Biskyrmion-based artificial neuron

A biskyrmion consists of two circular subskyrmions (figure 2) both of which are magnetized along the out-of-plane direction in their center, opposite to the magnetization direction of the surrounding. In between, the magnetization $\boldsymbol{m}(\boldsymbol{r})$ has an in-plane component. Since the two skyrmions overlap partially (middle of figure 2), their helicity, characterizing the in-plane profile, must differ by π. That means two distinct types of skyrmions must form a biskyrmion, which is why a biskyrmion is disfavored by the Dzyaloshinskii-Moriya interaction and is instead stabilized by the dipole-dipole interactions that favor Bloch skyrmions of helicity $+{{\pi}}/{{2}}$ and $-{{\pi}}/{{2}}$, likewise [27].

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The two subskyrmions of the biskyrmion behave differently when we drive them by spin-orbit torques, due to their opposite helicity and corresponding in-plane magnetization profiles. For positive currents that are larger than a critical current, the two skyrmions move away from each other; see orange trajectory in figure 3(a). However, they do not move perfectly (anti-)parallel with respect to the current direction but move at an angle partially towards the edge of the sample. This effect is called skyrmion Hall effect [12, 13, 35] and will be further clarified in the next section based on the Thiele equation [36]. Once the two skyrmions reach their respective edge after $9\,\mathrm{ns}$, the motion almost stops. However, both skyrmions then begin to creep along the edge towards each other along the ±x direction. The motion along the x direction has reversed even though the current remains unchanged. This motion is much slower compared to the initial separation process and after a total $180\,\mathrm{ns}$ the two skyrmions have the same x = 0 component but are still positioned at the opposite edges with respect to the y coordinate. This configuration is a steady state under the applied constant current. However, once the current is turned off, the two skyrmions attract each other again and merge to reestablish the initial biskyrmion. Note that this will occur automatically in the neuronal operation mode where the input current is received pulsed.

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Before we continue and explain this unique trajectory in detail we want to comment on three alternative scenarios that might occur in practice. First, even if the current is turned off at any other point of the trajectory, the two skyrmions still attract each other and form the initial biskyrmion. Second, if the current has the wrong sign (or if the two subskyrmions are reversed) the biskyrmion first rotates by π, effectively exchanging the two subskyrmions. Third, if the driving current is too small, the biskyrmion will only rotate and the two individual skyrmions do not form. Therefore, as long as the driving current is large enough, these scenarios are unproblematic and it is sufficient to focus on explaining the trajectory described above.

To understand the biskyrmion motion via SOT, we use the generalized Thiele equation [36]. It is an effective description of the motion of non-collinear textures with the velocity v . The essential assumption is that the (bi-)skyrmion spin texture does not change in profile while moving, so that it can be condensed into a single point, and that the total force in the system vanishes. Since we know from the micromagnetic simulations that the biskyrmion splits up into two subskyrmions, we continue to analyze the trajectory of the skyrmions. The right skyrmion (figures 1(c) and 2) is discussed if not stated otherwise because we will consider this skyrmion for the detection later on.

The Thiele equation consists of five force terms and can be written as [40]

$$\begin{equation} b \boldsymbol{G} \times \boldsymbol{v} - b \underline{D} \alpha \boldsymbol{v} - B j \underline{I} \boldsymbol{s} - \nabla U_{\mathrm{int}}(r_{12}) - \nabla U_{\mathrm{edge}}(y) = 0. \end{equation} \tag{ 2 }$$

The constants b and B are determined from the sample parameters (see Methods section): $b = M_s d_z / \gamma_e$ and $B = \hbar \Theta_\mathrm{SH} / 2 e$ where M_s is the saturation magnetization, d_z is the thickness of the ferromagnetic sample, γ_e is the gyromagnetic ratio and $\Theta_\mathrm{SH}$ is the spin Hall angle.

The topological charge of the (bi-)skyrmion $N_\mathrm{Sk} = \frac{1}{4 \pi} \int \boldsymbol{m} \cdot (\partial_x \boldsymbol{m} \times \partial_y \boldsymbol{m} ) \, \mathrm{d}^2 r$ gives rise to the first term. This so-called gyroscopic force is characterized by the gyroscopic vector $\boldsymbol{G} = 4 \pi N_\mathrm{Sk}\boldsymbol{e}_z$. The space-dependent magnetic profile $\boldsymbol{m}(\boldsymbol{r})$ has been condensed into a single vector by integrating over the whole extent of the skyrmion. Each skyrmion has a topological charge of $N_\mathrm{Sk} = +1$. The second term is the dissipative force, quantified by the Gilbert damping α. The dissipative tensor $D_{ij} = \int ( \partial_{x_i} \boldsymbol{m} \cdot \partial_{x_j} \boldsymbol{m} ) \, \mathrm{d}^2 r$ only has non-zero $D_{xx} = D_{yy}\equiv D_0$ elements, irrespective of the type of skyrmion, as long as there is no deformation. The third term accounts for the spin-orbit torque. The injected spins s interact with the skyrmion's magnetic moments. The torque tensor $I_{ij} = \int ( \partial_{x_i} \boldsymbol{m} \times \boldsymbol{m} )_j \,\mathrm{d}^2 r$ strongly depends on the skyrmion's in-plane magnetization profile; more precisely on its helicity. For the two Bloch skyrmions with opposite helicity $\gamma = \pm{{\pi}}/{{2}}$ only the $I_{xx} = I_{yy}\equiv \lambda I_0$ components are non-zero, as long as there is no deformation. Note that $\lambda = \pm 1$ has been introduced to distinguish the two skyrmions with positve and negative helicity, respectively.

The other two terms are the interaction between the two skyrmions, quantified by $U_\mathrm{int}(r_{12})$, and the interaction of a skyrmion with the edge $U_\mathrm{edge}(y)$. If we neglect them, for now, we are able to understand why the two skyrmions move away from each other once the current is turned on. Under the above explained symmetry considerations, the Thiele equation becomes

$$\begin{equation} 0=-4\pi b \left( \begin{matrix} -v_y\\ v_x\\ \end{matrix} \right)-bD_0\alpha\left( \begin{matrix}v_x\\ v_y\\ \end{matrix} \right)- \lambda BjI_0\left( \begin{matrix}0\\1\\ \end{matrix} \right). \end{equation} \tag{ 3 }$$

Both skyrmions move at the skyrmion Hall angle $\tan\theta_\mathrm{sk} = {{v_y}}/{{v_x}} = {{D_0\alpha}}/{{4\pi}}$ with respect to the current direction x and they move along opposite directions so that the biskyrmion splits like we have seen in the micromagnetic simulations.

The two skyrmions move at that angle away from each other until they approach their respective horizontal edge. This leads to a force $\nabla U_\mathrm{edge}$ along $\lambda y$. Since this force does not have an x component, the first component of equation (3) reveals that the skyrmions can still only move at the skyrmion Hall angle. This also means that once the force from the potential compensates the forces from the spin-orbit torque at $y = y_c$, the skyrmion cannot move anymore at all $v_x = v_y = 0$. Note that if the right skyrmion was somehow displaced beyond $y\gt y_c$, it would move along the opposite direction but along the skyrmion Hall angle until it reaches $y = y_c$. This is a special feature of Bloch skyrmions compared to Néel skyrmions, which have a different $\underline{I}$ tensor symmetry. While Néel skyrmions can creep along the edges of a confined geometry [41], Bloch skyrmions always get stuck.

The only remaining force we have not discussed yet, is the skyrmion-skyrmion interaction which is attractive for all points of the observed skyrmion trajectory. Even though it is weak compared to all other interactions, it has an x component which allows the two skyrmions to leave the straight course dictated by the skyrmion Hall angle.

We qualitatively reproduced the unique trajectory (figure 3(a)) by solving the Thiele equation after calculating and fitting D₀, I₀, $U_\mathrm{int}(r_{12})$ and $U_\mathrm{edge}(\boldsymbol{r})$ with the data from our micromagnetic simulations (see Methods). This approach allows us to determine the trajectory immediately and precisely calculate the five forces individually, which helps to understand why the skyrmions reverse their direction of motion under constant current.

We note that the forces in the Thiele equation are not to be understood in the Newtonian sense since the skyrmion does not move in the direction of the forces' sum, which is zero per definition (cf equation (2)). Instead, we have to find the velocity vector, such that all five forces compensate: The reader is reminded that the spin-orbit torque and edge related forces always point along ±y. The skyrmion moves always perpendicular to the gyroscopic force and anti-parallel to the dissipation force. The skyrmion-skyrmion attraction always points towards the center of the sample $\boldsymbol{r} = 0$ since the two skyrmions move symmetrically.

In the first part of the trajectory (I in figure 3(b)), the force related to the spin-orbit torque is larger than the force from the edge (cf figure 3(c)), as explained before. This means the velocity must be oriented such that an additional gyroscopic force occurs that compensates the spin-orbit force. Since the velocity is always perpendicular to the gyroscopic force, it must have a positive x component. At the reversal point (II), the velocity has drastically decreased since it would be zero if there was no skyrmion-skyrmion interaction due to the compensation of the spin-orbit related force and the edge force, as explained before. However, due to the consideration of the skyrmion-skyrmion interaction, y_c is not a stationary position anymore. The skyrmion consequently moves along −x in (III). This is possible because the dissipative force fully compensates for the interaction force's x component. Additionally, the gyroscopic force must not deliver any x component which is only fulfilled for this direction of motion: If v is along −x, the gyroscopic force is along y. Compared to the discussion without interaction potential, the skyrmion will even move a bit further along y, thereby increasing the edge force. The gyroscopic force is oriented along y to compensate for this additional edge force. For such a high value of the y component, all five forces can only compensate each other if the velocity is oriented along −x. Upon returning towards x = 0 the skyrmion-skyrmion interaction becomes stronger. Consequently, the dissipative and gyroscopic forces must also increase so that the skyrmion speeds up.

We close this section by noting that the Thiele equation assumes a rigid skyrmion structure. However, from the micromagnetic simulation, we note that the skyrmion can be deformed, and once it approaches the edge, its size changes. Mathematically this translates into non-diagonal elements of the tensors $\underline{D}$ and $\underline{I}$ different from zero. Since these terms are very small, the idealized trajectory based on the Thiele equation does not differ qualitatively from the trajectory based on the micromagnetic simulation (figure 3(a)). However, these non-diagonal components lead to slower resetting dynamics once the skyrmion has reached the edge.

The behavior under current pulses is very similar to the situation explained above. If a continuous sequence of short current pulses is applied, the trajectory looks almost identical to the case with constant currents (figure 4(a)): The right skyrmion is pushed along the positive x and y directions according to the skyrmion Hall angle. Once it reaches its maximum x coordinate, it slowly moves back along the −x direction towards the center of the sample. However, for a neuron device it is more important to understand the behavior under limited sequences of current pulses, as will be discussed next.

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In the following, we will present the neuronal functionalities of the artificial neuron by discussing three archetypal examples. The first case (figure 4(b)) allows to discuss the LIF functionality of our device. We apply a spike train of positive current pulses corresponding to $j\Theta_\mathrm{SH} = 15\,\mathrm{MA}\,\mathrm{cm}^{-2}$ and track the skyrmion's core position. The black lines represent the input pulses, each with a duration of $2\,\mathrm{ns}$ applied every $3\,\mathrm{ns}$. Six pulses are sufficient to drive the skyrmion into the detector beginning at $x_d = 100\,\mathrm{nm}$ (red dashed line). The red line indicates the output signal fired by the device. For this first example, once the artificial neuron has fired, the input signals are turned off, and the skyrmion-skyrmion interaction resets the device. However, compared to the constant current case, the reset period is much shorter since no force is still pushing the two skyrmions apart.

In the second example (figure 4(c)), we present that the biskyrmion adds a layer of bio-fidelity for skyrmion-based artificial neurons by incorporating the refractory signature. To test for refractoriness, $4\,\mathrm{ns}$ after firing, we apply the same input-spike train again. In a device with only the LIF functionality, as in the skyrmion-based neuron from figure 1(e), this should be more than enough to trigger another firing event. For the biskyrmion case, however, as shown in figure 4(c), the same spike train does not trigger another firing event. To understand this, we refer to the dynamics under constant current. While the device is being reset, the biskyrmion has not yet formed, and the individual skyrmion has a non-zero position component along the y-axis (the colors in figure 4(c)). The skyrmion moves back on a slightly different path compared to the initial motion towards the edge (figure 4(f)). Once the second train of pulses arrives, the skyrmion moves again towards the detector but cannot reach the detector because the motion stops at a value $x\lt x_d$. This type of motion remains unless this hysteresis is resolved by reestablishing the biskyrmion.

We present this particular case in the third archetypal example (figure 4(d)). This time, the second train of pulses arrives $25\,\mathrm{ns}$ after the firing event so that the biskyrmion has already reformed. The refractory period is overcome and the neuron is fully reset.

Before we conclude, we want to discuss three details about the refractory period. (i) In our simulations, one assumption was that all input current pulses were of the same magnitude. Only under this assumption do we have an absolute refractory period during which it is impossible to make the neuron fire again (similar to the red region of the biological neuron shown in figure 1(d)). If larger input currents are allowed, the neuron can fire again. It is a relative refractory period in this case (similar to the orange region of the biological neuron shown in figure 1(d)). (ii) For the neuron to enter the refractory period, there must be at least one input pulse missing after a firing event. If current pulses keep on being received as inputs, the device will fire several times in a short sequence before it enters the refractory state (figure 4(a)). Such a signature is called 'phasic bursting' and is also present in biological neurons [42]. (iii) Already a sub-threshold input without a firing event can initiate a refractory period in our device (figure 4(e)), which is also in good agreement with the analogous biological neuron.