $D/J=0.21,A/J=0.055$, and $\alpha =0.1$. We superimpose two vortex beams $(m=1$ and $m=20)$ with $T_0/J=0.5$ and $w=20a$, to realize temperature with a double-ring profile. We see that the heating leads to formation of an antiferromagnetic skyrmion quadplex. The system is a periodic square lattice with $200\times 200$ sites.

Figure 13

Success probability of creating antiferromagnetic skyrmion multiplexes by vortex beams with $p=0$ and $m=5$. We fix $J=1,D=0.205,A=0.055$, and $\alpha =0.1$, with which the system is in the AFMS region (see Fig. 10). The initial state at $t=0$ is a Néel ordered ground state and the temperature is varied in accord with $\left(\text{d}\right)$ for $\left(\text{a}\right)t_0=3000,\left(\text{b}\right)t_0=5000$, and $\left(\text{c}\right)t_0=7000$. The probability is high when the wavelength of the vortex beams is of the same order of the size of antiferromagnetic skyrmions and the period of the irradiation of the vortex beams is long. $\left(\text{e}\right)$ Skyrmion multiplexes observed after the irradiation of vortex beams. We show the $z$ component of staggered spins for visibility.