Magnetic skyrmions are candidates for information carriers in applications for racetrack memories and Brownian computers because of their topologically protected properties. The circuits to confine the skyrmions in the 1D track with the magnetic wire or the magnetic energy wall play a key role in realizing such novel devices1,2. The ratchet using magnetic energy gradient is also required for improving calculation speed in ultralow-energy-consumption operation3. Because the focused-ion-beam (FIB) technique can vary the dose of ions locally, FIB effectively makes the circuit with the energy gradient4. This study aims to verify the effect of the FIB irradiation on the confinement and diffusion of the skyrmions and to structure the magnetic-energy gradient driving the skyrmions.
We fabricated the stacking structure as Ta(5.9)|Co16Fe64B20(1.2)|Ta(0.2)|MgO(2.2)|SiO2(2.9) and microfabricated the surface of the sample by the FIB of the Ga+ source. The potential condition for skyrmion creation can be controlled spatially by adjusting the dose of ions. Fig.1 shows the observations of the irradiated region indicated with the white dash line by the magneto-optical Kerr effect (MOKE) microscope. The skyrmions in the unirradiated region appear at about 333K, and in the irradiated region at about 295.5K. Fig.2(a) shows the mean squared displacement (MSD) of the skyrmions in the irradiated region. Fig.2(b) shows the temperature dependence of the diffusion coefficient before and after FIB irradiation. FIB irradiation increases the activation energy. However, the size of the diffusion coefficient after irradiation is as large as that in the sample before irradiation as far as a higher temperature is utilized. We will discuss the possibility of sputtering and of Ga+ doping and the motion in the energy gradient. This work was supported by ULVAC, Inc., JSPS KAKENHI Grant Number JP20H05666, Japan and JST, CREST Grant Number JPMJCR20C1, Japan, and Nanotechnology Platform of MEXT, Grant Number JPMXP09 S-21-OS-0053. 
**References** [1] S. Woo, et al. Nat. Mater. 15, 501 (2016)
[2] Y. Jibiki, S. Miki, et al. Appl. Phys. Lett. 117, 082402 (2020)
[3] S. Miki, et al. J. Phys. Soc. Jpn. 90, 114703 (2021)
[4] K. Fallon, et al. small 16, 1907450 (2020) 
![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/d39a2a8539e87d2efebfe4fcc418a73f.png)

Spatial control of the nucleation and dynamics of magnetic skyrmions by FIB irradiation

Magnetic skyrmions are recently attracting attention from the viewpoint of application to next-generation spintronics devices such as ultra-low-power Brownian computers. Skyrmions are expected to function as information carriers in those devices because of their unique features such as Brownian motion at near room temperature, small size, easy detection and easy control (1-4). In order to realize such devices, electrical gating and amplification of the skyrmion signal are indispensable. The primary mechanism of the voltage control technique for skyrmions is the voltage-controlled-magnetic-anisotropy effect (VCMA effect) (5-6). In the previous studies using the VCMA effect, for example, modulation of the diffusion coefficient (2) and creation/annihilation of the skyrmions (3) have been demonstrated. However, “skyrmion gates/transistors” have not been realized experimentally yet. Obstacles might be that the conditions creating skyrmions are easily affected by the local stress introduced from the gate fabrication. Furthermore, a non-negligible energy barrier that appears at the edge of the gate electrode, traps the skyrmions there. In this talk, we will present our attempts to realize a device in which skyrmions are not trapped at the edge of the gate electrode.
A skyrmion film consisting of Ta|Co16Fe64B20|Ta|MgO|SiO2 was deposited on a thermally oxidized silicon substrate by the magnetron sputtering. In order to apply voltage to the magnetic layer, thin Ru gate electrodes with a thickness of 2 [nm] were deposited onto this film, and the MOKE microscope observed the motion of skyrmions. Narrow gate electrodes trapped skyrmion at the edge of the electrodes even though no voltage was applied. The wider Ru electrode enables a free passage of the skyrmions through the edge. The result is one of the fundamentals of realizing the skyrmion transistors. 

Acknowledgement: The authors acknowledge to Ono laboratory in Kyoto Univ. This research and development work was supported by ULVAC, Inc., JST CREST Grant number JPMJCR20C1, Japan and JSPS KAKENHI Grant Number JP20H05666. 

**References:** 
(1) J. Zázvorka et al., Nat. Nanotechnol., 14, 658–661 (2019), 
(2) T. Nozaki et al., Appl. Phys. Lett., 114, 012402 (2019), 
(3) M. Schott et al., Nano Letters., 17, 3006-3012 (2017), 
(4) B. W. Walker et al., Appl. Phys. Lett., 118, 192404 (2021), 
(5) M. Weisheit et al., Science 315, 349 (2007), 
(6) T. Maruyama et al., Nature Nanotech., 4, 158 (2009), 
(7) Y. Jibiki et al., Appl. Phys. Lett., 117, 082402 (2020)

Control of the skyrmion passage by a gate electrode

Magnetic skyrmions, which are topologically protected particle-like spin textures, exhibit Brownian motion at ambient temperatures in a ferromagnetic thin film. The skyrmion system can be an ideal platform to design unconventional computers like stochastic/Brownian computers. Therefore, investigation of the Brownian motion of the magnetic skyrmion is now the subject of scientific and technical interest [2-5]. The motion of skyrmions driven by the current [6] and temperature gradient [7] have the chirality depending on the skyrmion number. In this study, we explore the effects of the chiral property of a skyrmion on its Brownian motion experimentally and theoretically. We fabricated the stacking structure as Ta(5)|Co16Fe64B20(1.26)|Ta(0.22)|MgO(1.5)|SiO2(5.0) and observed the position of the skyrmion x(t) = (x(t), y(t)) by using MOKE microscope. Fig. 1 shows the simultaneous velocity-position correlation function &lt;v(t)●x(t)&gt; and &lt;(v(t)×x(t))z&gt;. After enough time has passed, &lt;v(t)●x(t)&gt; and &lt;(v(t)×x(t))z&gt; correspond to the diffusion coefficient Dxx and the angular momentum of the skyrmionic diffusion, respectively, and the latter also has a sizable value. Fig. 2 represents the simulation trajectory in the periodic potential obtained by integrating the Thiele-Langevin eq. [8,9] including the thermal fluctuation and the mass term. The chirality of the Brownian motion is affected by the fluctuated potentials of the samples. This research and development work was supported by ULVAC, Inc., the Ministry of Internal Affairs and Communications, JSPS Grant-in-Aid for Scientific Research (S) Grant Number JP20H05666, Japan and CREST (Non-classical Spin project, JPMJCR20C1) of the Japan Science and Technology Agency.

![](https://assets.underline.io/uploads/markdown_image/1/image/b2fe0462e46a0f673fe46f65f2392e00.jpg)

Fig. 1 velocity-position correlation function and <(v(t)×x(t))z>


![](https://assets.underline.io/uploads/markdown_image/1/image/6aa7d699583b7f1eef889ab6bb6a071e.jpg)

Fig. 2 The simulation result of the motion of the skyrmion in the periodic potential. The red arrows show the direction of the motion.

Chiral Brownian motion of the magnetic skyrmions

Since the magnetic skyrmion is a topologically protected particle-like domain in the ferromagnetic film, it can survive under thermal agitations and shows Brownian motion. The skyrmion system can be an ideal platform to design unconventional computers like stochastic/Brownian computers. Therefore, investigation of the Brownian motion of the magnetic skyrmion is now the subject of scientific and technical interest [1,2].
In this talk, experimental proof of a non-vanishing x-velocity to y-coordinate correlation will be shown. The result is explained by the Thiele equation including a stochastic field [3,4]. Description of the phenomena introduces the gyro-diffusion constant [5] that is special for the diffusion of the chiral particles like skyrmions. The existence of the x-velocity to y-coordinate correlation also evidences the perpetual rotation motion of the skyrmion. Theory shows that the result does not conflict with the 2nd law of thermodynamics and the van Leeuwen theorem.
Acknowledgments 
The authors acknowledge Mr. Y. Jibiki of Osaka University, Dr. S. Auffret, Dr. C. Baraduc, and Dr. Hélèn Béa of Spintech, France for their initial contributions. The authors also acknowledge Dr. M. Oogane and Dr. K. Saito. This research and development work was supported by the ULVAC, Inc., the Ministry of Internal Affairs and Communications, Basic research S (20H05666) of JSPS, and CREST (Non-classical Spin project) of JST.
References
[1] Y. Jibiki et al., Appl. Phys. Lett. 117, 082402 (2020).
[2] L. Zhao et al., Phys. Rev. Lett. 125, 027206 (2020).
[3] A. Thiele et al., Phys. Rev. Lett. 30, 230 (1973).
[4] C. Schuette et al., Phys. Rev. B 90, 174434 (2014)
[5] E. Tamura et al. https://arxiv.org/abs/2005.04860.

Hidden diffusion and perpetual motion of the skyrmions

Magnetic Skyrmion is a particle-like magnetic excitation in ferromagnetic material similar to the Dirac-monopole in the spin-ice systems. Since the magnetic Skyrmion is topologically protected [1], it can survive under thermal agitation and shows Brownian motion. The skyrmion system can be an ideal platform to design stochastic computer [2] and/or in Brownian computer [3], since magnetic Skyrmion is a special particle that may show Brownian motion in all-solid-state devices. Therefore, investigation of the Brownian motion of the magnetic Skyrmion is now the subject of scientific and technical interest. The dynamics of the Skyrmions in a two-dimensionally extended film at finite temperature is described by the Thiele equation [4]. According to the equation, the Skyrmion may pose a velocity perpendicular to the force's direction and tend to turn right or left depending on its chirality determined by the Skyrmion number. This gyro-motion reduces the (diagonal element of the) diffusion constant [5]. Naïvely thinking, we may suppose that the diffusion flow may pose a component perpendicular to the gradient of the skyrmion density because of the gyro-motion. The component should be proportional to the gradient of the Skyrmion density and newly defined “gyro-diffusion constant” (off-diagonal element of the diffusion constant) [6]. However, such flow is divergence-less and does not appear in the equation of diffusion. According to this context, we mention the gyro-diffusion constant as “the hidden diffusion constant”. In this talk, we show evidence of gyro-motion in diffusion and discuss the observation of the gyro-diffusion constant. Then, we mention recent progress to implement “Brownian computer” by use of Skyrmions. In our experiments, the magnetic Skyrmions are bubble-shaped magnetic domains generated in a ferromagnetic thin film having spontaneous magnetization perpendicular to the film surface. The Skyrmions have their diameter adjusted to 1 μm to allow the observation by an optical microscope with polarization analysis (MOKE microscope). To create the Skyrmions, Ta/CoFeB /Ta sandwiched films [7] possessing perpendicular magnetic anisotropy and the interfacial- Dzyaloshinskii-Moriya interaction (i-DMI) were fabricated. Changing the upper and lower Ta film thicknesses independently, the ratio of the i-DMI energy to the magnetic anisotropy energy was adjusted. Besides, by eliminating the annealing process, the generation of grain boundaries was suppressed. The annealing free process enhanced the diffusion coefficient of skyrmions by a factor of more than 10 compared to the previous study [2] and comparable to the recent observations [7,8]. Firstly, we have developed the Skyrmion channel by a slight modification of the SiO2 passivation layer thickness. By this method we could make degradation and dipole-field free channels. With this method the Skyrmions show free Brownian motion even in a complicated Skyrmion circuit like Hub (Y-cross) [7]. Secondly, we put a Skyrmion in one-dimensional channels to interfere with its gyro-motion. As a result, a significant increase in the diagonal diffusion constant was observed. This is clear evidence of the role of gyro-motion in the diffusion process. Thirdly, we have observed the diffusion of a Skyrmion in uniformly extended ferromagnetic films. In the uniform film, Skyrmions show free diffusion as it is shown in Fig. 1. The analysis of the simultaneous position-position correlation function &lt;X(t)*X(t)&gt;, which is a half of the mean square displacement (MSD), results that &lt;X(t)*X(t)&gt; is proportional to the elapsed time providing the diagonal diffusion constant as its slope. On the other hand &lt;X(t)*Y(t)&gt; is zero as it is expected form the symmetry. While, we have found that simultaneous velocity-position correlation functions (&lt;Vx(t)*X(t)&gt; and &lt;Vx(t)*Y(t)&gt;) are non-zero. From the theory, &lt;Vx(t)*Y(t)&gt; corresponds to the gyro-diffusion constant. The size of the observed gyro-diffusion constant is about 1/10 of the diagonal diffusion constant, and non-negligible. However, the sign of the observed gyro-diffusion constant is opposite to that expected from a sense of gyro-motion. Since the sense of the gyro-motion is determined by the sign of the Skyrmion number, it should not be changed easily. The change of the sign in the gyro-diffusion constant will be discussed in terms of the diffusion in a potential. Finally, essential devices to construct a Brownian computing circuit, i.e. hub, ratchet, and c-join [3], were tested. By now, ratchet and c-join need a control using MOKE observation and manual feedback. To overcome this difficulty, we are now developing autonomous devices using voltage control [9, 10] of the Skyrmion (Fig. 2). Acknowledgments The authors acknowledge Mr. Y. Jibiki of Osaka University, Dr. S. Auffret, Dr. C. Baraduc, and Dr. Hélèn Béa of Spintech, France for their initial contributions. The authors also acknowledge Dr. M. Oogane for his FMR measurements. This research and development work was supported by the ULVAC, Inc., the Ministry of Internal Affairs and Communications, Basic research S (20H05666) of JSPS, and CREST (Non-classical Spin project) of JST.

**References:**

[1] N. Nagaosa et al., Nat. Nanotech. 8, 899-911 (2013). [2] J. Zazvorka et al., Nat. Nanotech. 14, 658 (2019). [3] F. Peper et al., ACM J. Emerg. Technol. Comput. Syst, 9, 1, Article 3 (2013). [4] A. Thiele et al., Phys. Rev. Lett. 30, 230 (1973). [5] C. Schuette et al., Phys. Rev. B 90, 174434 (2014) [6] E. Tamura et al. https://arxiv.org/abs/2005.04860. [7] Y. Jibiki et al., Appl. Phys. Lett. 117, 082402 (2020). [8] L. Zhao et al., Phys. Rev. Lett. 125, 027206 (2020). [9] T. Nozaki et al., Appl. Phys. Lett. 114, 012402 (2019). [10] M. Scott et al., Nano Lett. 17, 3006 (2017).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/6533c8637225b7f1b2cad390443cbfe3.jpg)
Fig. 1 Trajectories of magnetic skyrmions observed at 316 K using a MOKE microscope.

![](https://assets.underline.io/uploads/markdown_image/1/image/4c8fd416dc80b83ea49919378f59a037.jpg)
Fig. 2 Voltage/current controlled Skyrmion ratchet (test device).

 Hidden skyrmion diffusion and Brownian computing

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Spatial control of the nucleation and dynamics of magnetic skyrmions by FIB irradiation

Control of the skyrmion passage by a gate electrode

Chiral Brownian motion of the magnetic skyrmions

Hidden diffusion and perpetual motion of the skyrmions

Hidden skyrmion diffusion and Brownian computing

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