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\title{Skyrmion Signal Reshuffler ~ ~ ~\\}
\author[ ]{Daniele Pinna}
\author[ ]{Awaiting Activation}
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\date{}
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\section*{ABSTRACT}\label{auto-label-section-381316}
The topologically protected magnetic spin configurations known as
\emph{skyrmions} offer promising applications due to their stability,
mobility and localization. In this work, we propose a skyrmion
gas-mediated signal reshuffler within the well-established framework of
micromagnetics, through which a detailed modelling of the long-range
skyrmion-skyrmion interactions paired with their individual thermal
diffusion is used to transform a telegraph noise signal into an
uncorrelated copy of itself. Our results serve as a proof-of-concept for
a compact device~inheriting all the scaling and low-energy advantages
afforded by the spintronics toolbox. Whereas its immediate application
to stochastic computing circuit designs will be made apparent, we argue
that its basic functionality, reminiscent of an integrate-and-fire
neuron, qualifies it as a novel bio-inspired building block.~\\
\section*{INTRODUCTION}\label{introduction}
Magnetic skyrmions promise unique opportunities for the processing,
storage and transfer of information by means of ultrathin metallic
nanostructures at the intersection of both spintronics and
nanoelectronics
{[}\cite{Fert_2013},~\cite{Nagaosa_2013},~\cite{Sampaio_2013},\cite{Iwasaki_2013},
\cite{Zhang_2015},~\cite{Koshibae_2015}{]}. ~They appear ubiquitously in
diverse systems whenever the competition between Dzyaloshinski-Moriya
interaction (DMI) and other magnetic energy~contributions result in an
equilibrium spin texture that is strongly adverse to deformations. As a
result, skyrmions are exceptionally stable structures capable of
withstanding room-temperature environments
{[}\cite{Jiang_2015},~\cite{Woo_2016}{]} and being manipulated at
extremely small current densities ( \selectlanguage{english}\(\sim 10^{6}\mathrm{A}/\mathrm{m}^2\)) with
negligible ohmic heating as compared to domain walls
{[}\selectlanguage{english}\cite{Jonietz_2010}, \selectlanguage{english}\cite{Yu_2012}, \selectlanguage{english}\cite{Parkin_2008},
\selectlanguage{english}\cite{Schulz_2012}{]}.~\\
The appeal of skyrmions is so wide that it has defined a field of its
own,~\emph{skyrmionics}, which refers to the emerging technologies based
on magnetic skyrmions as information carriers. In an effort to push for
the commercialization of skyrmion electronics, challenges ranging
from~their~creation and annihilation
{[}\cite{Nagaosa_2013},\cite{Zhang_2015a},\cite{Tchoe_2012}{]}, the
conversion of their topological properties
{[}\cite{Zhou_2014},~\cite{Zhang_2015}{]}, as well as their
efficient transmission and read-out
{[}\cite{Nagaosa_2013},\cite{Iwasaki_2013},\cite{Sampaio_2013},\cite{Tomasello_2014},\cite{Koshibae_2015}{]}
are being tackled and solved. Their nano-metric size and high mobility
{[}\cite{Muhlbauer_2009},\cite{Lin_2013},\cite{Yu_2010},\cite{Fert_2013}{]}
has successfully been exploited in the context of skyrmion-based
racetrack memories {[}\cite{Zhang_2015a},\cite{Kang_2016}{]}, logic
gates {[}\cite{Zhang_2015}{]} and as voltage-gated transistors\selectlanguage{english}
{[}\cite{Zhang_2015b}{]}.\\
In this paper, we~explicitly address the problem~of leveraging the
repulsions exhibited by interacting skyrmions with their thermal
diffusion to demonstrate how skyrmions may be employed to perform
probabilistic computing. Particularly we will show how a very compact
device can be modelled for the reshuffling of stochastic telegraph noise
signals and random number generation. In conclusion, we will argue how
the device in question, subject to trivial modifications, can serve a
much more general purpose by effectively working as an anlog
integrate-and-fire neuron. We believe this lays the groundwork for the
application of skyrmionic devices as bio-inspired building blocks for
non-conventional computing paradigms.\\
\section*{MODEL}\label{model}
\subsection*{Isolated Skyrmion Dynamics}
The motion of an isolated skyrmion in a two dimensional film can~be
described by a modified version of Newton's equation which tracks the
coordinate evolution of the skyrmion's center. For forces too weak to
alter the skyrmion profile significantly, a symmetry analysis imposing
translational and rotational invariance suggests the following
\emph{Thiele-}equation of motion
{[}\cite{Everschor_2011},\cite{Clarke_2008},\cite{Thiele_1973},\cite{Iwasaki_2013}{]}:\\
\begin{eqnarray}
&\mathbf{\hat{G}}^{\alpha}\cdot\mathbf{\dot{x}}=\mathbf{F}_C+\mathbf{F}_{\mathrm{Th}}\label{eqn:Evo}\\
&\hat{G}^{\alpha}_{ij} = G\,\epsilon_{ji}+\alpha D\,\delta_{ij}\label{eqn:G}
\end{eqnarray}
where $\alpha$ is the damping constant, $\mathbf{\hat{G}}$ the gyrotropic matrix, and $\mathbf{F}_C$, $\mathbf{F}_{\mathrm{Th}}$ are forces due to applied electric currents and thermal fields. Lastly, $\epsilon_{ij}$ and $\delta_{ij}$ are, resepctively, the Levi-Civita tensor and Kr\"{o}necker delta and time is in units of $\gamma M$, where $\gamma$ is the gyromagnetic ratio and $M$ the magnetic texture's local moment magnitude. Under the assumption of an invariant skyrmion profile, the gyrotropic matrix (\ref{eqn:G}) encodes the gyrocoupling $G$, a topological invariant arising from the unique {\it twist} in the spin-texture, and the dissipative diadic $D$ which, together with $\alpha$, characterizes the net friction acting on the skyrmion [\cite{Everschor_2012}]. Both are typically computed explicitly from the static magnetic profile:
\begin{eqnarray}
G &=& \int\mathrm{d}\mathbf{r}\,\mathbf{n}\cdot\left(\partial_x\mathbf{n}\times\partial_y\mathbf{n}\right)\label{eq:G}\\
D &=& \int\mathrm{d}\mathbf{r}\,\left(\partial_x\mathbf{n}\cdot\partial_x\mathbf{n}+\partial_y\mathbf{n}\cdot\partial_y\mathbf{n}\right)/2,\label{eq:D}
\end{eqnarray}
where $\mathbf{n}$ is the local, unit-normalized, magnetization orientation.
Both the current and thermally induced forces have their origin in the microscopic magnetization dynamics. The conduction of electrons through the spin-texture is known to induce both an adiabatic and non-adiabatic spin-torque [\cite{Tatara_2008}] on the local magnetization which, in the collective coordinate limit, gives rise to well-defined force components acting on the skyrmion profile [\cite{Schulz_2012},\cite{Everschor_2012}]. The net force acting on the skyrmion can thus be written as:
\begin{equation}
\mathbf{F}_C = \mathbf{F}_S+\mathbf{F}_{\mathrm{boundary}}+\mathbf{\hat{G}}^{\beta}\cdot\mathbf{v}_s,
\end{equation}
where $\mathbf{F}_S$ and $\mathbf{F}_{\mathrm{boundary}}$ are, respectively, a skyrmion-skyrmion and boundary repulsion which will be discussed shortly, $\mathbf{v}_s$ is the spin-drift velocity (directly proportional to the current density $j$) and $\beta$ is a small, dimensionless, material constant characterizing the non-adiabatic spin-torque effect. For highly symmetric skyrmion profiles moving on defect-less materials, $\beta=\alpha$ and, consequently, $\mathbf{\hat{G}}^{\beta}=\mathbf{\hat{G}}^{\alpha}\equiv\mathbf{\hat{G}}$ resulting in a skyrmion motion which will exactly follow the external current. In all other circumstances, the discrepancy between $\alpha$ and $\beta$ will lead to current-induced skyrmion flows which proceed at an angle to the current-induced force driving them.
Analogously, the thermal forces present in (\ref{eqn:Evo}) result from the collective action of independent random magnetic fields acting on the texture's local magnetic moments. The net effect results in an additive, homogeneous mean-zero stochastic term to the Thiele dynamics [\cite{Mertens}]:
\begin{eqnarray}
\langle\mathbf{F}_{\mathrm{Th}}\rangle &=& 0\\
\langle F_{\mathrm{Th},i}(t) F_{\mathrm{Th},j}(t')\rangle &=& \frac{k_BT}{\gamma M}\frac{\alpha D}{G^2+(\alpha D)^2}\delta_{i,j}\delta(t-t'),
\end{eqnarray}
where $k_BT$ is the thermal energy and $\langle\cdot\rangle$ represents averaging over noise realizations.
\subsection*{Skyrmion-Skyrmion Interactions}
Short range repulsions between skyrmions due to deformations in their reciprocal spin textures have already been theoretically and numerically discussed in the literature in the absence of magnetic dipole effects [\cite{Lin_2013}]. However, dense skyrmion populations are known to exhibit long range order leading to the formation of regular lattice structures [\cite{Muhlbauer_2009},\cite{Woo_2016},\cite{Heinze_2011}] for which dipole may play a relevant role. To capture the net sum of these effects on a single skyrmion's Thiele dynamics, we introduce a force term $F_{S}$ in (\ref{eqn:Evo}) which sums all the two-body interactions among skyrmions in a given ensemble (in the vein of [\cite{Lin_2013}]):
\begin{eqnarray}
\mathbf{F}_{S}&=&\sum_{i,j} \mathbf{F}_{S-S}(\mathbf{d}_{ij})\label{eq:addforce}\\
\mathbf{F}_{S-S}(\mathbf{d})&=&\exp\left[-\frac{a_1d^2+a_2d+a_3}{d+a_4}\right]\mathbf{\hat{d}}\label{eq:interparticle}
\end{eqnarray}
where $d_{ij}$ is the distance between particles $i,j$. The specific exponential form was chosen such that the repulsion behave gaussian-like at short range and and scale like a simple exponential at distances greater than the typical skyrmion diameter. To fit the phenomenological parameters $a_k$, we performed micromagnetic simulations of two interacting skyrmions absent of thermal and applied current effects (see Methods for details). Upon using (\ref{eq:G}-\ref{eq:D}) to compute the gyrotropic matrix elements (see Figure \ref{fig:TwoSkx_GD}) from the evolving magnetic profiles (where $G\equiv-4\pi$ due to topological constraints), the repulsion force between the two particles was extracted by computing $\mathbf{\hat{G}}\cdot\mathbf{\dot{x}}$ as the simulation progressed. Figure \ref{fig:TwoSkx} shows the particle trajectories (left) along with the extracted repulsion force as a function of inter-particle distance (right).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fig1/Fig1}
\caption{{Two skyrmions, initially separated $50\,n\mathrm{m}$, are allowed to evolve micromagnetically for $140\,n\mathrm{s}$ in the absence of thermal noise and applied currents. Their extracted trajectories (left) are employed to compute the inter-skyrmion repulsion force intensity (right), plotted as a function of the average observed skyrmion diameter $\sim 60\,n\mathrm{m}$, which was then fit phenomenologically according to equation (\ref{eq:interparticle}). Over the course of the simulation, the two skyrmions are seen to spiral away from each other as a direct consequence of the gyrotropic terms appearing in the Thiele-dynamical model. The initial scatter of the extracted force is an artifact of the magnetic relaxation proceeding from an initially imposed discontinuous texture towards its equilibrium configuration. \label{fig:TwoSkx}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fig2/Fig2}
\caption{{Two skyrmions, initially separated $50\,n\mathrm{m}$, are allowed to evolve, via micromagnetic simulation, for $140\,n\mathrm{s}$ in the absence of thermal noise and applied currents. The topological constants are shown as a function of time. After a very brief relaxation phase, both the gyrocoupling $G$ (left) and the dissipative dyadic $D$ (right) stabilize around fixed values. Numerical deviations from constancy are due to the limited magnetic texture that was considered when performing integrals (\ref{eq:G} and \ref{eq:D}). Between the initially imposed magnetic texture and its steady-state configuration, one can observe a $20\,n\mathrm{s}$ relaxation phase during which the gyrotropic terms reach their equilibrium values. \label{fig:TwoSkx_GD}%
}}
\end{center}
\end{figure}
\subsection*{Boundary Effects}
The radial size of isolated skyrmions in magnetic dots is known to be confined by an explicit condition on the orientation of magnetic moments at the boundary [\cite{Rohart_2013}]. This effect, a direct result of DMI in the material, is also responsible for repelling skyrmions from boundaries guaranteeing the transport properties which make them so useful for applications such as skyrmion-based memories [\cite{Sampaio_2013}]. Since skyrmions in rarefied ensembles (as opposed to lattices) are capable of moving about freely, their dynamics might eventually lead them close enough to their sample boundary where their behavior must be somehow quantified if we wish to capture it through the Thiele formalism.
We repeat our previous phenomenological analysis to model boundary effects by extracting the net force experienced by a solitary skyrmion initially placed $50 n\mathrm{m}$ from the boundary (see Figure \ref{fig:Boundrep}). The force experienced by the skyrmion results in a net drift both along and away from the dot's boundary (left) whose scaling behavior as a function of distance from the boundary is distinctly different from that observed for skyrmion-skyrmion repulsions.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fig3/Fig3}
\caption{{Net force experienced by a solitary skyrmion initially nucleated at a distance of $d=50\,n\mathrm{m}$ from the boundary of a $1024\,n\mathrm{m}$-diameter circular geometry and allowed to evolve micromagnetically for $1\,\mu\mathrm{s}$. Over the course of the simulation, the skyrmion's trajectory (left) is seen to move - clockwise - both along and away from the magnetic boundary (thick black curve) as a result of the gyrotropic effects captured by the Thiele-dynamical model. The red line (right) shows a cubic gaussian fit to the extracted force. The initial scatter of the extracted force is an artifact of the magnetic relaxation proceeding from an initially imposed discontinuous texture towards its equilibrium configuration. \label{fig:Boundrep}%
}}
\end{center}
\end{figure}
\subsection*{Gas Dynamics}
Having modelled both two-particle and boundary repulsions, we then proceeded to verify our model's assumptions by repeating the same numerical procedures for larger skyrmion numbers to confirm that effective forces perceived by single skyrmions can indeed be reconduced to an {\it n}-body sum of individual two-particle repulsions plus a boundary interaction term. We considered 30 skyrmions initially placed randomly inside a magnetic dot ($1024\,n\mathrm{m}$ diameter) and allowed them to evolve micromagnetically for $140\,n\mathrm{s}$ (see \ref{fig:20Thiele}).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fig5/Fig5}
\caption{{\label{fig:20Thiele} 30 skyrmions, initially placed randomly inside a $1024\,n\mathrm{m}$-diameter circular geometry, are allowed to evolve micromagnetically for $140\,n\mathrm{s}$ in the absence of thermal noise and applied currents. The topological constants are shown as a function of time. After a very brief relaxation phase, both the gyrocoupling $G$ (left) and the dissipative dyadic $D$ (right) stabilize around fixed values. Numerical deviations from constancy are due to the limited magnetic texture that was considered when performing integrals (\ref{eq:G} and \ref{eq:D}).%
}}
\end{center}
\end{figure}
Due to the effect originating directly from the DMI, the resulting interaction on skyrmions will necessarily be in the form of a short-range repulsion from the boundaries. Furthermore, considering a large enough skyrmion ensemble, only the particles closest to boundaries will perceive its effect as all the others will be screened from it. We verify this argument by simulating 30 randomly scattered skyrmions in a $1024\,n\mathrm{m}$-diameter nanodot micromagnetically in an attempt to reproduce the results from the previous section. As discussed for the two- and one-particle studies, upon extracting the particle trajectories and gyrotropic constants, we can reconstruct the net force experienced by each skyrmion (see Figure \ref{25Force_NOb}) and compare it to our phenomenological inter-particle force (\ref{eq:addforce}). The skyrmions shown exemplify the importance of the boundary interaction term to properly account for the entire effective force acting on them. Upon including this correction $\mathbf{F}_{\mathrm{Boundary}}$ to our fits for the skyrmion forces experienced in the 30-particle ensemble, we obtain a significant better fit to the forces experienced by particles close to the boundary (see Figure \ref{25Force_YESb}).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fig6/Fig6}
\caption{{\label{25Force_NOb} The extracted trajectories of 30 skyrmions initially placed randomly inside a $1024n\mathrm{m}$-diameter circular geometry and allowed to evolve micromagnetically for $140n\mathrm{s}$ in the absence of thermal noise and applied currents were used, in conjunction with their extracted gyrotropic constants (see Figure \ref{fig:20Thiele}), to compute the effective forces acting on each particle. In the figure, the extracted forces of two such skyrmions are shown (identified by a red dot in each inset) and compared to the additive two-skyrmion fit (\ref{eq:addforce}) in the absence of a boundary interaction term to exemplify the importance of all the effects discussed in the text.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fig7/Fig7}
\caption{{\label{25Force_YESb} The extracted trajectories of 30 skyrmions initially placed randomly inside a $1024n\mathrm{m}$-diameter circular geometry and allowed to evolve micromagnetically for $140n\mathrm{s}$ in the absence of thermal noise and applied currents were used, in conjunction with their extracted gyrotropic constants (see Figure \ref{fig:20Thiele}), to compute the effective forces acting on each particle. In the figure, the extracted forces of two such skyrmions are shown (identified by a red dot in each inset) and compared to the additive two-skyrmion fit (\ref{eq:addforce}) plus boundary correction term (Figure \ref{fig:Boundrep}).%
}}
\end{center}
\end{figure}
\section*{Device}
\subsection*{Motivation}
As feature sizes in semiconductor systems are scaled down to the point where their deterministic behavior cannot be guaranteed, stochastic computing seeks to embrace probabilistic computing elements capable of achieving tremendous gains in signal processing efficiency at the cost of degradation in computational precision. Instead of defining operations as precise bitwise manipulations of stored values, one instead aims to encode numerical values in the statistical properties (ratio of up-time to signal length, also known as the {\it p-value}) of a random telegraph noise signal and operate on such signals in physical time via cascaded logic gates. The output of such operations will then also be a telegraph noise signal whose statistical properties can then be read out accordingly.
Cascading gates operating on random signals sequentially will however propagate unwanted correlations very rapidly even after a few elementary operations. It is therefor crucial to be able to regularly reshuffle the signals so that they stay uncorrelated. We call this element a stochastic reshuffler, capable of {\it copying} an input stream by producing an uncorrelated random stream coding for an identical p-value. As of present, no efficient stochastic reshufflers are known to exist. In CMOS, reshuffling signals can be done by combining a pseudo random number generator with a shift register or counter (both requiring long term memories). This means that each reshuffling operation has a large surface imprint, and consumes a lot of energy. It is therefore impossible to insert these reshufflers after each calculation stage in a stochastic computing circuit, preventing the realization of any large scale demonstration of stochastic computing on chip.
In this section we will propose a device capable of overcoming such crucial obstacles in stochastic computing.
\subsection*{Skyrmion Reshuffler}
The Skyrmion Reshuffler (see diagram in Figure \ref{Reshuffler}) consists of multiple chambers [MATERIALS HERE??] - two are considered for simplicity - with input/output conduits through which current flows capable of transporting nucleated skyrmions both into and out of the chamber. Skyrmions can be nucelated either singularly via injection of localized spin-currents with magnetic tunnel junctions (MTJ) [\cite{Romming_2013}] or by rapid proliferation with appropriately constructed input conduit geometries [\cite{Jiang_2015}]. For the purpose of our proof-of-concept, however, it will suffice that each chamber in our device presents both dedicated skyrmion injection and detection elements at the input and output conduits resepctively. For later convenience, we also include a reading element {\it inside} the chamber as well as a voltage-gate capable of impeding the skyrmion flow [\cite{Kang_2016}] to the output conduit if activated.
A bit-stream or any generic two-state telegraph noise signal can effectively be thought of as the combination of two separate spike trains coding the activated and ground state transitions. We employ these to selectively nucleate skyrmions to the input conduit of one of the two chambers depending on which transition the incoming signal codifies (up-to-down or down-to-up). Steady current flows will ensure a net drift of the skyrmions across the entire chamber whereupon they will be read once the output conduit is reached, and used to reconstruct the output bit-stream. If the chamber is significantly larger than the transverse size of the input/output conduit tracks, current densities inside the chamber may become small enough, allowing skyrmions to interact and diffuse thermally before reaching the output. By virtue of these interactions, skyrmion particle number conservation will allow for the reconstruction of an uncorrelated output signal with identical ratios of time spent in the active and ground states.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.42\columnwidth]{figures/Nchamber/Nchamber}
\caption{{\label{Reshuffler} The proposed device consists of two magnetic chambers into which skyrmions are injected depending on the state of an input telegraph noise signal. The net drift of the skyrmion particles due to a constant current flow along with the thermal diffusion in the chambers will lead to an exit order that can be significantly different from that of entry. This is employed to reconstruct a new outgoing signal with the same statistical properties as the first as well as being uncorrelated from it.%
}}
\end{center}
\end{figure}
We simulate the device by considering two identical circular $1024\,n\mathrm{m}$ chambers into which we inject skyrmions every $50\,n\mathrm{s}$ from $100\,n\mathrm{m}$-wide input conduits with currents of varying intensity. The skyrmions are injected into one chamber or the other depending on the activation state of a $0.5\,m\mathrm{s}$-long, randomly generated, telegraph noise signal and diffuse in the chamber due to a $300\,K$ thermal field as well as the inter-particle interactions modeled in the previous sections. The skyrmions are then {\it read} at the output conduit in the order they arrive and used to reconstruct an output telegraph noise signal. The output signal's correlation to the input is then checked by computing the product-moment correlation coefficient of the two signals whose result we show in Figure \ref{ReshufflerSim}.
As expected, the correlation coefficient decreases rapidly with the current intensity since the skyrmions are given more time to diffuse and scatter in the chamber before reaching the output conduit (Figure \ref{ReshufflerSim}-left). In all cases considered, from the moment the output signal reconstruction commences, the correlation reaches its steady state value on a $\mu\mathrm{s}$ timescale (Figure \ref{ReshufflerSim}-right).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fig9/Fig9}
\caption{{\label{ReshufflerSim} Simulation of the Skyrmion Reshuffler operating at a temperature of $300\,K$ employing two identical chambers ($1024\,n\mathrm{m}$ diameter) to scramble randomly generated telegraph noise ($0.5\,m\mathrm{s}$-long) input streams for different current intensities as measured at the $100\,n\mathrm{m}$-wide conduits. Figure shows both the final product-moment correlation coefficient at each current intensity (left) as well as its temporal evolution as the output stream is generated (right).%
}}
\end{center}
\end{figure}
\subsection*{Device Limits}
Ideally, one would like the signal scrambler to work as fast as possible. The speed with which an output signal can be sampled will depend directly on the intensity of the driving current moving the injected skyrmions through the chambers. However, as shown in Figure \ref{ReshufflerSim}, higher currents lead to highly correlated output signals due to the fact that skyrmion particles do not spend enough time diffusing and interacting in the chambers.
Increasing the system temperature may seem like an obvious approach to increasing diffusion but is untenable due it implying a higher energetic consumption as well as potentially destroying the topological stability of the skyrmion particles and, as such, the particle number conservation which the device's functioning relies on.
A potential improvement can theoretically be found in anti-ferromagnetic (AFM) materials where the possibility of stabilizing AFM skyrmions has recently been suggested [\cite{Barker_2016},\cite{Zhang_2016}]. At the core, the only difference in the material parameters between AFM and FM devices lies in the sign of the exchange interaction. This difference however has large repercussions on the Thiele modeling of skyrmions in the two systems. Particularly, it leads to a vanishing gyrocoupling $G\equiv 0$ implying two very important consequences. Firstly, the AFM skyrmion will always have zero transverse velocity relative to the direction of current flow. This is true in FM skyrmions exclusively when $\alpha\equiv\beta$ in the case of perfectly symmetric magnetic profiles. Secondly, and most importantly however, a vanishing gyrocoupling will imply a larger diffusion constant as can be seen by inspecting the denominator of (\ref{eq:D}). Depending on the value of the damping constant $\alpha$ in the materials employed ($\alpha\sim 0.01-1$), the difference in magnitude of the diffusion constant in FM and AFM skyrmions can easily span an order of magnitude difference. This in turn would allow operation of our proposed device at higher currents and faster speed.
\section*{Discussion}
The device presented seems solely capable of uncorrelating stochastic signals. Its operational principles can, however, be tuned for it to perform different tasks. The first obvious one is to scramble a regular input signal with fixed {\it p-value}, thus generating a telegraph noise signal which can successively be fed into a stochastic computing architecture. In this sense, the signal scrambler allows for a direct encoding of p-values in telegraph signals.
Another truly novel application can be surmised by leveraging the particle conservation properties of skyrmionic ensembles. If the passage of skyrmions through the output conduit were to be precluded (such as with a voltage gate), skyrmions would progressively accumulate inside their respective chambers. Each chamber can then be thought of as a reservoir collecting the memory of whatever input signal was employed to populate it. Furthermore, an upper limit exists to how many skyrmions may be crammed into each such chamber. In fact, as skyrmions become compressed with their neighbors to length scales comparable to the typical skyrmion diameter, the strain placed on the individual magnetic profiles of each skyrmions may cause them to annihilate pairwise. The maximal particle density inside a chamber can hence be thought of as a saturation threshold beyond which memory of the input signal will certainly be destroyed.
Another source of potential skyrmion annihilation already discussed is due to thermal effects. The topological stability of the skyrmion profile is only guaranteed under continuous deformations of the magnetic texture. The introduction of a thermal field in the magnetization dynamics can hence make a skyrmion collapse with a certain probability. As such, room temperature skyrmions can potentially annihilate (and nucleate) simply due to thermal fluctuations. This also would lead to an effective loss of memory of the input signal used to populate the chamber.
The combination of memory due to particle conservation and memory-loss from annihilations allow one to view our proposed skyrmionic chamber as a lossy memory storage device which integrates an incoming signal while allowing a certain decay of the accumulated information. If the voltage gate used to constrain the particles inside the chamber is then allowed to turn off whenever a certain threshold skyrmion density is reached (thus releasing the collected skyrmions into the output conduits), the functioning of our device becomes very reminiscent of an integrate-and-fire neuron where, instead of integrating discrete current pulses by accumulating electric potential differences between pre- and post-synaptic membranes, our device collects skyrmions instead.
In conclusion, we have proposed a skyrmionic signal scrambler and studied its properties by employing both micromagnetic simulations to characterize particle interactions among each other as well as boundaries, and molecular dynamic-like {\it n}-body simulations for efficient verification of the functioning of our device. The proper functioning of our proposed device depends on a subtle trade-off between large driving current intensities selecting for fast operation and long particle diffusion times in the device chambers to ensure proper uncorrelation of the reconstructed signals. Our device offers a novel application of the skyrmionic toolbox towards unconventional computing paradigms. Furthermore, the
\section*{Methods}
The three-dimensional micromagnetic simulations were performed by using the MuMax3 GPU-accelerated micromagnetic simulation program. The average energy density contains the exchange energy, the anisotropy energy, the applied field (Zeeman) energy, the magnetostatic (demagnetization) energy and the DMI energy terms. In all simulations, the thickness of the magnetic nanotracks is $3\,n\mathrm{m}$. The length of the input/output conduits was $50\,n\mathrm{m}$, while the width was set to $100\,n\mathrm{m}$. Magnetic parameters used in the simulations: saturation magnetization $M_S=1400\,k\mathrm{A/m}$, exchange stiffness $J_{ex}=27.5\,p\mathrm{J/m}$, interface-induced DMI constant $D=2.05\,m\mathrm{J}/\mathrm{m}^2$, perpendicular magnetic anisotropy constant $K_u=1.45\,M\mathrm{J}/\mathrm{m}^3$ and gyromagnetic ratio $M_S=-2.211\cdot 10^5\,\mathrm{m}/\mathrm{A}\mathrm{s}$. The Gilbert damping coefficient $\alpha$ was set to $1$. All models are discretized into tetragonal cells with the constant cell size of $2\times 2\times 3\,n\mathrm{m}^3$ in the simulations, which is smaller than the fundamental length scale $J_{ex}/D\simeq 13.4\,n\mathrm{m}$.
The {\it n}-body Thiele dynamics were solved using a specialized GPU-accelerated CUDA/C++ solver developed by the authors. The stochastic dynamical system of equations was solved employing a Heun scheme thus ensuring proper convergence to the proper Stratonovich solution. As described in the text, the gyrocoupling $G$ and gyrodamping $D$ used were extracted numerically from the micromagnetic simulations be performing surface integrals over the magnetic profiles of individual skyrmions.
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