Topologically-mediated energy release by relativistic antiferromagnetic solitons

Magnetic solitons offer functionalities as information carriers in multiple spintronic and magnonic applications. However, their potential for nanoscale energy transport has not been revealed. Here we demonstrate that antiferromagnetic solitons, e.g., domain walls, can uptake, transport, and release energy. The key for this functionality resides in their relativistic kinematics; their self-energy increases with velocity due to Lorentz contraction of the soliton and their dynamics can be accelerated up to the effective speed of light of the magnetic medium. Furthermore, their classification in robust topological classes allows us to selectively release this energy back into the medium by colliding solitons with opposite topology. Our work uncovers important energy-related aspects of the physics of antiferromagnetic solitons and opens up the attractive possibility for spin-based nanoscale and ultrafast energy transport devices.

Figure 1

(a) A magnetic domain wall (DW) is a region of width ${\mathrm{\Delta }}_0$ at rest, which separates two magnetic domains. In AFMs DWs can be driven up to relativistic speeds $v_s$ resulting in the Lorentz contraction of the DW, $\mathrm{\Delta }={\mathrm{\Delta }}_0{\gamma }_{\mathrm{L}}$, and the increase of the self-energy, $E\left(v_s\right)=E_0/{\gamma }_{\mathrm{L}}$, where ${\gamma }_{\mathrm{L}}=\sqrt{1-{(v_s/c)}^2}$. (b) DWs can hold two spin winding numbers $Q=\pm 1$, which are degenerated in energy. The winding number is defined by the integral over the $x$ axis of the topological density, $-{\nabla }_x\varphi \left(x\right)$, where $\varphi \left(x\right)$ is the magnetization azimuthal (in-plane) angle and $x$ is the DW propagation direction. For $Q=1$, at $x\to -\infty$, starting at $\varphi =\pi /2$, the path (blue arrow) to $\infty$, $\varphi =-\pi /2$ defines its topology. For $Q=-1$, $\varphi (-\infty )=-\pi /2$ and $\varphi \left(\infty \right)=\pi /2$. (c) Schematic representation of two DWs with the same winding number, $Q=+1$ and $Q=+1$, approaching each other in an accelerated motion until collision. After collision DWs stop. (d) During the motion of the DWs the self-energy increases due to Lorentz contraction caused by the increase of their speed. After collision, the self-energy stored at the DWs is partially discharged. (e) Schematic representation of two DWs with the opposite winding number, $Q=+1$ and $Q=-1$, approaching each other in an accelerated motion until collision. After collision DWs annihilate and a breather appears. (f) During the motion of the DWs the self-energy rises due to Lorentz contraction caused by the increase of their speed. After collision, the self-energy stored at the DWs is completely discharged.

Figure 2

Time dependence of the self-energy of the two antiferromagnetic domain walls (DWs) when subject to a spin-orbit field $H_{\text{SO}}$, which ramps up from $H_{\mathrm{SO}}=0$ to $H_{\mathrm{SO}}=60$ mT in 10 fs, in the case of two solitons with opposite topological charges, which entails their annihilation when they are conducted towards each other, which implies the liberation of the self-energy of both textures. The red dashed line represents the sum of the static self-energy of the two DWs, $E_0=2E_{\mathrm{exc}}^i(H_{\mathrm{SO}}=0)=4a_0^{2}A/{\mathrm{\Delta }}_0$, where $A$ represents the effective exchange stiffness of the medium, $a_0$ is the in-plane lattice constant, and ${\mathrm{\Delta }}_0$ expresses the DW width at rest, according to Eq. (9), which is taken as a zero reference for the time evolution of the exchange energy of the system.

Figure 3

Model simulations of the collision of two domain walls (DWs) in ${\mathrm{Mn}}_2\mathrm{Au}$. (a),(b) Spatiotemporal diagram of the dynamics of the process showing the DWs collision and the corresponding output. (a) Two DWs with opposite topological charges result after the collision in a dispersing breather. (b) Two DWs with the same topological charges after the collision stop and increase their size. A finite $x$ component of the magnetization, $m_x$, represents the position and extension of the DWs. Subplots (a1),(b1), (a2),(b2), and (a3),(b3) are cuts of the color map in (a) and (b) for three characteristic times $t=14,16$, and 18 ps, respectively.

Figure 4

(a) Sketch of two domain walls (DWs) with the same chirality $Q=+1$ under the action of an SO field $H_{\mathrm{SO}}$. They move driven by SO field antiparallel to the central magnetic domain (blue box). (b) DWs move until collision. After the collision, both DWs remain at an equilibrium distance from each other. (c) Schematic illustration of the role played by the exchange $E_{\mathrm{exc}}^{1,2}$ and Zeeman $E_{\mathrm{Zee}}$ energies. While the exchange energy tries to separate ${\mathrm{DW}}_1$ and ${\mathrm{DW}}_2$, the Zeeman energy forces them to stay as close as possible. (d) Zeeman (blue line) and exchange (orange line) energies as a function of the relative distance between ${\mathrm{DW}}_1$ and ${\mathrm{DW}}_2$ for an applied SO field of 20 mT. (e) Comparison between analytical Eq. (21) and numerically extracted stable distances between ${\mathrm{DW}}_1$ and ${\mathrm{DW}}_2$ as a function of the applied SO field.

Figure 5

(a) Stored energy at two domain walls (DWs) with opposite topology can be transferred to the electron system by colliding them. Besides ${\dot{q}}_{\mathrm{dyn}}$, an additional topologically-mediated transfer mechanism opens up, ${\dot{q}}_{\mathrm{topo}}$, when the winding of the DWs are opposite. The electron and lattice are coupled via electron-phonon coupling, $g_{\mathrm{el}\text{−}\mathrm{ph}}$. (b) Color map of the spatiotemporal distribution of the rate of heat dissipation caused by the motion of the DWs. At the collision (zoomed in inset), an explosion of heat occurs due to the recombination of DWs. Subplots (b1)/(c1), (b2)/(c2), and (b3)/(c3) represent the spatial distribution of $\mathrm{\Delta }{T}_{\mathrm{el}}$ along the track for three characteristic times: 14, 16, and 18 picoseconds, respectively.

Figure 6

(a) Time evolution of the stored energy at the DWs. Before domain wall (DW) collision (yellow circle), energy increases as the velocity of the DWs accelerates. After collision (yellow circle), for $Q_1=-Q_2$, there is a complete release of stored energy, while for $Q_1=Q_2$, a partial release of stored energy takes place. The difference of stored-energy release is defined as $\mathrm{\Delta }{E}_{\mathrm{topo}}=\mathrm{\Delta }E(Q_1,-Q_1)-\mathrm{\Delta }E(Q_1,Q_1)$. (b) The maximum (at the collision) temperature increase of the electron system as a function of the reduced DW width, $\mathrm{\Delta }/{\mathrm{\Delta }}_0$. The shaded area defines the topological effect $\mathrm{\Delta }{T}_{\mathrm{topo}}^{\mathrm{el}}=\mathrm{\Delta }{T}_{\mathrm{el}}(Q_1,-Q_1)-\mathrm{\Delta }{T}_{\mathrm{el}}(Q_1,Q_1)$. (c) The maximum (at the collision) temperature increase of the phonon system, as a function of the reduced DW width. The shaded area defines the topological effect $\mathrm{\Delta }{T}_{\mathrm{topo}}^{\mathrm{ph}}=\mathrm{\Delta }{T}_{\mathrm{ph}}(Q_1,-Q_1)-\mathrm{\Delta }{T}_{\mathrm{ph}}(Q_1,Q_1)$.