France



Microwave-assisted magnetic recording (MAMR) is a new technology where a microwave magnetic field having a frequency of few tens of GHz can assist switching of the magnetic domains in the recording media[1]. This microwave magnetic field is generated by using a spin-torque oscillator (STO) under the influence of the magnetic field from the write pole of the recording head and an applied dc bias current. When certain conditions for bias current and magnetic field are met, spin-transfer torque causes the field generating layer (FGL) of the STO to precess, generating an oscillating magnetic field. Although the oscillation frequency of the magnetic field generated by the STO is a key parameter in MAMR, its conventional measurement using the radio-frequency (RF) electrical signal arising from the magnetoresistance (MR) of STO has the following concerns: the RF electrical signal is sometimes very weak and contains several frequency components that do not correspond to the frequency of the actual magnetic field generated by the STO because of the complex magnetization dynamics of the ferromagnetic layers inside STO; the RF signal attenuates heavily in the write head-STO assembly. Therefore, it is strongly desired to find a way to characterize the oscillation frequency of the STO which is already a part of the write head assembly.&lt;br/&gt;&lt;br/&gt;In this study, we have proposed and successfully demonstrated a new technique to characterize the actual frequency of the magnetization precession in an STO using injection locking from externally applied radio-frequency (RF) magnetic field. Using our method, there is no need to electrically connect the STO to a spectrum analyzer via an RF-compatible electrical circuit.&lt;br/&gt;&lt;br/&gt;Injection locking is a phenomenon occurring when an oscillator (in this case, an STO) is forced to couple and synchronize with an external oscillating signal (in this case, oscillating magnetic field at GHz frequency range)[2]. When the FGL of an STO is precessing at a given frequency (&lt;i&gt;f&lt;/i&gt;&lt;sub&gt;FGL&lt;/sub&gt;), it can undergo injection locking to an externally applied magnetic field whose frequency(&lt;i&gt;f&lt;/i&gt;&lt;sub&gt;ext&lt;/sub&gt;) is close to the &lt;i&gt;f&lt;/i&gt;&lt;sub&gt;FGL&lt;/sub&gt; which changes the trajectory of magnetization precession. Since the dc resistance of the STO depends on the trajectory of precession, the effect of injection locking is reflected in the dc resistance of the STO. We can use this phenomenon to detect the &lt;i&gt;f&lt;/i&gt;&lt;sub&gt;FGL &lt;/sub&gt;without measuring conventional RF output signal from STO.&lt;br/&gt;&lt;br/&gt;To demonstrate this experimentally, we used the design of all-in-plane STO which was shown previously as suitable to the MAMR application[3]. We fabricated an STO pillar of around 40nm in diameter and also a co-planar waveguide on top of it to act as an RF antenna that generates an oscillating magnetic field, electrically separated by SiO&lt;sub&gt;2&lt;/sub&gt;. As shown in fig. 1, a bias current and magnetic field were applied to the STO as usual, and the STO underwent oscillation at the suitable values of bias current and magnetic field. Then, we applied a pulsed RF electrical excitation to the co-planar waveguide from a signal generator, with frequency &lt;i&gt;f&lt;/i&gt;&lt;sub&gt;ext&lt;/sub&gt; and pulsed at 1MHz, and monitored the change of dc resistance between off and on parts of the pulse (ΔR) by using a lock-in amplifier with the pulse frequency as reference. Note that although we measured the resistance oscillation of the STO at a frequency of 1 MHz, the measurement result reflected the dc resistance. Because 1 MHz is much smaller than the STO oscillation frequency of few tens of GHz, the RF resistance change due to the STO oscillation is averaged over several precession cycles of the magnetization. Fig. 2 shows an example of such measurement. Fig. 2(a) shows a typical spectrum of the RF output signal with a tiny peak at 32GHz from STO under dc bias voltage (&lt;i&gt;V&lt;/i&gt;&lt;sub&gt;dc&lt;/sub&gt;) = 50mV and applied magnetic field (&lt;i&gt;H&lt;/i&gt;&lt;sub&gt;ext&lt;/sub&gt;) = 1200mT, from which we could not identify whether or not the frequency of this peak corresponds to the actual magnetization precession. Fig. 2(b) shows the value of ΔR as a function of &lt;i&gt;f&lt;/i&gt;&lt;sub&gt;ext&lt;/sub&gt; for the same values of &lt;i&gt;V&lt;/i&gt;&lt;sub&gt;dc&lt;/sub&gt; and &lt;i&gt;H&lt;/i&gt;&lt;sub&gt;ext&lt;/sub&gt;. We clearly detected the change in ΔR as a function of &lt;i&gt;f&lt;/i&gt;&lt;sub&gt;ext&lt;/sub&gt;, corresponding to injection locking. This informs us of the actual frequency of magnetization precession without measuring the RF output signal.&lt;br/&gt;&lt;br/&gt;In conclusion, we demonstrated that the dc resistance of the STO changes measurably upon injection locking to an externally applied magnetic field. This can be used to characterize the oscillation frequency of the magnetic field generated by the STO which is already a part of the write head assembly. This will be beneficial for developing an STO-based microwave magnetic field generator for MAMR.

**References:**

[1] J. G. Zhu, X. Zhu, and Y. Tang, &lt;i&gt;IEEE Trans. Magn.&lt;/i&gt;, vol. 44, no. 1, pp. 125–131, (2008)&lt;br/&gt;[2] S. Urazhdin, &lt;i&gt;et al.&lt;/i&gt;, &lt;i&gt;Phys. Rev. Lett.&lt;/i&gt;, vol. 105, no. 10, p. 104101, (2010)&lt;br/&gt;[3] H. Sepehri-Amin &lt;i&gt;et al.&lt;/i&gt;, &lt;i&gt;J. Magn. Magn. Mater.&lt;/i&gt;, vol. 476, pp. 361–370, (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/dd3c256437e821279ec173fbd1dae0ce.jpg)
Fig. 1 A schematic of our measurement. The STO is connected to a bias tee setup which allows a mix of dc voltage input (V&lt;sub&gt;dc&lt;/sub&gt;), a high-frequency component and a low-frequency component. The high-frequency component is not to be used in our newly proposed method

![](https://assets.underline.io/uploads/markdown_image/1/image/301904ba6e5647975c3894317f292714.jpg)
Fig. 2 (a) PSD of the RF electrical signal from the STO (b) Change of resistance △R as a function of externally applied RF field frequency &lt;i&gt;f&lt;/i&gt;&lt;sub&gt;ext&lt;/sub&gt; under the same bias conditions

INTERMAG 2021

 Characterizing the oscillation frequency of a spin-torque oscillator (STO) by measuring the change of dc resistance upon injection locking to an external RF magnetic field

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Microwave-assisted magnetic recording (MAMR) is a new technology where a microwave magnetic field having a frequency of few tens of GHz can assist switching of the magnetic domains in the recording media[1]. This microwave magnetic field is generated by using a spin-torque oscillator (STO) under the influence of the magnetic field from the write pole of the recording head and an applied dc bias current. When certain conditions for bias current and magnetic field are met, spin-transfer torque causes the field generating layer (FGL) of the STO to precess, generating an oscillating magnetic field. Although the oscillation frequency of the magnetic field generated by the STO is a key parameter in MAMR, its conventional measurement using the radio-frequency (RF) electrical signal arising from the magnetoresistance (MR) of STO has the following concerns: the RF electrical signal is sometimes very weak and contains several frequency components that do not correspond to the frequency of the actual magnetic field generated by the STO because of the complex magnetization dynamics of the ferromagnetic layers inside STO; the RF signal attenuates heavily in the write head-STO assembly. Therefore, it is strongly desired to find a way to characterize the oscillation frequency of the STO which is already a part of the write head assembly. In this study, we have proposed and successfully demonstrated a new technique to characterize the actual frequency of the magnetization precession in an STO using injection locking from externally applied radio-frequency (RF) magnetic field. Using our method, there is no need to electrically connect the STO to a spectrum analyzer via an RF-compatible electrical circuit. Injection locking is a phenomenon occurring when an oscillator (in this case, an STO) is forced to couple and synchronize with an external oscillating signal (in this case, oscillating magnetic field at GHz frequency range)[2]. When the FGL of an STO is precessing at a given frequency (fFGL), it can undergo injection locking to an externally applied magnetic field whose frequency(fext) is close to the fFGL which changes the trajectory of magnetization precession. Since the dc resistance of the STO depends on the trajectory of precession, the effect of injection locking is reflected in the dc resistance of the STO. We can use this phenomenon to detect the fFGL without measuring conventional RF output signal from STO. To demonstrate this experimentally, we used the design of all-in-plane STO which was shown previously as suitable to the MAMR application[3]. We fabricated an STO pillar of around 40nm in diameter and also a co-planar waveguide on top of it to act as an RF antenna that generates an oscillating magnetic field, electrically separated by SiO2. As shown in fig. 1, a bias current and magnetic field were applied to the STO as usual, and the STO underwent oscillation at the suitable values of bias current and magnetic field. Then, we applied a pulsed RF electrical excitation to the co-planar waveguide from a signal generator, with frequency fext and pulsed at 1MHz, and monitored the change of dc resistance between off and on parts of the pulse (ΔR) by using a lock-in amplifier with the pulse frequency as reference. Note that although we measured the resistance oscillation of the STO at a frequency of 1 MHz, the measurement result reflected the dc resistance. Because 1 MHz is much smaller than the STO oscillation frequency of few tens of GHz, the RF resistance change due to the STO oscillation is averaged over several precession cycles of the magnetization. Fig. 2 shows an example of such measurement. Fig. 2(a) shows a typical spectrum of the RF output signal with a tiny peak at 32GHz from STO under dc bias voltage (Vdc) = 50mV and applied magnetic field (Hext) = 1200mT, from which we could not identify whether or not the frequency of this peak corresponds to the actual magnetization precession. Fig. 2(b) shows the value of ΔR as a function of fext for the same values of Vdc and Hext. We clearly detected the change in ΔR as a function of fext, corresponding to injection locking. This informs us of the actual frequency of magnetization precession without measuring the RF output signal. In conclusion, we demonstrated that the dc resistance of the STO changes measurably upon injection locking to an externally applied magnetic field. This can be used to characterize the oscillation frequency of the magnetic field generated by the STO which is already a part of the write head assembly. This will be beneficial for developing an STO-based microwave magnetic field generator for MAMR.

**References:**

[1] J. G. Zhu, X. Zhu, and Y. Tang, IEEE Trans. Magn., vol. 44, no. 1, pp. 125–131, (2008) [2] S. Urazhdin, et al., Phys. Rev. Lett., vol. 105, no. 10, p. 104101, (2010) [3] H. Sepehri-Amin et al., J. Magn. Magn. Mater., vol. 476, pp. 361–370, (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/dd3c256437e821279ec173fbd1dae0ce.jpg)
Fig. 1 A schematic of our measurement. The STO is connected to a bias tee setup which allows a mix of dc voltage input (Vdc), a high-frequency component and a low-frequency component. The high-frequency component is not to be used in our newly proposed method

![](https://assets.underline.io/uploads/markdown_image/1/image/301904ba6e5647975c3894317f292714.jpg)
Fig. 2 (a) PSD of the RF electrical signal from the STO (b) Change of resistance △R as a function of externally applied RF field frequency fext under the same bias conditions

technical paper

Magnetic skyrmions are a peculiar kind of chiral magnetic textures stabilized by chiral interactions like the Dzyaloshinskii-Moriya interaction (DMI) [1]. Their topological properties offer interesting robustness qualities but, as a counterpart, add a complexity to their dynamics. Since their first observation in a bulk chiral magnet in 2009 [2], rapid experimental advances demonstrated their stability at room temperature without any magnetic field in ferromagnet/heavy metal bilayers and multi-stacks [3]. The experimental demonstrations of smaller and smaller skyrmions at room temperature (down to few 10s of nm) suggest that they are the perfect candidates for information storage and logic devices. Before building such devices, we first need to master skyrmions' static and dynamical properties, namely the nucleation and stabilization of small skyrmions that can move at high speed inside tracks. One significant issue with skyrmion dynamics is the topological deflection. Also known as Skyrmion Hall effect, it is caused by the topological charge of skyrmion and consists in a deflecting force proportional and perpendicular to the skyrmion velocity. Deflection can lead to the expulsion of skyrmions from the tracks that contain them, and thus imposes a limit to the skyrmion velocity. Experiments and calculations found that the deflection angle varies with skyrmion velocity and diameter probably due to its interaction with pinning disorder [4]. It has been suggested that antiferromagnetically-coupled systems, such as antiferromagnets, synthetic antiferromagnets (SAFs) or ferrimagnets, may be used to overcome these issues. The low net magnetization of these systems reduces the size of the skyrmions, and their low angular momentum density should suppress the troublesome topological deflection. Ferrimagnetic Rare-Earth/Transition Metal alloys possess two anti-parallel sublattices whose relative magnetic moments can be changed by temperature or alloy composition. Ferrimagnets share properties from ferromagnets and antiferromagnets (at the magnetic compensation) with which they present many similarities. They are perfect systems to study the advantages of antiferromagnetically-coupled materials, while preserving the effects that have already been used to stabilize and drive skyrmions in ferromagnets. Recent results on SAF showed the promising advantages of using multi-lattices coupled antiferromagnetically [5], and in 2018 a huge breakthrough was achieved in Rare-Earth/Transition Metal ferrimagnets where the smallest skyrmions were observed (10-30nm) [6] as well as a reduced topological deflection [7]. Using co-evaporation in Ultra-High-Vacuum [8], we grew ferrimagnetic thin film of Ta(1nm)/Pt(5nm)/Gd0.3Co0.7(5nm)/Ta(5nm) with an amorphous structure that decreases pinning of skyrmions. By optimizing the material stack and the process conditions, we were able to strengthen the interfacial DMI while keeping a balanced interfacial anisotropy to obtain out of plane magnetized samples (around room temperature) that can host skyrmions. We then observed the skyrmions by Magneto-Kerr Microscopy with sizes smaller than the technique’s resolution limit (around 500 nm) (Fig. 1). After saturating the sample, we were able to nucleate skyrmions in the tracks and to tune their overall density by applying magnetic fields. Skyrmions were also nucleated on the edges of the track by nanosecond-long current pulses. Thanks to the tunability of the magnetic properties of the ferrimagnet, we studied the sample in a range of magnetic field and temperature (0–40 mT, and 295–320 K). We observed different regimes: saturated magnetic textures, skyrmions with different densities, and stripes textures. Skyrmions were observed between ~295-310 K. The field necessary for nucleating skyrmions increased with temperature (8 mT at 295 K, 15 mT at 305 K), probably due to a decreasing Ms (as the compensation temperature of this sample is above the tested range). We also observed their dynamics under the application of nanosecond current pulses. The current going through the adjacent Pt layer generates a spin orbit torque (SOT) due to the spin Hall effect, which drives the skyrmions. We observed speeds higher than 150 m/s (for ~10 GA/m2) in a denser skyrmion regime and up to 40 m/s in the most diluted regime (Fig. 2), limited for higher current density by edge nucleations. The speed seems to follow a linear dependence with current. In the conditions of higher density, the velocity was higher and the depinning current lower, probably due to a change of the skyrmion diameter. We also studied the evolution of the deflection angle of the skyrmions, which increased from 40° at low current to finally saturate around 60°. This work shows that ferrimagnetic skyrmions can be nucleated, stabilized and propagated by SOT in amorphous rare earth/transition metal thin films. In these results, we stabilized skyrmions far away from the magnetic compensation of the sample (also far away from the angular compensation). We have since prepared new samples with accessible magnetic and angular compensation in order to study the dynamical properties of our sub-micron skyrmions in these very peculiar regimes.

**References:**

 [1] – A. Fert et al., Nature Nanotechnology, Vol 8, p 152–156 (2013) [2] – X. Z. Yu et al. Nature, Vol 465, p 901–904 (2010) [3] – A. Hrabec et al., Nature Com. Vol 8, 15765 (2017) [4] - Joo-Von Kim et al. Appl. Phys. Lett., Vol 110, 132404 (2017) [5] – W. Legrand et al., Nature Materials, Vol 19, p 34–42 (2020) [6] – L. Caretta et al., Nature Nanotechnology, Vol 13, p 1154–1160 (2018) [7] – S. Woo et al. Nature Communications Vol 9, 959 (2018) [8] – E. Haltz et al. Physical Rev. Materials Vol 2, 104410 (2018)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/4deb0696a0fd43f90ed46a6cc224f887.jpg)
Fig.1 : Magneto-Kerr Microscopy image of skyrmions in GdCo micro-structured sample.

![](https://assets.underline.io/uploads/markdown_image/1/image/25864383d3eb7d0bd0916c7804a27862.jpg)
Fig. 2 : SOT-driven velocity at room temperature in diluted and denser regime.

 Spin Orbit Torque Dynamics of Magnetic Skyrmions in GdCo Ferrimagnetic Thin-Films

The spintronics devices have been considered as promising candidates for future artificial neural networks by mimicking the functionality of biological neurons with nonvolatile control, which can enable effective reduction of energy consumption and heat generation during the information processing and transmission. Among them, voltage-controlled magnetic skyrmions have attracted special attentions because that they will satisfy the demands of well-controlled, high efficient, and energy-saving for future skyrmion-based neuron device applications. In this work, we propose a compact leaky-integrate-fire (LIF) spiking neuron device by deploying the voltage-driven skyrmion dynamics in a multiferroic nanodisk structure. The relationship between the skyrmion dynamics and the characteristics of voltage induced piezostrains have been investigated, where the skyrmion radius can be effectively modulated by the applied piezostrain pulses. Similar to the biological neuron, the proposed skyrmionic neuron will accumulate a membrane potential as skyrmion radius varying by inputting the continuous piezostrain spikes, and the skyrmion radius will return to the initial state in the absence of stimulus. Therefore, this skyrmion radius based membrane potential will reach a definite threshold value by the strain stimuli and then reset by removing the stimuli. The skyrmion size variation can be transformed to resistance detection signal by the magnetoresistance effect, which means that the skyrmionic neuron could fire an output resistance spike with controlling the input strain pulses. Such LIF neuronal functionality and behaviors of the proposed skyrmionic neuron device are elucidated through the comprehensive micromagnetic and theoretical studies. Our results provide a new route for realizing voltage-controllable skyrmionic neuron in future energy-efficient and tunable neuromorphic computing hardware.

**References:**

1. P. A. Merolla, J. V. Arthur, and D. S. Modha, Science, Vol. 345, p. 668 (2014). 2. X. Liang, G. Zhao, and Y. Zhou, Applied Physics Letters, Vol. 116, p. 122402 (2020). 3. X. Chen, Y. Zhou, and W. Zhao, Nanoscale, Vol. 10, p. 6139 (2018).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/624573e1b7b93a13d5c0eb82c5b43516.jpg)
(a) Schematic of the proposed spiking neuron and demonstration of the LIF behaviors of the skyrmion-based artificial neuron. (b) Skyrmion radius varying with the spiking strain signals under the homogeneous square wave with the frequency of 1.09 GHz. (c) The creation of LIF functionality based on the Resistance-Strain characteristic by magnetoresistance effect of the skyrmionic neuron device.

 Voltage-controllable Magnetic Skyrmion Dynamics for Spiking Neuron Device Applications

Magnetic skyrmions (Sks) and antiskyrmions (aSks) are topologically protected non-trivial spin textures1, 2. Those objects can be used as magnetic bits in spintronics devices3, 4. However, the presence of the skyrmion Hall effect in the Sk always hinders its movement along the straight path, and the transverse defection of it will cause a problem on reading data in devices. The unwanted Hall effect can be eliminated in aSk host systems2, 5 such that it can be moved in a straight path along the current direction6. Therefore, it is highly desired to prepare nano-stripes along specific crystallographic directions of the aSk host compound. Here we stabilize a single-chain of aSks in nano-stripes of Mn1.4Pt0.9Pd0.1Sn prepared along different crystallographic directions within the tetragonal basal plane7. We observe the enhanced stability of aSks compared to bulk systems without providing in-plane magnetic field components. Additionally, the presence of dipole-dipole interactions with the anisotropic Dzyaloshinskii-Moriya exchange interaction (DMI) in the Heusler compound Mn1.4Pt0.9Pd0.1Sn allows the stability of distinct spin textures called elliptical Bloch skyrmions7,8. The co-existence of elliptical skyrmions (eSks) and aSks is found in nano-stripes at room temperature7. The simultaneous existence of different spin textures in a racetrack has great advantages. It is not needed to write the distances between the magnetic bits anymore, unlike the racetrack of only single type magnetic objects. Our results provide a gateway for the remaining steps towards realizing an advanced version of racetrack storage devices based on magnetic aSks and eSks.

**References:**

1. Mühlbauer et. al., Science 323, 915 (2009). 2. Nayak et. al., Nature 548, 561 (2017). 3. Parkin et. al., Science 320, 190 (2008). 4. Fert et. al., Nat. Nanotechnol. 8, 152 (2013). 5. Jena et. al., Nano Lett. 20, 59 (2019). 6. Huang et. al., Phys. Rev. B 96, 144412 (2017). 7. Jena et. al., Sci. Adv. 6, 49, eabc 0723 (2020). 8. Jena et. al., Nat. Commun. 11, 1115 (2020).

**Images:**

 Evolution and competition between chiral spin textures in nano-stripes with D2d symmetry

1. Introduction As quasiparticle-like swirling topological spin textures in nanostructures1, magnetic skyrmions are expected to be widely used in future spintronic applications2. The creation and transmission of an isolated skyrmion in magnetic films are important for skyrmionic applications, which utilize skyrmions as information carriers3. An exemplary application is the racetrack-type storage memory4. Recently, it has been reported that spin-polarized current, spin wave, magnetic field, and temperature gradients, can effectively modulate the dynamics of skyrmions5. As the most widely used method, the magnetic field has been used to manipulate magnetic spin textures in studies6. Slonczewski and Berger first proposed theoretically spin transfer torque (STT)7 can be induced when spin polarized current passes through the magnetic layer/non-magnetic layer. STT is an effect in which the orientation of a magnetic layer in a spin valve can be modified by applying a spin-polarized current. On the other hand, in magnetic multilayered structures, metallic ferromagnets in contact with heavy metals exhibit strong effects due to spin-orbit coupling. Such spin-orbit coupling effects give rise to spin-orbit torque (SOT)8 on the ferromagnetic (FM) materials. In this work, we demonstrate that a skyrmion guide on the nanotrack under both in-plane magnetic field and spin-transfer torque or spin-orbit torques driven. In applying both in-plane magnetic field and STT conditions, we focus on spatially sinusoidal-varying Gilbert damping coefficients and non-adiabatic STT coefficients with in-plane magnetic field. For in-plane magnetic field and SOT conditions, we discuss the effect on damping-like toque, field-like torque with in-plane magnetic field. The results provide a guide for designing and developing the skyrmion transporting nanotrack channel in spintronic devices based on the manipulation of skyrmions. 2. Simulation Methods The skyrmion dynamics guide is based on an ultrathin FM nanotrack structure. All simulations are processed by using the Object Oriented Micromagnetic Framework (OOMMF). The time-dependent magnetization dynamics are governed by the LLG equation including STT (adiabatic and non-adiabatic) and SOT (field-like and damping-like). Note that we consider the effect of both in-plane magnetic field and STT or SOT driven. In the first part, the magnetization dynamics is based on the Landau-Lifshitz-Gilbert (LLG) equation augmented with in-plane magnetic field, adiabatic and non-adiabatic STT: dm/dt=-|γ|m×Heff+α(m×dm/dt)+u(m×dm/dt×m)-βu(m×dm/dt) where m represents the reduced magnetization, M/MS. γ and α are the gyromagnetic ratios and Gilbert damping constant, respectively. β is the strength of the non-adiabatic STT coefficient and u represents the adiabatic STT coefficient. The Gilbert damping coefficient α and the non-adiabatic STT coefficient β are defined as functions of the longitudinal coordinate x as follows: α(x)=αamp{1+sin[2π(x/λα)]} β(x)=βamp{1+sin[2π(x/λβ)-θ]} where αamp is the amplitude of α function and βamp represents the amplitude of β function, respectively. λβ and θ denote the wavelength and phase of β function. In the second part, the magnetization dynamics is based on the Landau-Lifshitz-Gilbert (LLG) equation augmented with in-plane magnetic field, and spin-orbit torque (SOT): dm/dt=-|γ|m×Heff+α(m×dm/dt)+ζDLT m×(mp×m)-ζFLT m×mp where ζSOT, ζDLT, and ζFLT stand for the spin-orbit torque, damping-like torque and field-like torque, respectively. 3. Results and Discussion The effects of varying damping coefficient α and non-adiabatic coefficient β on velocity in both x and y direction, and skyrmion hall angle φ are illustrated in Fig. 1, by fixing external magnetic field as 800 mT. We set αamp and βamp from 0.1 to 0.5 in our simulation to obtain the discipline, by fixing βamp and αamp in Fig. 1(a)-(c) and Fig. 1(d)-(f), respectively. The contour lines in Fig. 1(g)-(i) result from the combinations of αamp and βamp. The results show that the damping coefficient α exhibits declining effect on velocity, while non-adiabatic STT coefficient β exhibits increasing influence. The polarization rate is fixed at 0.4 and the value of field-like torque is modified to change the ratio of K=ζFLT/ζDLT . Applying both magnetic field and SOT to drive skyrmion simultaneously at magnetic field H = 800 mT. The ratio of field-like torque and damping-like torque K and damping-like torque ζDLT both indicates increasing effect on skyrmion velocity, as shown in Fig. 2(a)-(b). However, damping-like torque ζDLT provides more contribution on skyrmion motion.

**References:**

1. X. Zhang, Y. Zhou, K. M. Song, T.-E. Park, J. Xia, M. Ezawa, X. Liu, W. Zhao, G. Zhao and S. Woo, J. Phys. Condens. Matter 32 (14), 143001 (2020). 2. A. Fert, N. Reyren and V. Cros, Nat. Rev. Mater. 2 (7), 1-15 (2017). 3. W. Kang, X. Chen, D. Zhu, S. Li, Y. Huang, Y. Zhang and W. Zhao, presented at the 2018 Design, Automation & Test in Europe Conference & Exhibition (DATE) (unpublished). 4. C. Song, C. Jin, J. Wang, H. Xia, J. Wang and Q. Liu, Appl. Phys. Lett. 111 (19), 192413 (2017). 5. C. Schütte, J. Iwasaki, A. Rosch and N. Nagaosa, Phys. Rev. B 90 (17), 174434 (2014). 6. J. Müller, New J. Phys. 19 (2), 025002 (2017). 7. M. D. Stiles and A. Zangwill, Phys. Rev. B 66 (1), 014407 (2002). 8. Y. Li, K. W. Edmonds, X. Liu, H. Zheng and K. Wang, Adv. Quantum Technol. 2 (1-2), 1800052 (2019).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/fca6ea5a3e90fff1a8d05d79a4845325.jpg)
Figure 1. Both magnetic field and STT driven skyrmion are applied to study the effect of damping coefficient α on velocity in both x and y direction, and skyrmion hall angle φ: (a) The Vx as a function of α at a fixed β = 0.2-0.5. (b) The Vy as a function of α at a fixed β = 0.2-0.5. (c) The φ as a function of α at a fixed β = 0.2-0.5. The effect of non-adiabatic β on velocity in both x and y direction, and skyrmion hall angle φ. (d) The Vx as a function of β at a fixed α = 0.2-0.5. (e) The Vx as a function of β at a fixed α = 0.2-0.5. (f) The Vx as a function of β at a fixed α = 0.2-0.5. (g) Vx, (h) Vy, and (i) skyrmion hall angle φ at magnetic field H = 800 mT.

![](https://assets.underline.io/uploads/markdown_image/1/image/338c7353053a46418aa778ff47c803f2.jpg)
Figure 2. Applying both magnetic field and SOT to drive skyrmion at magnetic field H = 800 mT: (a) Vx as the function of the ratio of field-like torque and damping-like torque from 0.1 to 10. (b) Vx as the function of damping-like torque , which indicates the polarization.

 Dynamic property of ferromagnetic skyrmion in an in-plane magnetic field

Topological non-trivial whirling spin textures such as magnetic skyrmions have recently triggered a lot of interest by virtue of their potential nanosized solitonic nature, stability and tunability. All these attributes, if combined with the efficient current-induced motion via spin orbit (SO) torque, make them attractive to build elementary units of digital information for spintronic logic devices and memories. In ultra-thin ferromagnetic films with Perpendicular Magnetic Anisotropy (PMA) sandwiched between a heavy metal with strong SO and an insulator (i.e. HM/FM/I), the chirality of this spin arrangements is ascribable to an antisymmetric exchange, Interfacial Dzyaloshinskii-Moriya Interaction (iDMI) [1,2]. Among the possible topological magnetic quasi-particles, which can be stabilised in such peculiar trilayer systems, a lower dimension (1D) non-trivial magnetic texture, named Domain Wall Skyrmion (DWSk), has been newly reported and experimentally observed using Lorentz Transmission Electron Microscopy [3]. The DWSk is a 1D topological excitation sharing the same integer topological charge, noted Q, as conventional 2D skyrmions (Q = 1). It is describable as a localized 2π rotation of the In-Plane (IP) component of magnetization within a Néel domain wall which stands between two perpendicular magnetic domains. It can be intuitively viewed as the counterpart of a 2π-vertical Bloch lines coming about in Bloch instead of Néel domain walls. Skyrmions with chiral kinks inside the otherwise homochiral domain wall, hereinafter called Double Skyrmion (DSk), have been numerically predicted as a soliton solution [4,5]. In our work, we investigated the stability and the spin-driven dynamics of the DSk in a sample of the type Pt/Co(0.9nm)/MgO [6] using micromagnetic simulations [7]. Interestingly, as shown in Fig.1a, we are able to stabilise simultaneously various spin-textures including skymion (Sk, Q=1), double skyrmion (DSk, Q=2) and domain wall skyrmion (DWSk, Q=1). It is found that such magnetization pattern is robust against both thermal fluctuations and normally distributed anisotropy, for different grain-like region sizes. We also proved the resilience of double skyrmion to material parameters variation (DMI, anisotropy) and applied field. The static analysis is complemented with the investigation of the dynamic regime of motion under electrical current. Fig.1b shows the trajectory of the double skyrmion induced by spin-orbit torque. In the limit of moderate injected current the skyrmion and the double skyrmion behave similarly (Fig.1c). However, depending on the geometry of the injected current the double skyrmion can be annihilated or splits in two skyrmions, opening the path to conceive logical devices.

**References:**

[1] I.E. Dzyaloshinskii Sov. Phys. JETP 5, (1957), J.Phys. Chem Solids, 4, 241 (1958) [2] T. Moriya Phys. Rev Lett., 4, 228, (1960), Phys. Rev. 120, 91, (1960) [3] Li at al., https://arxiv.org/abs/2004.07888 (2020) [4] Kuchkin et al., Phys. Rev. B 102, 144422 (2020) doi.org/10.1103/PhysRevB.102.144422 [5] S.-G. Je Current Applied Physics 21, 175 (2021) doi.org/10.1016/j.cap.2020.10.021 [6] Juge et al., Phys. Rev. Applied 12, 044007 (2019) doi.org/10.1103/PhysRevApplied.12.044007 [7] Vansteenkiste et al. AIP Adv. 4, 107133 (2014). doi/10.1063/1.4899186

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/469f5808fec70a1e12fc572d65ebf2ad.jpg)
Fig. 1: a) Multidomain equilibrium state of 0.5µmx0.5µm ideal Pt/Co(0.9nm)/MgO dot at 0K and no applied magnetic field. The color scale is associated to the perpendicular component of magnetization. b) Sequence of micromagnetic simulation snapshots illustrating the current-induced SOT driven DSk dynamics at 0K with Japp=50x1010 A/m2. The skyrmion Hall angle of DSk is 64°. c) Comparison of DSk and Sk velocities (orange) and effective diameters (black) as a function of injected current density.

 Static and dynamic behaviour of double skyrmions in Pt/Co/MgO trilayer

Chiral interactions between magnetic moments in ferromagnetic systems with broken inversion symmetry has attracted lots of attention in the recent years. Dzyaloshinskii-Moriya interaction[1] (DMI) present in these systems leads to an ordering of the spins that forms various exotic spin configurations such as chiral domain walls[2] and skyrmion[3]. DMI is an anti-exchange interaction which prefers to cant the neighbouring spins orthogonal to each other that forms whirling structures such as skyrmions. A magnetic skyrmion is a topological object having particle-spin like configuration, which can be stabilized in ferromagnetic systems with DMI. Recently magnetic skyrmions are proposed as promising candidates for non-volatile memory and logic applications due to their nanoscale size, high stability, high density and small driving currents required for their motion. Besides memory applications, skyrmions also hold great potential in the emerging area of magnonics where collective spin excitations (magnons) are used as a medium of information processing. In addition, observation of skyrmions on low Gilbert damping materials may allow the use of the skyrmions for manipulations of magnon transport. Recently skyrmion generation at an antidot is demonstrated using micromagnetic simulations[4]. In this work, we present the various routes for obtaining giant tunability in microwave properties based on skyrmion dynamic via micromagnetic simulations using Mumax3. We have investigated microwave properties of magnetic skyrmions in a circular nanodisk having antidot at the centre. An in-plane ac magnetic field excites gyrotropic dynamics with a clockwise and a counter-clockwise mode. We have systematically investigated skyrmion resonant modes as a function of DMI and perpendicular magnetic anisotropy (K). Large variation of resonant modes as a function of small out-of-plane bias magnetic field has been demonstrated. Simulated microwave spectrum of this Néel type skyrmion in an antidot nanodisk has been shown in figure 1(a) and it is compared with an antidot nanodisk having uniform OOP magnetization without any skyrmions. It is to be noted that there is no external bias field applied for these results in fig 1(a). Microwave responses are found in 13-22 GHz frequency range. The most prominent modes are found at 13 GHz and 17 GHz for the disk with a skyrmion and no skyrmion, respectively. Therefore, the modes at 13.1 GHz can be associated to the skyrmion resonances whereas the other higher frequency modes arise from the bulk magnetization (OOP) of the disk. Next, we demonstrate the tunability of the skyrmion resonance modes by applying a small bias field (Hext) along the OOP direction of the nanodisk with a single skyrmion. We have varied the polarity and the magnitude of the OOP field. The results are shown in fig 1(b) where the microwave resonance spectra are shown at different external fields: 0 to 35 mT. A relatively large frequency shift of 3.3 GHz has been observed for a change of 35 mT bias field. To further explore the functionalities, we have investigated skyrmion stabilization with various combinations of the parameters, DMI and K. Néel skyrmion is found to be stabilized at K = 0.25×106 J/m3 and DMI = 1.8×10-3 J/m2. Next, we have varied the K from 0.22 to 0.34 (×106) J/m3 for a fixed value of DMI = 1.8 mJ/m2 and we have varied the parameter DMI from 1.6 mJ/m2 to 1.9 mJ/m2, for a fixed value of K = 0.25×106 J/m3 shown in fig 2(a & b) respectively. Furthermore, a broad microwave tunability of 9.8 GHz has been shown with a variation of magnetic anisotropy of 1×105 J/m3. The manifestation of inertial mass of the skyrmion, its dependence on the resonance responses and the size of the skyrmion are explained by using Thiele’s equation. The inertial mass of the skyrmion is estimated to be in the order of 10-23 kg. We believe these results will have potential implications in the area of skyrmion based nano-scale reconfigurable microwave devices.

**References:**

[1] Dzyaloshinsky 1958 A Thermodynamic Theory of Gadolinium Molybdate J. phys.chem.solids 46 763–72 [2] DeJong M D and Livesey K L 2017 Domain walls in finite-width nanowires with interfacial Dzyaloshinskii-Moriya interaction Phys. Rev. B 95 054424 [3] Fert A, Reyren N and Cros V 2017 Magnetic skyrmions: Advances in physics and potential applications Nat. Rev. Mater. 2 17031 [4] Kumar Behera A, Murapaka C, Mallick S, Bhusan Singh B and Bedanta S 2021 Skyrmion racetrack memory with an antidot J. Phys. D. Appl. Phys. 54

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/b61b65d775f8963b8b27c1292b1525e5.jpg)
Fig 1. (a) Dynamic responses of a disk with a skyrmion and without a skyrmion in an antidot and (b) Skyrmion resonance modes for different OOP bias magnetic field ranging from 0-35 mT.

![](https://assets.underline.io/uploads/markdown_image/1/image/4ce77cfb5ab580ce27e4122c11df437d.jpg)
Fig 2. (a) Variations of the gyrotropic modes of skyrmion as a function of K for a fixed DMI = 1.8 mJ/m2 and (b) variations of the gyrotropic modes of skyrmion as a function of DMI for a fixed K = 0.25×106 J/m3.

 Tunable microwave properties of a skyrmion in an antidot nanodisk structure

Magnetic skyrmions are topologically protected spin texture with diameters in the nanometer length scale. Thus, skymions possess profound applications for next-generation spintronics, such as racetrack memory and logic gates. In order to be used in devices, it is necessary to consider how to manipulate magnetic skyrmions. By reading literature，we learned that skyrmion has two driving methods, one is current drive and the other is voltage-controlled magnetic anisotropy (VCMA) drive. The method of using the potential well generated by the VCMA can induce the motion of the skyrmion, which can not only avoid the Joule heating caused by the current-induced methods, but also simplify the existing complex structure. Therefore, studying the movement properties of skyrmion induced by potential wells generated by VCMA alone is necessary. First we study the possibility of the potential well induced skyrmion movement within a certain distance. Then, the relevant factors that affect the distance of skyrmion induction, such as the D value of DMI, the perpendicular magnetic anisotropy (PMA) constant Ku, the external magnetic field H, and the relation between the size of skyrmion and the inductive distance, are studied. The simulation result is shown in Fig.1. All the parameters that cause the radius of skyrmion to change will affect the maximum sensing distance of skyrmion. Finally, the continuous motion scheme of skyrmion induced only through the potential well generated by VCMA explored in Fig.2. In the picture, the green area represent the voltage-controlled potential well and 1, 2 are two sets of potential energy wells driven by inverted square wave voltage signals. In this way, setting a spaced potential well with alternating voltage control on a long track could enable the skyrmion to achieve continuous motion on the track. Overall, the method of using the potential well generated by the VCMA can realize the drive control of skyrmion continuous movement. This work provide guidance for the design of low energy consumption voltage-driven skyrmion driving method and a new concept for track memory.

**References:**

[1] Wang, X. et al. Efficient skyrmion transport mediated by a voltage controlled magnetic anisotropy gradient. Nanoscale 10, 733-740, doi:10.1039/c7nr06482a (2018). [2]Zhang, X., Ezawa, M. & Zhou, Y. Magnetic skyrmion logic gates: conversion, duplication and merging of skyrmions. Scientific Reports 5, doi:10.1038/srep09400 (2015).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/cc2100a2c7bf12d613fce3df0996541b.jpg)
Fig.1

![](https://assets.underline.io/uploads/markdown_image/1/image/6a2127e6f4db009ab5c521b6e9a3b34b.jpg)
Fig.2

 Potential Well Inducing the Motion of Skyrmion and Sensing Distance

Magnetic skyrmion is a non-trivial topological object which originates due to the interplay among various types of interactions such as magneto-static, anisotropy, dipolar, exchange interaction, and Dzyaloshinskii-Moriya interaction (DMI). In magnetic thin films, skyrmions originate through the competition between interfacial DMI and exchange interaction1. Patterned nanostructures provides a pathway to predictably stabilize and thus allow understanding their behavior in presence of different important parameters in a more controlled way (ref.). Moreover, due to the potential application of skyrmions in spintronic devices such as race-track memory etc., several studies of their creation, stability, and dynamics in nano wires have been reported1. Skyrmions in nanostructures can be dynamically controlled by magnetic field gradiant3 and spin-polarized current1. In case of high-speed skyrmion-based magnetic storage devices, large spin-polarized current density is required to control skyrmion dynamics. This in turn causes Joule heating and skyrmion Hall effect4 which limits the efficiency of skyrmion-based devices. These issues can be resolved by employing magnon spin current as a tool to control skyrmion propagation. However, very few studies have so far reported dynamics of skyrmions in presence of temperature gradient in a nanotrack5. L. Kong et al. reported the systematic investigation of the dynamics of Bloch skyrmions in presence of the temperature gradient. Skyrmions are found to move from region of lower to higher temperature. Moreover, the velocity of skyrmions is found to increase with the increase of thermal gradient5. Magnon spin current generated due to thermal gradient diffuses from hot to the cold region. This magnon spin current applies a negative torque on the skyrmion resulting in movement of skyrmion cold to hot region4. This study also reports that the skyrmion Hall component is absent or very small with the magnon spin current. A few other studies report the role of temperature on the current induced dynamics which demonstrate the velocity of skyrmion for the constant current density increases with the temperature 7. Although the dynamical behavior of Bloch skyrmions in presence of a temperature gradient has been reported in literature, there is no comprehensive studies performed to understand the dynamics of Neel skyrmion in presence of temperature gradient. In this work, we performed micromagnetic simulations to investigate the dynamics of Néel skyrmions in a race track of Co/Pt in presence of the temperature gradient. We considered a Co/Pt track with dimensions of 100 nm x40 nm x 1nm. Individual skyrmions were stabilized due to the i-DMI of ~3 mJ/m2 in the nanotrack and thereafter, their dynamics in presence of a temperature gradient7,8 was studied. Our studies reveal that similar to Bloch skyrmions, the temperature gradient plays a significant role on the velocity of Neel skyrmions in the nanotrack. The skyrmions move from cold to the hotter region and the velocity increases as the temperature gradient is increased (Fig. 1(b)). Furthermore, we have also investigated the role of Gilbert damping constant on the skyrmion velocity in presence of thermal gradient on the track. Our results show that the velocity is inversely proportional to the values damping constant. The details of these results are discussed in this work.

**References:**

1 J. Iwasaki, et al. Nat. Nanotechnol. 8, 742 (2013). 2 A. Casiraghi, et al. Nat. Commun Phys. 2, 145 (2019). 3. J. J. Liang et al. New J. Phys. 20, 053037 (2018). 4 G. Chen, et al. Nat. Phys 13, 112–113 (2017). 5 L. Kong et al. Phys. Rev. Lett. 111, 067203, (2013). 6 M T Islam et al. J. Phys.: Condens. Matter 31, 455701 (2019). 7 Litzius, K. et al. Nat Electron 3, 30–36 (2020). 8 Everschor, K. et al. Phys. Rev. B 86, 054432 (2012).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/33fa4a9ffb9f74e344b2669dae0c9fd4.jpg)
Fig.1.(a) Position of a skyrmion in a nanotrack in presence of temperature gradient at three different times. (b) The velocity of Skyrmion as a function of the temperature gradient in K/nm.

 Thermal gradient driven dynamics of Neel skyrmions in a nanoracetrack

Skyrmions have been proposed for racetrack memory [1], but there are still many questions about the best way to detect them electronically. The most promising approaches so far have used the transverse voltage due to the skyrmion Hall effect, but the signal is still very weak [2, 3]. Here we use simulations consider alternative vertical detection methods using magnetoresistance. A spin polarized current is used to move a Néel skyrmion down a racetrack that contains a circular magnetic dot above it. When the dot is flush with the racetrack, exchange interactions dominate, while when it is separated by a 1 nm thick tunnel barrier, the interactions are mainly magnetostatic. The trajectory of the skyrmion over a period of ~ 4 ns is calculated using mumax3micromagnetics simulation software. Results for different dot and skyrmion core magnetization directions are used as a guide to predict the magnetoresistance as a function of time, which would be used to detect individual skyrmions. For these model studies, a saturation magnetization of 5.8 x 105 A/m, exchange stiffness 1.5 x 10-12 J/m, anisotropy 8 x 105 J/m3, and damping parameter 0.1 were used for both the racetrack and 20 nm diameter detection dot. In addition, to stabilize 20 nm diameter skyrmions, the racetrack also had a Dzyaloshinskii-Moriya Interaction strength of 3 x 10-3 J/m2. A current density of 1 x 1012 A/m2 with spin polarization 0.4 was used to move the skyrmions. In all cases there was a transverse deflection due to the skyrmion Hall effect. When the dot is flush with the racetrack, exchange coupling to the skyrmion spin texture background can be treated as an effective magnetic field with an antiskyrmion texture [4]. When there is a 1 nm MgO tunnel barrier separating the dot and racetrack, magnetostatic effects dominate and the magnitude of the interaction field is much smaller. However, simulations show that the vertical tunnel current is dominated by the volume directly below the dot. There will be a transient change in the magnetoresistance as a skyrmion passes by the dot detector. With magnetostatic coupling both the trajectory and skyrmion size as a function of time differ, depending on whether the dot and skyrmion core magnetizations are parallel or antiparallel. When they are parallel (Figure 1), the skyrmion diameter grows slightly as it approaches the dot, since the dipolar field tends to orient misaligned spins. The spatial field gradient due to the dot causes a repulsive deflection around the dot. When the skyrmion core and dot magnetiation are antiparallel (Figure 2), the skyrmion slightly shrinks in diameter as it approaches the dot, and there is an attractive interaction. The magnitudes of the deflection depend on the dot and skyrmion stack diameters and thicknesses, but as long as part of the skyrmion passes under the dot, it should also be possible to differentiate core up and core down skyrmions. For Figure 1 there would be a transient drop in resistance, and for Figure 2 a transient rise. If in-plane magnetization dot detectors were used, there would also be characteristic magnetoresistance transients that are sensitive to the in-plane component of the skyrmion spins. This could enable Néel and Bloch skyrmions to be differentiated, if magnetized along the racetrack axis, and chirality detection of Bloch skyrmions, if magnetized perpendicular to the racetrack axis.

**References:**

1. A. Fert, V. Cros, J. Sampaio, Nat. Nanotech. 8 152 (2013). 2. D. Macciarello, et al., Nat. Nanotech. 13, 233 (2018). 3. K. Zeissler, et al., Nat. Commun, 11, 428 (2020). 4. K. Hamamoto, M. Ezawa, and N. Nagaosa, Phys. Rev. B 92, 115417 (2015).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/169f90678345b92ecec0d1db3eed2d01.jpg)
20 nm Néel skyrmion trajectory along a 200 nm x 80 nm racetrack, passing by a 20 nm diameter dot separated by a 1 nm tunnel junction barrier. The skyrmion core and dot have parallel magnetizations. a) at beginning, b) repulsion near the dot, and c) at the end of the racetrack.

![](https://assets.underline.io/uploads/markdown_image/1/image/1fd3076c0db495724367847365e0bfa1.jpg)
Similar conditions to those of Figure 1 except that the dot magnetization direction has been reversed, so that it is antiparallel to that of the skyrmion core. Here the interaction near the dot is attractive.

 Simulations of Magnetoresistive Detection of Skyrmions

Second harmonic Hall voltage (SHV) method [1] has been widely used to measure spin-orbit torques (SOTs) in heavy metal/ferromagnet (HM/FM) bilayers. It is typically applied for the single domain regime while the signals in multi-domain regimes are almost always ignored due to the complication from domain structures. In this study, we analyze the SHV data taken in a magnetic multilayer made of W/Pt/Co/SiO2. First harmonic planar Hall effect (PHE) and anomalous Hall effect (AHE) measurements reveal weak in-plane anisotropy of the Co layer. At low temperatures, the PHE loops become asymmetric, which indicate sizable Dzyaloshinskii-Moriya interaction (DMI) at the Pt/Co interface due to inversion symmetry breaking and strong spin-orbit coupling [2]. Typical sheared hysteresis for labyrinth domains is observed in AHE measurements. Interestingly, the second harmonic Hall voltage loops show “humps” in the intermediate field range where DMI-induced topological Hall effect and magnetic bubbles are typically observed. We explain such enhancement in SHV as a result of chiral Néel wall formation and efficient SOT driven domain wall motion [3]. Our results demonstrate that second harmonic voltage measurement can be utilized to detect chiral spin structures in magnetic multilayers.

**References:**

[1] M. Hayashi et al, Phys. Rev. B 89, 144425 (2014). [2] M. Heide et al, Phys. Rev. B 78, 140403(R) (2008). [3] A. V. Khvalkovskiy et al, Phys. Rev. B 87, 020402(R) (2013).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d4f5c98afb392aa4644f93209f07ec09.jpg)
Fig.1 Planar Hall effect measurement under in-plane field results for a W(3)/Pt(1)/Co(1.2)/SiO2(5) device.

![](https://assets.underline.io/uploads/markdown_image/1/image/69bc664348995b3c5e32aa62bbd1c7ca.jpg)
Fig.2 First and second harmonic Hall voltages under a nearly out-of-plane field scan. FM, Néel, and HTH denote ferromagnet, Néel domain wall, and head-to-head domain wall, respectively.

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