Single skyrmion true random number generator using local dynamics and interaction between skyrmions

Magnetic skyrmions are of great interest to both fundamental research and applications in post-von-Neumann computing devices. The successful implementation of skyrmionic devices requires functionalities of skyrmions with effective controls. Here we show that the local dynamics of skyrmions, in contrast to the global dynamics of a skyrmion as a whole, can be introduced to provide effective functionalities for versatile computing. A single skyrmion interacting with local pinning centres under thermal effects can fluctuate in time and switch between a small-skyrmion and a large-skyrmion state, thereby serving as a robust true random number generator for probabilistic computing. Moreover, neighbouring skyrmions exhibit an anti-correlated coupling in their fluctuation dynamics. Both the switching probability and the dynamic coupling strength can be tuned by modifying the applied magnetic field and spin current. Our results could lead to progress in developing magnetic skyrmionic devices with high tunability and efficient controls.

(S1) In Eq. S1, = � x , y , z � is the normalized magnetization, z is the perpendicular magnetic field, S is the saturation magnetization, is the exchange stiffness, u is the perpendicular magnetic anisotropy (PMA), Int is the DMI constant and demag is the demagnetization field. Supplementary Figure 2 displays the magnetic-configuration dependence on the perpendicular magnetic field z , which is illustrated by both the Hall-resistance ( H ) measurement ( Supplementary Fig. 2a) and p-MOKE images (Supplementary Fig. 2b). We observe a shift in the Hall curve from zero field which is the classic signature of exchange bias ( Supplementary Fig. 2a). The exchange bias most likely arises from the CoFeB/MgO interface magnetization as discussed in detail in previous studies 1,2 . At the exchange bias field of -2. 21 Oe, a labyrinthine multi-domain state is observed ( Supplementary Fig. 2b). The labyrinthine domain phase transforms into the state with multiple skyrmions by either increasing or decreasing the field, which finally transforms into the uniform magnetization state at even higher or lower magnetic fields. The state with multiple skyrmions appears in the field range where an irreversible Hall curve is observed ( Supplementary Fig. 2a).

II. The evidence of magnetic skyrmions -skyrmion Hall effect
Magnetic skyrmions are topologically protected quasi-particles with non-zero topological charges = ±1.
Due to its topology, the skyrmion experiences a skyrmion Hall effect under a driving force. On biasing with a current, in addition to the longitudinal motion along the current direction, the skyrmion also acquires a transverse velocity component through a Magnus force induced by the topology 3,4 . The phenomenon of skyrmion Hall effect can be used to distinguish skyrmions from other trivial magnetic structures.
Here we consider the motion of a general magnetic texture. The centre of mass motion of the magnetic texture can be approximately described by the Thiele equation 5,6 × + The first and second terms are functions of the velocity of the magnetic texture and are known as Magnus force and dissipative force, respectively. The gyro coupling vector only has a z-component = 4 = 4 ∫ ( ) which is an integration of the topological charge density ( ). The third term is the currentinduced driving force for the motion of a magnetic texture, with the coefficient  Figure 3a, b, illustrates the current-driven motion of skyrmions with both polarities which experience the skyrmion Hall effect. We also see that the skyrmion in Fig. 1b 8 . We have demonstrated that | Int | ≥ thr and thus concluded purely Néel-type domain walls. We expect that the parameters of the magnetic thin film used for this study are close to our previous measurements as the multilayer structure and growth conditions are similar, and | Int | ≥ thr remains true.
To confirm the type of domain walls, we track evolutions of magnetic domains under an in-plane magnetic field, as presented in Supplementary Fig. 4. It has been shown that the Bloch-type domain walls would expand perpendicularly to the in-plane field while the expansion of the Néel-type domain walls is parallel to the in-plane field 9 . Domain walls with opposite vorticities also expand conversely. We first apply a perpendicular magnetic field, z = −6.43 Oe, to shrink the labyrinthine multi-domain phase into a mixed skyrmion and labyrinthine domain phase, and then apply an in-plane magnetic field IP . The asymmetric expansion of magnetic domains along the in-plane field direction suggests the left-handed Néel-type domain walls.

III. The skyrmion size and pinning size
Supplementary Figure  Ta/CoFeB systems observed in previous studies 8, 10 .
We calculate the stable size of skyrmions Skyr ( Skyr = 2 ) as a function of z through 11 The skyrmion radius and the domain-wall width are two unknown parameters. Supplementary  In the magnetic film under study, the pinning can be accounted for by the inherent inhomogeneities such as the surface roughness, grain structures, and/or material composition variations as a result of atomic diffusions at the interfaces. The inherent inhomogeneities may have a characteristic length comparable with the grain size of the polycrystalline structure of the magnetic film. According to our x-ray diffraction measurements 12 , the average grain size in the Ta layer is on the order of 10 nm which is much smaller than the skyrmion size. This suggests that a magnetic skyrmion can interact with a cluster of pinning centres, leading to the local dynamics as observed in this work.

IV. Effects of the sputter rate on the pinning of magnetic skyrmions
We regulate the DC power ( Ta = 3, 4, and 5 Watt) for the deposition of the Ta layer to implement moderate pinning strengths. A lower DC power induces a lower sputter rate for the Ta layer. The Ta is known as a getter material. One expects that a lower sputter rate would induce more contaminations in the Ta layer.
This could affect local atomic arrangements and increase spatial non-uniformities of the magnetic proper-

VII. Demagnetization-field variation owing to the fluctuation in skyrmion size
The coupling between two neighbouring skyrmions is most likely mediated by the demagnetization-field variation that results from the fluctuation in skyrmion size. In this section, we calculate the demagnetization-field variation owing to the skyrmion-size variation. Supplementary Figure 12a value is comparable to the field that is required for the switching-probability transition between L ≈ 0 and L ≈ 1 (Fig. 2 in the main text), indicating that the demagnetization-field variation due to the fluctuation in skyrmion size plays an important role in the anti-correlated coupling between the two skyrmions shown in Fig. 3 in the main text.

VIII. Potential of local dynamics of skyrmions for applications in computing
Supplementary Figure 13 schematically shows some potential applications of the skyrmion dynamics in computing. An isolated skyrmion can serve as a robust true random number generator for probabilistic computing (Supplementary Fig. 13a). Both the field and current are efficient parameters to control the switching probability of the probabilistic bit (Fig. 2 in the main text). The stochastic computer has been proven successful in addressing the issues of optimization and invertible logic that von Neumann computers fail to address efficiently 14  If one can control each skyrmion using a local magnetic field or current, two neighbouring skyrmions with an anti-correlated coupling can serve as a strong candidate for logic devices (Supplementary Fig.   13b). Local magnetic fields can be applied through multiple methods such as through the stray field generated from a magnetized tip 15,16 or by applying an electric current to a strip patterned nearby the skyrmion locations 17 . The local current can be applied to individual skyrmions through patterning two Hall crosses with each Hall cross hosting one skyrmion. The magnetized tip can be integrated in the device like the readwrite head in the hard disk drive. Additionally, advanced patterning techniques such as the e-beam lithography allow us to fabricate nanometre-sized structures such as the well-defined strip and Hall crosses. The local field or current can serve as inputs and the resultant Hall resistance serves as the output and can be detected through electronic measurements. In a logical AND operation, for example, inputs 1 and 0 can be currents for L = 1 and L = 0.5, respectively. When both inputs are 1, both skyrmions are at the L state, which gives a high resistivity of 〈∆ xy2 〉 ≈ 0.016 μΩ cm corresponding to an output of 1. When the input for one skyrmion is 1 and the other is 0, one skyrmion would be stabilized at the L state while the other has an increased probability at the S state, leading to a lower resistivity of 〈∆ xy2 〉 ≈ 0.008 μΩ cm which corresponds to an output of 0. When both inputs are 0, due to their mutual coupling, the two skyrmions are either at the LS or SL state, which also leads to a resistivity of 〈∆ xy2 〉 ≈ 0.008 μΩ cm corresponding to an output of 0. In a logical OR operation, on the other hand, inputs 1 and 0 can be currents for L = 0.5 and L = 0, respectively. Similarly, when both inputs are 1, the two skyrmions are either at the LS or SL state, which leads to a resistivity of 〈∆ xy2 〉 ≈ 0.008 μΩ cm corresponding to an output of 1. When the input for one skyrmion is 0 and the other is 1, one skyrmion would be stabilized at the S state while the other has an increased probability at the L state, leading to a resistivity of 〈∆ xy2 〉 ≈ 0.008 μΩ cm which corresponds to an output of 1, as well. When both inputs are 0, both skyrmions are at the S state, giving a low resistivity of 〈∆ xy2 〉 ≈ 0 corresponding to the output of 0. Therefore, the logical operations can be achieved by local dynamics of two neighbouring skyrmions with an anti-correlated coupling. In previous studies, skyrmionic logics have been proposed on the basis of dynamic motion of skyrmions 18,19 . Implementing the skyrmion motion-based devices experimentally, however, encounters difficulties in the precise control over skyrmion motion, in geometric and operational complexities. Comparatively, the logic devices based on local dynamics of skyrmions proposed here are much easier to operate and more spatially compact.
In addition to skyrmionic logics, two neighbouring skyrmions can serve as an element of a ternary numeral system with three discrete SS, LS/SL and LL states.
Two neighbouring skyrmions form the simplest skyrmion network. A more complex skyrmion network has further potential in computing. Skyrmion networks in naturally grown magnetic films have great potential in reservoir computing applications ( Supplementary Fig. 13c). The reservoir computing paradigm is inspired by the human brain and is a type of recursive neural network which exhibits capabilities in recognition and prediction of spatio-temporal events. Reservoir computing does not require any knowledge of the node weights for training purposes, only the output weights, and can thus utilize the naturally existing skyrmion network in magnetic films. In skyrmion network-based reservoir computing, the applied fields or currents in particular regions serve as the inputs. Local magnetic fields can be in principle applied by multiple "monopole" writing elements and the local current can be applied through multiple nanocontacts. The skyrmion network with mutual interactions is the reservoir whose weighting function can be unknown, and the matrix of the node weights is generally fixed for reservoir computing. In a skyrmion device, a fixed matrix is generated under a given global field and current. When the field or current is varied, the matrix changes and a new reservoir computer is formed, thereby allowing the skyrmion network to have higher tunability and functionality compared to memristor and atomic-switch networks 20,21 . Electronic signals from selected regions can then be converted into output neurons.
In addition to naturally existing skyrmion networks, skyrmion lattices with different structures can also be implemented via artificial pinning centres. These may exhibit more interesting dynamics for potential in computing applications such as Ising computing. Overall, local dynamics of skyrmions may provide effective functionalities for versatile computing. Implementing these devices experimentally with demonstrations of skyrmion manipulations by using the local field and current remains to be further explored.

IX. Fluctuation rate in the local dynamics of a skyrmion
The fluctuation rate in the local dynamics of a skyrmion increases with increasing temperature, reducing energy barrier and the distance ∆ between two weaker pinning sites. These dependencies can be theoretically understood through a forward flux sampling method which has been demonstrated to be effective in exploring thermal stability of skyrmions 22 and thermally activated magnetization reversals 23,24 . Although the energy profile can be described by two local minima at the two weaker pinning sites and an energy barrier in between, the recrossings will necessarily occur as the skyrmion approaches one state from the other, even in the vicinity of the barrier. The path sampling method relies on a series of interfaces in con- and thus the fluctuation rate. On the other hand, Φ L,0 is expected to be larger at an elevated temperature and with a reduced barrier height, which will thereby increase the fluctuation rate. A deeper understanding of how the fluctuation rate scales with temperature, barrier height, and ∆ remains to be further explored which is beyond the scope of this paper.

X. Micromagnetic simulations of local dynamics of skyrmions
We perform micromagnetic simulations using finite-difference solver MUMAX3 based on the graphic pro-  Supplementary Fig. 14a, e. The mobile part of both skyrmions can fluctuate in time between two sites, leading to the fluctuation of the perpendicular magnetization z in the blue box region as shown in Supplementary Fig. 14c, g.
For the skyrmion shown in Supplementary Fig. 14a-c, the mobile part of the skyrmion fluctuating in time between two sites separated by approximately 50 nm and the average residence time is as low as 30 ns. On the other hand, for the skyrmion presented in Supplementary Fig. 14e-g, this time is about 540 ns while the distance between the two weaker pinning sites is larger and about 60 nm. We summarize the average residence time of both skyrmions as functions of ∆ u / u , the random PMA variation, and temperature , in Supplementary Fig. 14d, h. ∆ u / u reflects the energy barrier height and thus a lower is observed at a lower ∆ u / u . The simulation results indicate that a fluctuation rate beyond the MHz range may be experimentally achieved by more elaborate control of the energy landscape, which remains to be further studied. Supplementary