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In bit patterned media recording (BPMR), to increase the areal density (AD), we must reduce the distance between the magnetic islands. This leads to severe two-dimensional (2D) interference. The 2D interference includes inter-symbol interference (ISI) and inter-track interference (ITI) from the down- and across-track directions, respectively. To combat the 2D interference, we can decompose the ISI and ITI into the serial model [1]. However, in [1], the authors just use the Viterbi algorithm for detection, the detection lacks prior information from the inner detector to the outer detector. In this article, we use the maximum a posteriori (MAP) algorithm to supply the prior information from the inner detector to the outer detector. Our proposed model is presented in Fig. 1. We use the serial GPR target in [1]. The received signal is feed to the horizontal detection (using MAP algorithm). Each output data of the horizontal detection &lt;i&gt;r&lt;/i&gt;[&lt;i&gt;j&lt;/i&gt;,&lt;i&gt;k&lt;/i&gt;] includes six probability values of 6 symbols, respectively. We use the argmax(&lt;i&gt;r&lt;/i&gt;[&lt;i&gt;j&lt;/i&gt;,&lt;i&gt;k&lt;/i&gt;]) to convert the signal &lt;i&gt;r&lt;/i&gt;[&lt;i&gt;j&lt;/i&gt;,&lt;i&gt;k&lt;/i&gt;] into the hard output &lt;i&gt;b&lt;/i&gt;[&lt;i&gt;j&lt;/i&gt;,&lt;i&gt;k&lt;/i&gt;]. &lt;i&gt;b&lt;/i&gt;[&lt;i&gt;j&lt;/i&gt;,&lt;i&gt;k&lt;/i&gt;] is used to extract the noise information and create the soft output signal [2] (i.e., &lt;i&gt;soft&lt;/i&gt;(&lt;i&gt;b&lt;/i&gt;[&lt;i&gt;j&lt;/i&gt;,&lt;i&gt;k&lt;/i&gt;])). The signal &lt;i&gt;soft&lt;/i&gt;(&lt;i&gt;b&lt;/i&gt;[&lt;i&gt;j&lt;/i&gt;,&lt;i&gt;k&lt;/i&gt;]) is detected by the MAP algorithm in the vertical detection and creates the probability of -1 or +1 in &lt;i&gt;s&lt;/i&gt;[&lt;i&gt;j&lt;/i&gt;,&lt;i&gt;k&lt;/i&gt;]. Then, &lt;i&gt;s&lt;/i&gt;[&lt;i&gt;j&lt;/i&gt;,&lt;i&gt;k&lt;/i&gt;] and &lt;i&gt;r&lt;/i&gt;[&lt;i&gt;j&lt;/i&gt;,&lt;i&gt;k&lt;/i&gt;] are entered to the log detection. We calculate the product of the probability of +1 from &lt;i&gt;s&lt;/i&gt;[&lt;i&gt;j&lt;/i&gt;,&lt;i&gt;k&lt;/i&gt;] and the sum of the probability from the symbols, which contains the input +1, in &lt;i&gt;r&lt;/i&gt;[&lt;i&gt;j&lt;/i&gt;,&lt;i&gt;k&lt;/i&gt;]. This product is assigned to a numerator. Then, we calculate the product of the probability of -1 from &lt;i&gt;s&lt;/i&gt;[&lt;i&gt;j&lt;/i&gt;,&lt;i&gt;k&lt;/i&gt;] and the sum of the probability from the symbols, which contains the input -1, in &lt;i&gt;r&lt;/i&gt;[&lt;i&gt;j&lt;/i&gt;,&lt;i&gt;k&lt;/i&gt;]. This product is assigned to a denominator. Finally, we calculate &lt;i&gt;d&lt;/i&gt; = log(numerator/denominator). If &lt;i&gt;d&lt;/i&gt; ≥ 0, â[&lt;i&gt;j&lt;/i&gt;,&lt;i&gt;k&lt;/i&gt;] = 1 and if &lt;i&gt;d&lt;/i&gt; &amp;lt; 0, â[&lt;i&gt;j&lt;/i&gt;,&lt;i&gt;k&lt;/i&gt;] = 0. The results of our proposed model are presented in Fig. 2. We compare our model with the serial detection in [2]. Our proposed model improved the bit error rate (BER) performance of the serial detection.

![](https://assets.underline.io/uploads/markdown_image/1/image/31b45afd062763fbaee54fb1167b8174.jpg)

Fig.1 Our proposed model.

![](https://assets.underline.io/uploads/markdown_image/1/image/0ef7ea1ba565b056af38a507b0793812.jpg)

Fig. 2 BER performance of the proposed model.

2022 Joint MMM-INTERMAG

Improving Serial Detection with MAP Algorithm for Bit Patterned Media Recording

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INTERMAG 2021

In bit patterned media recording (BPMR), to increase the areal density (AD), we must reduce the distance between the magnetic islands. This leads to severe two-dimensional (2D) interference. The 2D interference includes inter-symbol interference (ISI) and inter-track interference (ITI) from the down- and across-track directions, respectively. To combat the 2D interference, we can decompose the ISI and ITI into the serial model [1]. However, in [1], the authors just use the Viterbi algorithm for detection, the detection lacks prior information from the inner detector to the outer detector. In this article, we use the maximum a posteriori (MAP) algorithm to supply the prior information from the inner detector to the outer detector. Our proposed model is presented in Fig. 1. We use the serial GPR target in [1]. The received signal is feed to the horizontal detection (using MAP algorithm). Each output data of the horizontal detection r[j,k] includes six probability values of 6 symbols, respectively. We use the argmax(r[j,k]) to convert the signal r[j,k] into the hard output b[j,k]. b[j,k] is used to extract the noise information and create the soft output signal [2] (i.e., soft(b[j,k])). The signal soft(b[j,k]) is detected by the MAP algorithm in the vertical detection and creates the probability of -1 or +1 in s[j,k]. Then, s[j,k] and r[j,k] are entered to the log detection. We calculate the product of the probability of +1 from s[j,k] and the sum of the probability from the symbols, which contains the input +1, in r[j,k]. This product is assigned to a numerator. Then, we calculate the product of the probability of -1 from s[j,k] and the sum of the probability from the symbols, which contains the input -1, in r[j,k]. This product is assigned to a denominator. Finally, we calculate d = log(numerator/denominator). If d ≥ 0, â[j,k] = 1 and if d &lt; 0, â[j,k] = 0. The results of our proposed model are presented in Fig. 2. We compare our model with the serial detection in [2]. Our proposed model improved the bit error rate (BER) performance of the serial detection.

![](https://assets.underline.io/uploads/markdown_image/1/image/31b45afd062763fbaee54fb1167b8174.jpg)

Fig.1 Our proposed model.

![](https://assets.underline.io/uploads/markdown_image/1/image/0ef7ea1ba565b056af38a507b0793812.jpg)

Fig. 2 BER performance of the proposed model.

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One of the tools to move a skyrmion is an injected current[1,2], which gives rise to the risk of producing excessive heat. On the other hand, spin-waves (SW) are a promising way to drive skyrmions without producing such high temperatures and hence affecting the skyrmionic device's performance. Most recent work on SW-driven skyrmion motion is based on micromagnetic simulations on single-layers with either small magnetic track width (W), high SW excitation rf field (Arf), or low Gilbert damping constant α[3,4]. However, to complement the realization of skyrmion motion experimentally, simulations should meet experimentally achievable conditions at the same time: W ≥100nm, Arf ≤100mT and α &gt;0.05. The dataset and characteristic analysis we present concern SW driven skyrmion simulations by OOMMF[5] with fixed parameters, D=1.5 mJ/m2, MS=1004 KA/m, A=3.6 pJ/m, Ku=1 MJ/m3. We varied the α=0.02-0.06, W=40-100 nm and Arf=100-800 mT. The size of the relaxed skyrmion is from 20nm to 23nm when W increases from 40nm to 100nm, so there is more than 5 nm between the skyrmion to each sample edge at initial stage. Fig.1 shows the skyrmion displacement (dsky) with time along the SW propagation direction for various W is shown. When W &gt;50 nm, the skyrmion will initially move opposite to the SW propagation direction before moving forward at longer times. Fig.2 shows that with larger α, the skyrmion will have smaller initial velocity when moving backward. If the leading edge of the skyrmion is regarded as a planar wall on the sample, the domain wall (DW) plane is predicted to rotate when linear momentum transfer between the SW and DW occurs. The angle between the DW wall plane and the out-of-sample direction is defined as DW wall plane angle (ΦDW)[6]. Also, the inset shows that ΦDW of skyrmion leading edge has similar behaviour to the dsky, but with an offset. The skyrmion backward motion gives a possible observation result in the future and guidance about finding a balance between W and magnetic properties to have skyrmions with aiming motion type.

![](https://assets.underline.io/uploads/markdown_image/1/image/20d3a9acd993b9c5372ac498eb39b2f1.jpg)

Fig.1 Time-dependent skyrmion displacement in nm for various W.

![](https://assets.underline.io/uploads/markdown_image/1/image/3a0f9f542c8a882ca49dfb2dc79008fd.jpg)

Fig.2 Skyrmion displacement with time for various α. Inset: comparison between ΦDW and skyrmion displacement when α=0.06.

Transient retrograde motion of spin wave-driven skyrmions in multilayer wave guides

Magnonic devices are based on the wave properties of spin wave (SW), so that one can transport more information in a single device if modes of SWs are freely controllable. SWs in a narrow waveguide have eigenmodes due to standing waves formed in the width direction and the thickness direction [1, 2]. However, excitation of one specific mode of propagating SW has not been realized, yet. Here, we report that high SW mode can be selectively excited by employing SW conversion. We numerically investigated SW propagation in a T-shaped structure using MuMax3 [3]. A 5-nanometer-thick Py film is assumed as the device and a magnetic field of 70 mT is applied along the y-direction (Fig. 1). By applying an rf field to the cells 1 μm away from the junction, Damon-Eshbach spin waves (DESWs) are excited and travel along the x-direction. The DESWs induce a magnetization precession in the junction, which excites SWs in the magnetostatic backward volume wave configuration (BVSWs). First, we varied the SW resonant frequency (f=12, 18 GHz) using a device having w1=500 nm and w2=100 nm. As shown in Fig. 2(a) and (b), the wavelength of the DESW (λDESW) depends on f which is well known [2]. In addition, the BVSWs clearly change their wave front. To extract the mode number, we applied a Fourier transformation (FT) along the y-direction. The FT results indicate that the 1st mode is dominantly excited at 12 GHz but the 2nd mode is dominant at 18GHz (Fig. 2(d) and (e)). We next changed the device to have w2=500 nm and repeated the simulation at 18 GHz (Fig. 2(c)). The result indicates the 12th mode was dominant at this condition (Fig.2 (f)). To further understand the high-mode-excitation, we varied the frequency and plotted the FT peak height of each mode as a function of the ratio of the width to λDESW (Fig.2 (g)). The result implies that modes can be controlled by optimizing a T-shaped waveguide.

![](https://assets.underline.io/uploads/markdown_image/1/image/29fb2eeaedfa09fa59c4d092aa944455.jpg)

Fig.1. Overview of the calculation area. SWs are excited at the orange area.

![](https://assets.underline.io/uploads/markdown_image/1/image/fc18c5114e3e73123e5e447b234cee46.jpg)

Fig.2. (a)-(c) The z-component of the magnetization and (d)-(f) their corresponding FT of BVSWs. (f, w2) is (a, d) (12 GHz, 100 nm), (b, e) (18 GHz, 100 nm), and (c, f) (18 GHz, 500 nm). (g) The peak height of FT of each mode obtained from the device with w2=500 nm.

Mode selective excitation of spin waves

Spin torque nano-oscillators (STNO) generate high frequency output voltage signals upon injection of a spin polarized DC current into a magnetic tunnel junction. For their applications in wireless communications or neuromorphic computing schemes, a central aspect is the control of their phase dynamics via injection locking to an external rf signal [1, 2]. Many interesting, non-linear effects may arise at higher excitation levels that manifest themselves in the Arnold tongue [3, 4]. For the in-plane magnetized STNO (in-plane magnetized polarizer and free layer) [2], we provide here analytical expressions for (i) the asymmetric Arnold tongue boundaries, arising from the dependence of the forcing on the STNO power; (ii) the Takens-Bogdanov bifurcation threshold [4] which leads to a reduced locking range and (iii) the hysteresis boundaries [3,4]. These effects are discussed within a common model for ‘pure’ rf signals arising from either a spin polarized current or a magnetic field. ‘Pure’ refers to a well-defined direction ui (i=x,y,z ; ux the STNO easy axis) of the spin polarization or a magnetic field [2]. In a further study we also investigate synchronization at harmonic order n=1 (rf signal frequency close to the STNO frequency) for mixed spin polarization with x- and y- components. The important and unexpected finding here is that the STNO phase does not follow an arcsine (Adler-type) behavior [5] within the locking range, but shows a discrete jump that can be as large as Pi/2. Our analysis shows that the phase jump results from a competition between locking to the ux- and uy-components of the rf signal, where locking to the ux-component at n=1 is enabled through the fact that the free running oscillation power is non-constant with frequency components at f. This demonstrates that, for mixed rf signals, injection locking can be more complex, and that the results can be of interest for applications such as the phase jump for phase shift keying concepts [6]. MIG was supported by the French Space Agency (CNES) and the CEA NUMERICS program, which has received funding from the European Union's Horizon 2020 program (No 800945). Partial support was provided by the French National Research Agency (ANR-20-CE24-0002).

Arnold tongue of injection locked spin torque nano-oscillators under high and mixed driving signals

We theoretically investigate the transmission of exchange-dominated spin waves in a one-dimensional magnonic crystal (MC) with a periodic exchange bias field. By recasting the Landau-Lifshitz-Gilbert equation into an effective Schrödinger equation and establishing spin-wave functions, it is found that MCs with upward (up) and downward (down) magnetization, respectively, correspond to the rectangular N-fold barriers and wells for magnons. The broadband transmission spectra in up and down states are systematically investigated. We show the phenomena of the magnon resonant transmission in both states and calculate the resonant transmission wave functions, which are related to the magnon density. Our results also show a transmission spectra shift effect (TSSE) between up and down states, which is found to be general in this system. Furthermore, the TSSE is useful to design a type of magnon valve, the magnonic-crystal-based magnon valve (MCMV), which has a large on/off ratio and bandwidth. By high-throughput screening, 125 000 groups of parameters of the MC are calculated, and 1948 parameter groups of high-performance MCMVs are screened out. Our work clarifies the physical details of the exchange-dominated spin-wave transmission in rectangular N-fold barriers and wells and also provides a promising route for designing novel magnonic devices.

![](https://assets.underline.io/uploads/markdown_image/1/image/06606510a3d9fad6f0a47ccb6c39f8f6.jpg)

The magnon transmission in a one-dimensional MC. The schematic diagrams of the MC in (a) the up state and (b) the down state. (c) The broadband transmission spectra in up and down states, corresponding to the black curve (T↑) and red curve (T↓) respectively. The parameters of the MC are the following: N = 14, a = 10 nm, and b = 25.2 nm. (d) The difference between T↑ and T↓ as a function of ω/2π. ω/2π is shown in logarithm scale.

Magnon valve effect and resonant transmission in a one-dimensional magnonic crystal

The study of high-frequency spin dynamics in three-dimensional (3D) magnetic nanostructures may lead to paradigm-shifting, next generation devices including high-density spintronics and neuromorphic systems. Despite remarkable progress in fabrication [1], the measurement and interpretation of spin dynamics in complex 3D magnetic structures remains exceptionally challenging. Here we present the measurement of coherent spin waves within a 3D artificial spin ice (ASI) system using Brillouin light scattering. The 3D-ASI was fabricated by using a combination of two-photon lithography and thermal evaporation. Two spin-wave modes were observed in the experiment whose frequencies showed nearly monotonic variation with the applied magnetic field strength. Numerical simulations qualitatively reproduced the observed spin-wave modes. The simulated mode profiles revealed the collective nature of the modes extending throughout the complex network of nanowires, while showing spatial quantization with varying mode quantization numbers [2]. The study shows a well-defined means to explore high-frequency spin dynamics in complex 3D spintronic and magnonic structures. &lt;!--![endif]----&gt;

![](https://assets.underline.io/uploads/markdown_image/1/image/03908b10f9b4a86f48a8879cc94464fd.jpg)

Figure 1: Schematic of BLS measurement geometry. The measurement was performed at θ = 0°. (a) A scanning electron micrograph image of the sample (scale bar: 2 µm) with applied field (H) direction, (b) BLS spectra at H = 1.8 kOe. (c) Schematic of 3D-ASI network, (d) simulated equilibrium state of magnetisation at H = 1.6 kOe and (d) mode profiles of M1 at 1.6 kOe are shown. Respective legends are shown on right side.

Study of Coherent Spin Waves in a Three-Dimensional Artificial Spin Ice Structure

Magnonic and phononic crystals are artificial nanostructures with periodic lateral variations in their magnetic and acoustic properties [1,2]. Recently, there has been growing interest in the fundamental understanding of spin waves propagation in magnonic crystals because of their huge potential in a wide range of applications such as microwave resonators, filters and spin wave logic devices [3]. On the other hand, phononic crystal are periodic arrays of nano-objects with alternating elastic properties and mass densities, they have promising applications such as acoustic microsensing or signal processing. Besides these possible applications, the wealthy physics of the acoustic waves is one of the main reasons for the attention to their propagation. Finally, a magphonic crystal is also an artificial crystal that simultaneously presents magnonic and phononic behaviors [4,5]. The phononic and magnonic dispersions of Ni80Fe20 antidot arrays deposited on a Si substrate have been investigated by using Brillouin light scattering (BLS) and ferromagnetic resonance (FMR). The periodicity of the arrays is 400 nm with a nanohole diameter of 180 nm and with a thickness of 40 nm. The FMR technique has been employed to first investigate the magnetostatic modes in the center of the first Brillouin zone. We have thus highlighted the presence of several magnetostatic modes and we conducted micromagnetic simulations (OOMMF) that allowed us to determine the spatial location of these different modes. In a second step, we have used BLS to probe the phononic and magnonic features of the magphonic crystal. We have experimentally demonstrated the simultaneous presence of magnonic and phononic bandgaps. The different dispersion curves (both magnonic and phononic) as well as the potential applications of these magphonic crystals will be presented and discussed during the conference.

Magnonic and phononic dispersion in Ni80Fe20 array of antidots

In recent years, the research of long-distance transport of magnon in magnetic insulator showed a negative voltage in thermal excited magnons signal [1,2]. Subsequent research investigated the negative voltage and find its dependence on the thickness of YIG, which raising a challenge for traditional magnon diffusion model. Similar phenomenon of viscous electrons’ backflow in graphene inspired the magnon study. In graphene at low temperature where the scale of electron-electron scattering is much smaller than the scale of electron mean free path, the momentum-conserving process dominated will give make electrons behave as a viscous liquid and exhibit hydrodynamic phenomena as well as classical liquids [3], which could present a negative voltage at two edge of system. In magnons system, same momentum-conserving process would happen, even though magnon as a quasiparticle, people don’t emphasize the conservation laws of particular numbers. The conservation of magnon momentum will lead whole system relaxed to an collective equilibrium N(λ)=1/exp[(εk-k●λ)/kBT] , which is called displaced Planck distribution [4]. The collective velocity λ of the ordered motion as a function of coordinates must satisfy a Navier-Stokes type hydrodynamic equation. The backflow of viscous magnon naturally produce the negative signal in magnetic insulator system, as similar behaviors in graphene. In this work, based on Boltzmann method and considering momentum-conserving dominated process, we developed a collective hydrodynamic diffusion equation of magnons’ viscous fluid, without vanishing k-space scattering. We also found nonlinear properties of hydrodynamics magnon system. Two- dimensional calculations of magnon flux showed the viscosity of magnon fluid created two eddies near the heat source. And our result could not only fit experimental data precisely, but also reconstruct physical scattering process.

Collective Hydrodynamic Magnon Transport Study in Magnetically Ordered Insulator Based on Boltzmann Approach

Magnetic losses in ferromagnetic laminations have been studied for years [1][2]. But, magnetization processes are highly complex, and the ideal model is still in need. Classic methods working in the time domain combine the magnetic diffusion equation solved in 1D (Eq. 1) to a viscosity-based magneto dynamic model (differential equation). This concurrent resolution has already been made through a strong formulation: the diffusion equation temporal term (dB/dt) is replaced by its differential equation expression [3][4]. This elegant method leads to a fast resolution but unfortunately incorrectly considers the excess losses linked to the domain wall motions kinetic. Improved differential equations have been proposed for a better losses consideration [5], but dB/dt can’t be isolated analytically, and the resolution always goes through complex and unstable iterative methods like the fixed point or the Newton-Raphson algorithm [6]. In this study, we propose to use a fractional viscosity-based magneto dynamic differential equation (Eq. 2) [7][8]. Isolation of the dB/dt term is possible by expressing the differential equation remaining terms using a fractional operator. Strong formulation is obtained, which can be solved without an iterative method. This alternative method gives access to local magnetic information (excitation and induction fields)(see Fig. 1). The number of dynamic parameters is limited (ρ a constant and n the fractional order) but large enough to add a degree of freedom in the simulation process and leads to a perfect fit of the experimental results.

![](https://assets.underline.io/uploads/markdown_image/1/image/a409d1a89a2335aef121586c9f4ab912.jpg)

Simulated local hysteresis cycles, imposed sinus Hsurf, 150 A/m, 150 Hz



Diffusion equation and fractional viscosity-based magneto dynamic model for electrical steel ferromagnetic hysteresis

Spin-orbit torque (SOT)-induced magnetization switching provides a potentially efficient alternative to spin-transfer torque switching in spin valves or magnetic tunnel junctions. Current induced SOT-switching of perpendicular magnetization is observed in an external magnetic field collinear with the current (but non-collinear with the magnetization), which, however, is impractical in device applications. In this work, we study magnetization switching induced by spin-orbit torque in W (Pt)/Co/NiO heterostructures with variable thickness of W and Pt heavy-metal layers, a perpendicularly magnetized Co layer, and an antiferromagnetic NiO layer [1]. Using current-driven switching, magnetoresistance and anomalous-Hall-effect (AHE) measurements, we determine the perpendicular and in-plane exchange-bias field. Several Hall-bar devices possessing in-plane exchange bias from both systems are selected and analyzed in relation to our analytical switching model of the critical current density as a function of Pt and W thickness, resulting in an estimation of the effective spin Hall angle and perpendicular effective magnetic anisotropy. We demonstrate in both the W(Pt)/Co/NiO systems deterministic Co magnetization switching without an external magnetic field, which is replaced by an in-plane exchange-bias field. Moreover, we show that due to a higher effective spin Hall angle in the W-based system than in Pt, the relative difference between the resistance states in the magnetization current switching to the difference between the resistance states in magnetic field switching determined by AHE (ΔR/ΔRAHE) is about twice as high in W-based devices than in Pt, while the critical switching-current density in W-based devices is one order lower than in Pt. The current switching stability and the training process are discussed in detail. This work is funded by National Science Centre, Poland, Grant no. 2016/23/B/ST3/01430 SpinOrbitronics. KG acknowledge support for conference participation by the EU Project POWR.03.02.00-00-I004/16.

Current-Induced Magnetization Switching of Exchange-Biased NiO Heterostructures Characterized by Spin-Orbit Torque

In magnetic recording media, to overcome superparamagnetism, bit patterned media recording (BPMR) is proposed as the potential technology. In BPMR, to increase the areal density (AD), the distance between the islands must be reduced. As the magnetic islands become closer, two-dimensional (2D) interference appears violently. The 2D interference includes inter-symbol interference (ISI) from the down-track direction and inter-track interference (ITI) from the across-track direction. We use detection, which applies the Viterbi algorithm (VA) or the maximum a posteriori (MAP) algorithm, to handle the 2D interference. Because the VA and MAP algorithm are designed to remove the 1D interference, when detecting ISI, we need to remove ITI and vice versa. In this article, we proposed an estimator to predict ITI. This estimator uses the 2D VA with reducing the states. We consider each column of the channel as a symbol like [1]. Our channel is symmetric and has a size of 3x3. Therefore, we have 36 states and 6 input branches for the trellis of the 2D VA. The received signal is feed to the ITI estimator. We detect along the across-track direction to estimate the ITI. Then, we remove the ITI from the received signal and apply the 1D VA to detect along the down-track direction. With the signal is detected by the 1D VA from the down-track direction (horizontal detection), we achieve the estimated original signal. Therefore, we can use the estimated original signal to estimate the interference along the down-track direction (ISI) and remove it from the received signal. Finally, we detect the received signal, which removed the ISI, with 1D VA along the across-track direction (vertical detection). Our proposed model is presented in Fig. 1. In Fig. 1, we use the serial GPR target form in [2]. In the simulation, we compare the serial detection [3], which is designed as the 2D VA, with our proposed model on the BPMR channel at 3 Tb/in2. The results are presented in Fig. 2. We can see that our proposed model achieves a 1 dB gain compared to serial detection at BER of 10-4.

![](https://assets.underline.io/uploads/markdown_image/1/image/9e9b914afc7cd2be0b36066f4345deee.jpg)


Fig. 1. Our proposed model.

![](https://assets.underline.io/uploads/markdown_image/1/image/7fb38bf6e2f0335e40674b0b98121c9a.jpg)

Fig. 2 BER performance of the proposed model.

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Transient retrograde motion of spin wave-driven skyrmions in multilayer wave guides

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