Step-like dependence of memory function on pulse width in spintronics reservoir computing

Figures 2(a) and 2(b) show the dynamics of the centre position of the vortex core $s(t)$ from the random binary input with pulse widths of $0.5$ $\mu$s and $3.0$ $\mu$s, respectively. Six pulses, divided by dotted lines, represent the binary data ”001100” as shown by the black line for example. The core position saturates to the value shown in Fig. 1(b) when the binary pulse of ${\rm bi}=0$ is inputted, whereas it saturates to a large value when a binary pulse of ${\rm bi}=1$ is inputted because the spin-transfer torque due to the additional current, $\nu J_{\rm dc}$, pushes the vortex core away from the disc centre. As a result, the relaxation dynamics between the two states is observed when the value of the binary pulse changes. When the pulse width is shorter than the time to saturate the vortex core position, a gradual change in the vortex-core position can be observed, as shown in Fig. 2(a). On the other hand, the vortex core remains almost constant during a pulse when the pulse width is longer than the time to saturate the vortex core position, as shown in Fig. 2(b).

As a figure of merit for reservoir computing, we evaluate the STM and PC capacities. The memory capacity characterises the number of data points the reservoir can store and is a dimensionless quantity. The STM capacity is the memory capacity for a linear combination of the input data as defined in Eq. (5) below. The PC capacity is the memory capacity for nonlinear data, where a nonlinear transformation is applied to the input data, as shown in Eq. (6) below. In general, the memory capacity depends on the sequence of the input data, therefore, it is defined as an average of the number of stored data with respect to random input.

The time-multiplexing method is applied to construct a virtual many-body system from the dynamical response of a single ferromagnet. The dynamical response $s(t)$ during a pulse is divided into $N_{\rm node}$, and $s_{k,i}$ corresponding to $s(t)$ at the $i$th node in the presence of the $k$th pulse is regarded as the $i$th neuron at a discrete time $k$ [12, 15, 17]. By applying $N=1000$ pulses and dividing the output $s(t)$ into $N_{\rm node}=250$ nodes, we determined the weight $w_{D,i}$ minimising the error between the system output and the target data given by

$$\sum_{k=1}^{N}\left(\sum_{i=1}^{N_{\rm node}+1}w_{D,i}s_{k,i}-v_{k,D}\right)^{2},$$

(4)

where the bias term is $s_{k,N_{\rm node}+1}=1$ [12]. The integer $D(=0,1,2,\cdots)$ represents the delay [12, 16, 17, 22]. Since reservoir computing calculates the time sequence of input data, the capability of a reservoir is characterised by the number of past input data stored in it. The delay $D$ characterises such past input data; for example, it shows whether the $(k-D)$th input data can be reproduced from the $k$th output. Accordingly, the delay $D$ is a quantity related to the order of the input data, and thus, is an integer. It does not relate to any time scale of physical systems, such as physical delay time. In this paper, the symbol $D$ appeared in quantities related to reservoir computing and is used as an integer represents the delay, whereas the symbol $\mathscr{D}$ that appeared in the Thiele equation represents the damping strength. The data the reservoir should reproduce is called target data or output, and is constructed from the input data. The target outputs for the STM and PC tasks are, respectively, given by

$$v_{k,D}^{\rm STM}={\rm bi}_{k-D},$$

(5)

$$v_{k,D}^{\rm PC}=\sum_{j=0}^{D}{\rm bi}_{k-D}\ \ ({\rm mod}\ 2),$$

(6)

where ${\rm bi}_{k}$ is the $k$th random binary input data.

After determining the weight, we apply different $N^{\prime}=1000$ random binary pulses as a test set and evaluate the reproducibility of the target output $v_{n,D}^{\prime}$ ($n=1,2,\cdots,N^{\prime}$) from the system output defined as

$$v_{{\rm R},n,D}^{\prime}=\sum_{i=1}^{N_{\rm node}+1}w_{D,i}s_{n,i}^{\prime},$$

(7)

where $s_{n,i}^{\prime}$ is the vortex-core position in the presence of $N^{\prime}$ random pulses. Figure 2(c) shows the target output (red) and the system output (blue) for the STM task with a pulse width of $3.0$ $\mu$s, where the delay varies as $D=1$, $2$ and $3$ from left to right. The system output well reproduces the target output when the delay is small, whereas the reproducibility becomes low as the delay increases. The reproducibility is quantitatively characterised by the correlation coefficient ${\rm Cor}(D)$ between the target output and system output defined as

$$\begin{split}&{\rm Cor}(D)\equiv\frac{\sum_{n=1}^{N^{\prime}}\left(v_{n,D}^{\prime}-\langle v_{n,D}^{\prime}\rangle\right)\left(v_{R,n,D}^{\prime}-\langle v_{{\rm R},n,D}^{\prime}\rangle\right)}{\sqrt{\sum_{n=1}^{N^{\prime}}\left(v_{n,D}^{\prime}-\langle v_{n,D}^{\prime}\rangle\right)^{2}\sum_{n=1}^{N^{\prime}}\left(v_{{\rm R},n,D}^{\prime}-\langle v_{{\rm R},n,D}^{\prime}\rangle\right)^{2}}},\end{split}$$

(8)

where $\langle\cdots\rangle$ is the averaged value. Figures 3(a) and 3(b) show the square of the correlation coefficients for the STM and PC tasks, respectively, at pulse widths of $t_{\rm p}=0.5$, $3.0$ and $5.0$ $\mu$s. The correlation coefficient becomes small as the delay $D$ increases and becomes nearly zero as shown in Fig. 3. This is also indicated in Fig. 2(c) where the reproducibility of the input data decreases as the delay increases.

The STM and PC capacities, denoted as $C_{\rm STM}$ and $C_{\rm PC}$, respectively, are defined as

$$C=\sum_{D=1}^{D_{\rm max}}\left[{\rm Cor}(D)\right]^{2}.$$

(9)

The maximum delay $D_{\rm max}$ in this work is set to $20$, which is sufficient to evaluate the saturated values of the capacities. We also note that the number of the training data, $N=1000$, is also sufficient to saturate the values of the capacities; see Supplemental Information. Figure 3(c) shows the dependences of $C_{\rm STM}$ (red) and $C_{\rm PC}$ (blue) on the pulse width. It should be emphasised that both $C_{\rm STM}$ and $C_{\rm PC}$ show step-like behaviour with respect to the pulse width. When the pulse width is relatively short ($\lesssim 2.0$ $\mu$s), the capacities remain at large values over $2.0$. In a middle range of the pulse width ($2.0\lesssim t_{\rm p}\lesssim 4.5$ $\mu$s), on the other hand, the capacities suddenly drop to $1.5$. The capacities again drop to $1.0$ when the pulse width becomes longer ($t_{\rm p}\gtrsim 4.5$ $\mu$s).

The step-like behavior of the capacities in Fig. 3(c) can be attributed to the current-dependent relaxation phenomenon of the vortex core. To explain this point, we first focus on an analytical solution of the Thiele equation. Neglecting the small spin-transfer torque due to the in-plane component of the reference layer’s magnetisation and higher order terms of $s$ ($|s|\leq 1$) and $\alpha$ ($\alpha\ll 1$), the Thiele equation for the vortex centre position $s$ becomes

$$\dot{s}=as-bs^{3},$$

(10)

where $a$ and $b$ are defined as

$\displaystyle a=\frac{|\mathscr{D}|\kappa}{G^{2}}\left(\frac{Ga_{J}Jp_{z}}{|\mathscr{D}|\kappa}-1\right),$

$\displaystyle b=\frac{|\mathscr{D}|\kappa}{G^{2}}\left(\xi+\zeta\right).$

(11)

Equation (10) is the Stuart-Landau equation [40, 41] for the real variable $s$. The vortex core is stabilised at the disc centre when $a<0$, whereas a limit-cycle oscillation appears when $a>0$. The solution of Eq. (10) for $a>0$ with the initial condition $s(t=0)=s(0)$ is given by

$$s(t)=\frac{s(0)e^{at}}{\sqrt{1+s(0)^{2}(b/a)(e^{2at}-1)}}.$$

(12)

The solution saturates to $\lim_{t\to\infty}s(t)=\sqrt{a/b}$. The normalised core position should satisfy $0\leq s\leq 1$. Therefore, the condition to excite the limit cycle oscillation is $J_{\rm c1}<J<J_{\rm c2}$, where $J_{\rm c1}=(|\mathscr{D}|\kappa)/(Ga_{J}p_{z})$ and $J_{\rm c2}=J_{\rm c1}(1+\xi+\zeta)$ are determined by the conditions of $a>0$ and $a/b<1$, respectively. We note that the saturated core position depends on the disc radius $R$ through the parameter $\kappa\propto 1/R$. The parameter $\kappa$ determines the magnetic potential energy of the vortex core, as shown in Eq. (2), and becomes small as the disc radius $R$ increases because, in a large disc, a tiny displacement of the vortex core does not change the magnetic energy significantly. Since the spin-transfer torque should overcome the damping torque, which helps keep the vortex core close to the disc centre to minimise the magnetic energy, in driving the vortex-core dynamics, the critical current density $J_{\rm c1}$ is proportional to the parameter $\kappa$. Accordingly, the saturated core position $\sqrt{a/b}$ at a given current density $J$ depends on the disc radius $R$ through the parameter $\kappa$ in the critical current density. Roughly speaking, the saturated core position becomes large as the disc radius $R$ increases because the critical current density becomes small for a large $R$.

Equation (10) indicates that the time scale for the exponential evolution of the vortex core, given by $1/a$, depends on the input data through the current density; a large current density results in a fast relaxation. For example, $1/a$ for the current density $J$ with ${\rm bi}=0$ is $0.36$ $\mu$s for the present parameters, whereas it becomes $0.14$ $\mu$s when ${\rm bi}=1$.

Before proceeding to the discussion, we give a brief comment on the relation between the parameter $1/a$ and the time scale for reservoir computing. Reservoir computing estimates the information of the past input from the temporal response of the reservoir. Therefore, the relaxation phenomenon is key to performing physical reservoir computing. The parameter $1/a(\simeq 0.36\ {\rm or}\ 0.14\ \mu{\rm s})$ characterises the exponential relaxation of the vortex core. We, however, note that nearly $e^{-1}\simeq 0.37$, i.e., $37\%$ of the relaxation process to the saturated value $\sqrt{a/b}$ is not completed even after a time scale of $1/a$ has passed. The vortex dynamics, such as those at the last part of the relaxation process, can also be used in reservoir computing. As a result, the time scale of the relaxation dynamics of the vortex core applicable to reservoir computing is longer than $1/a$. For example, the pulse width ($\sim 4.5$ $\mu$s) at which the capacities drop from $1.5$ to $1.0$ shown in Fig. 3(c) is not exactly the same with the parameter $1/a$, although such a drop of the capacities is related to the relaxation phenomenon of the vortex core, as explained below. Therefore, we use the word ”relaxation time” in the following as a time scale to determine the performance of reservoir computing. However, defining the relaxation time quantitatively is difficult because it depends on the number of significant figures in the calculations. We also note that the relaxation time depends on the current magnitude, as in the case of the parameter $1/a$, because of the current-dependent relaxation dynamics. If the relaxation time is faster than the pulse width, the system rapidly relaxes to a steady state and loses the history of the time-series data. In other words, the pulse width should be shorter than the relaxation time of the reservoir. An excessively short pulse width is, however, also not preferable to reservoir computing because the change in dynamical response with respect to a pulse input becomes small for a short pulse width.

Regarding these points, the identification of the data inputted one sequence before the present pulse can be explained as follows. As an example, let us assume that the vortex core remains in a steady state under a binary pulse of ${\rm bi}=0$. For convenience, we name this pulse as the $(k-1)$th pulse. If the next ($k$th) pulse is ${\rm bi}=0$, the vortex state is unchanged, as shown in Fig. 4(a). On the other hand, the vortex core changes the position if the next pulse is ${\rm bi}=1$, as shown in Fig. 4(b). In other words, the response with respect to the $k$th pulse is affected by the $(k-1)$th pulse. Therefore, the value of the $(k-1)$th pulse can be estimated from the response under the $k$th pulse, corresponding to $[{\rm Cor}(D=1)]^{2}=1$. This fact results in capacities of $1.0$ in a long pulse-width limit shown in Fig. 3(c).

Another saturated value, $1.5$, of the capacities in the middle pulse-width range is explained in a similar way, by taking into account the current-dependent relaxation time. Let us consider the injection of three pulses, named as the $(k-2)$th, $(k-1)$th and $k$th pulses, as shown in Figs. 4(c) and 4(d). The solid and dotted lines in the figures show the vortex dynamics when the $(k-2)$th and $(k-1)$th binary pulses are different and the same, respectively. In Fig. 4(c), the $(k-1)$th pulse is ${\rm bi}=1$. In this case, the vortex core saturates rapidly, and thus, the vortex core is in a steady state at the end of the $(k-1)$th pulse, independent of the $(k-2)$th pulse. This fact indicates that the $(k-2)$th input data cannot be identified from the vortex dynamics under the presence of the $k$th input. On the other hand, when the $(k-1)$ th pulse is ${\rm bi}=0$, the vortex dynamics under the presence of the $k$th pulse depends on the $(k-2)$th pulse. This is because the relaxation time is slow, and thus, the vortex state at the end of the $(k-1)$th pulse reflects the state under the $(k-2)$th pulse, as shown in Fig. 4(d). As a result, the $(k-2)$th input can be identified when the $(k-1)$th pulse is ${\rm bi}=0$. Therefore, the reproducibility of the $(k-2)$th data is $50\%$, resulting in $[{\rm Cor}(D=2)]^{2}=0.5$ and the capacity value of $1.5$. As can be seen in this explanation, the step-like behaviour of the memory function in Fig. 3(c) can be attributed to the current-dependent relaxation mechanism of the vortex dynamics.

The role of the relaxation time of the vortex core on the memory function for reservoir computing can be discussed from a different viewpoint. Figure 5(a) shows the dynamics of the centre position of the vortex core in the presence of six pulses with widths of $0.5$ $\mu$s. The situation is the same with that shown in Fig. 3(a), but here, we use a relatively weak input pulse characterised by $\nu=0.05$. Because of the small input-pulse strength $\nu$, the change of $s(t)$ with respect to the input current is small compared with that shown in Fig. 2(a).

Figure 5(b) shows the dependences of the STM and PC capacities on the pulse width for $\nu=0.05$. Note that we argue above that the value of $1.5$ in the memory capacity reflects the difference in the relaxation times between two states. Due to the small parameter $\nu$, the relaxation time from the ${\rm bi}=0$ to ${\rm bi}=1$ state becomes close to that of the opposite case, compared with the system studied in the previous sections. As a result, the reservoir can distinguish both signals in a relatively large pulse-width range. Therefore, the range corresponding to the value of $C=2$ for the memory capacities is enlarged, compared with that found in Fig. 3(c) in the main text, and the step-like behavior is observed even in a short pulse-width range. In addition, the pulse width at which the jump from $C=2$ to $C=1.5$ occurs is shifted to a long pulse-width region. The results also indicate that the step-like behaviour originates from the current-dependent vortex relation, which is controlled by the parameter $\nu$. In Supplemental Information, we also provide the data showing the STM and PC capacities for various values of $\nu$, which also support the argument here.