United States

Recently, it was discovered that without any external magnetic field, only by a single laser pulse, selective and reversible photo-magnetic switching in Co-doped yttrium iron garnet films (YIG:Co) can be obtained. This recording was characterized by an extremely short switching time of about 20 ps at room temperature. Moreover, because of its resonant character the write–read events are accompanied by an unprecedentedly low heat load [1]. Microscopically, in this mechanism incident linearly polarized pump pulse excites strongly anisotropic garnet ions, generating an effective field of photo-induced magnetic anisotropy [2],[3].&lt;br/&gt;Here, we demonstrate the tunability of both magnetocrystalline and photo-induced anisotropies by the engineering of Co-doped thin iron garnet films and temperature variation. All experimental results were obtained using the time-resolved two-color pump and probe setup at Faraday geometry. The setup was tailored to perform the time-resolved femtosecond single-shot imaging on a CCD camera assuring obtain high temporal and spatial sensitivity.&lt;br/&gt;Through modifying the pump light fluence we were able to introduce a change in the light-induced anisotropy field. It allowed us to observe the change of switching times of magnetization and decrease it even more by two times but at the expense of the greater energy required to achieve the switching threshold using a single pump laser pulse. On the other hand, by steering the sample temperatures in the 100-450 K, the range we triggered a change in the magnetocrystalline anisotropy field. Increasing it allowed us to decrease the energy dissipation threshold required for switching over five times. However, decreasing the switching threshold comes at the expense of a longer switching time. Moreover, we demonstrate that the photo-magnetic recording-based garnet medium operates within an extremely large temperature range between -100°C to 100°C. This finding reveals the principles to be employed in achieving cold and ultrafast magnetic recording in dielectrics far beyond today’s state-of-the-art.

2022 Joint MMM-INTERMAG

Control of switching time and energy dissipation for ultrafast photo-magnetic recording in dielectrics

The 15th Joint MMM-INTERMAG Conference (2022 Joint) is sponsored jointly by AIP Publishing and the IEEE Magnetics Society. Members of the international scientific and engineering communities interested in recent developments in fundamental and applied magnetism are invited to attend and contribute to the technical sessions. The technical program will include invited and contributed papers in oral and poster sessions, invited symposia, a plenary session, and an evening session, with about 1500 presentations overall. This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

The 2022 Joint Conference will offer both in-person and prerecorded on-demand content.

For tickets to the event, please register at the following link: https://magnetism.org/register

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This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

Recently, it was discovered that without any external magnetic field, only by a single laser pulse, selective and reversible photo-magnetic switching in Co-doped yttrium iron garnet films (YIG:Co) can be obtained. This recording was characterized by an extremely short switching time of about 20 ps at room temperature. Moreover, because of its resonant character the write–read events are accompanied by an unprecedentedly low heat load [1]. Microscopically, in this mechanism incident linearly polarized pump pulse excites strongly anisotropic garnet ions, generating an effective field of photo-induced magnetic anisotropy [2],[3]. Here, we demonstrate the tunability of both magnetocrystalline and photo-induced anisotropies by the engineering of Co-doped thin iron garnet films and temperature variation. All experimental results were obtained using the time-resolved two-color pump and probe setup at Faraday geometry. The setup was tailored to perform the time-resolved femtosecond single-shot imaging on a CCD camera assuring obtain high temporal and spatial sensitivity. Through modifying the pump light fluence we were able to introduce a change in the light-induced anisotropy field. It allowed us to observe the change of switching times of magnetization and decrease it even more by two times but at the expense of the greater energy required to achieve the switching threshold using a single pump laser pulse. On the other hand, by steering the sample temperatures in the 100-450 K, the range we triggered a change in the magnetocrystalline anisotropy field. Increasing it allowed us to decrease the energy dissipation threshold required for switching over five times. However, decreasing the switching threshold comes at the expense of a longer switching time. Moreover, we demonstrate that the photo-magnetic recording-based garnet medium operates within an extremely large temperature range between -100°C to 100°C. This finding reveals the principles to be employed in achieving cold and ultrafast magnetic recording in dielectrics far beyond today’s state-of-the-art.

poster

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.

Estimating Interference with Two-Dimensional Viterbi Algorithm for Bit Patterned Media Recording

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.

Improving Serial Detection with MAP Algorithm for Bit Patterned Media Recording

Since conventional HDDs that use perpendicular magnetic recording suffer from the problems of thermal stability, superparamagnetic limit, and so on, bit-patterned media recording (BPMR) system is a promising candidate for the next generation of data-storage system to achieve areal density (AD) beyond one terabit per square inch (Tb/in2) [1]. To increase the AD of the BPMR, the space between islands in both the down- and cross-track directions must be reduced. Consequently, data detection is interrupted by inter-symbol interference (ISI) and inter-track interference (ITI). In this paper, we propose a bit-flipping (BF) scheme using k-means algorithm for BPMR system to improve a bit error rate (BER) performance. A partial response maximum likelihood (PRML) detector which consists of a two-dimensional minimum mean square error (MMSE) equalizer and a one-dimensional PR target soft-output Viterbi algorithm (SOVA) detector is used for equalization and detection, respectively. Fig. 1 shows a block diagram of the proposed BF scheme. We use sj,k = [sj-2, sj-1, sj, sj+1, sj+2] = [vj-1,k, vj,k-1, vj,k, vj,k+1, vj+1,k] as the input sequence of k-means algorithm in SOVA output sequence vj,k = [vj-1,k-1, vj-1,k, ..., vj+1,k, vj+1,k+1] to consider the effect of ISI and ITI and decrease the number of clusters. After k-means algorithm, we compare a sign between pj of the predicted value in predicted sequence pj,k = [pj-2, pj-1, pj, pj+1, pj+2] and vj,k in in SOVA output sequence vj,k. When the signs are different, we change the sign of vj,k. In the simulation, we compare the PRML detector with BF and no BF schemes on the BPMR channel [2]. Fig. 2 shows the BER performance of the PRML detector with BF scheme according to SNR. When the ADs are 2.5Tb/in2 and 3Tb/in2, the performance of the proposed scheme is 1.1dB and 2.8dB better than that of one LDPC code at 10-4 BER, respectively.

![](https://assets.underline.io/uploads/markdown_image/1/image/cac33bc2d083c848fe5e6236fb342fe3.jpg)

Fig. 1. Block diagram of the proposed bit-flipping scheme.

![](https://assets.underline.io/uploads/markdown_image/1/image/fd9bb39b0553ae861daced494f20aa6e.jpg)

Fig. 2. BER performance of the PRML detector with BF scheme according to SNR.

Bit-Flipping Scheme Using K-Means Algorithm for Bit-Patterned Media Recording

If hard disc drives were able to store information in more than one recording layer, or structure, their storage capacity should be greatly increased. In recent years microwave-assisted and heat-assisted magnetic recording (HAMR) on dual structure media have been investigated [1][2]. Techniques for deconvoluting the readback signals from two recording structures have also been developed [3]. Here we focus on dual structure, bit patterned media for use with HAMR. Each recording medium dot contained two, discrete storage structures with different Curie temperatures, Tc, allowing data to be written to one, or both structures depending on the maximum temperature to which the dots were heated. The effect of the dot diameter, D, and the thickness of the recording structures, z, was investigated, with the aim of maximising the user areal density (UAD). The Curie temperatures were optimised to maximise the joint switching probability (i.e. P = P1 × P2) for D = 6 nm and z1 = z2 = 5 nm. The switching probability peaked when (Tc1, Tc2) were around (460 K , 550 K), or (530 K, 460 K). Next, the magnetostatic field distributions from surrounding dots were calculated for a dot pitch of 10 nm. The switching probability as a function of magnetostatic field, Hd, and D is shown in fig. 1. The switching probability increased with D (volume). The switching probability was also calculated as a function of recording structure thickness, z. Fig. 2 shows P versus KuV, where KuV was evaluated at the point where the coercivity, Hc, was equal to the head field Hhead. In fig. 2 both D and z were varied in each data set. Hc was calculated from M-H loops, whilst Hhead depended on the distance from the write head to the recording structure. Higher KuV resulted in higher switching probabilities for both recording structures, but RL1 showed a much greater variation than RL2. To maximise the switching probabilities of both recording structures RL1 should have a larger volume than RL2. The UAD was calculated using Shannon's formula for channel capacity. Values of around 8.6 Tbits/in2 were predicted for dot areal densities of about 4.5 Tdots/in2.

![](https://assets.underline.io/uploads/markdown_image/1/image/2a94c3c7e71339febbff789a66d45499.jpg)

Fig 1 P vs. magnetostatic field from surrounding dots and D

![](https://assets.underline.io/uploads/markdown_image/1/image/bd241e8cdc22543bc4f245ec032c65c8.jpg)

Fig 2 P vs. KuV when Hc = Hhead. Inset: Hc vs. T

Optimising Dual Structure Patterned Media for Heat Assisted Magnetic Recording

Spin-torque oscillators (STO) with two field generation layers (dual FGL), have superior characteristics compared with conventional STOs with a single FGL used for microwave-assisted magnetic recording (MAMR) [1], [2]. As shown in Fig. 1, we used an integrated model with a STO inserted into the gap between the main pole (MP) and the trailing shield (TS) of a write head. The dual FGL STO was composed of two FGLs and one spin injection layer with negative polarization (nSIL). The thicknesses were FGL1 = FGL2 = 5 nm, IL1 = 2 nm, IL2 = IL3 = 1 nm, and nSIL = 3 nm. The STO was tilted by 25° with respect to the x axis. The MP – TS gap = 18.8 nm at the air bearing surface (ABS). We assumed an injected current from MP to TS, and no spin-torque between FGL1 and FGL2. Antiferromagnetic coupling, Jex, was assumed between FGL1 and FGL2. Fig. 2 shows the in-plane component of the magnetization of FGL1 and FGL2 (Min-plane/Ms) averaged over the FGL volume, relative to a unit vector lying in the plane of the FGL. When Min-plane/Ms = 1 the FGL rotates perfectly in the FGL plane. Fig. 2(a) shows Min-plane/Ms for Jex = -1.33 erg/cm2 [2] and (b) Jex = -10 erg/cm2. The average values of Min-plane/Ms between zero and 16.0 ns were 0.653 for (a) and 0.808 for (b), demonstrating a remarkable improvement in FGL rotation. We also performed calculations for Jex = +1.33, 0, -5, -50, and -100 erg/cm2. We confirmed that Jex = +1.33 did not work; there was a significant improvement between Jex = 0 and -10; while there was little change from -10 erg/cm2 for Jex = -50 and -100 erg/cm2. We also varied some other parameters, e.g., FGL thickness, exchange coupling between nSIL - TS, however, no distinct effect was found. The most dominant factor affecting the FGL rotation was the strength of the antiferromagnetic coupling between FGL1 - FGL2. We acknowledge financial support from ASRC.

![](https://assets.underline.io/uploads/markdown_image/1/image/f66b1d4affcc87fdccb31a87caeec52a.jpg)

Fig. 1 Schematic of dual FGL STO.

![](https://assets.underline.io/uploads/markdown_image/1/image/46ec27ce246774c0fd194facedc6a8ce.jpg)

Fig. 2 In-plane component of volume-averaged FGL rotation vs. time. nSIL Po = -0.3, J = 4.0×108 A/cm2, coil current = 0.2 ATpp. (a): Jex = -1.33 erg/cm2, (b): Jex = -10.0 erg/cm2. Degauss of write head was carefully done, then STO calculations were performed. Data from 14 – 16 ns are shown.

Micromagnetic Analysis of Dual FGL STO

Racetrack memory (RM) is a promising device for a new class of non-volatile memory because of its large storage capacity, fast data access, and low power consumption. Although many studies have been focused on improving the domain wall motion (DWM), only a few studies have been considered the effect of Joule heating on DWM. We first measured the domain wall velocity in the GdFeCo wires by applying single voltage pulses (Picosecond) and then observed the DWM by using a Kerr microscope. Figure 1. (a) shows the DWM in the GdFeCo wire as a function of current density, where the pulse widths of the applied current are 3 ns and 30 ns, respectively. By increasing the current density, DW velocity increases. It is clearly observed that the DW velocity becomes faster as the pulse duration becomes shorter [1][2][3]. In order to clarify the effect of Joule heating on the magnetic wire, we performed a numerical simulation based on the heat diffusion on the surface of the GdFeCo wire when both short and long pulse width currents were applied to the wire. Besides, we found that the temperature of the magnetic wire is higher when the pulse width was longer, as shown in Fig. 1(b). Therefore, the slower DWM in the magnetic wire was observed for a long pulse current of 30 ns due to the higher temperature of the magnetic wire. Consequently, we have experimentally demonstrated and theoretically explained that a short pulse current of 3 ns is faster than a long pulse width of 30 ns. ACKNOWLEDGEMENT This work was supported by the JSPS KAKENHI (grants 918023, 918043,918055)

![](https://assets.underline.io/uploads/markdown_image/1/image/3326716efb14c75c90173084fc43ba5c.jpg)

Figure 1.(a) The velocity of the magnetic domain wall in the Pt(5)/Gd24(Fe88Co22)76(20)/SiN(10) sample as a function of current density

![](https://assets.underline.io/uploads/markdown_image/1/image/9dd719a62293a5500fcf19a587c87e9f.jpg)

Figure 1.(b) Temperature distribution of GdFeCo magnetic wire in the short hand direction when pulsed current of 3ns and 30ns widths is applied

Influence of pulse width and Joule heating on current-induced domain wall motion

Microwave-assisted magnetic recording (MAMR) [1] is a technology which can potentially enable the areal recoding density to further increase. Notched media have been proposed [2] for improving MAMR writability as well as MAMR recording characteristics. In the present study, dependence of microwave assisted magnetization switching (MAS) and magnetic recording characteristics on layer anisotropy structure were investigated by micromagnetic simulations regarding the interlayer exchange constants for notched media. Fig. 1(a) illustrates the layer structure assumed in this study. Recording characteristics were evaluated in terms of signal-to-noise ratios (SNR) derived from the signal power for 200 kbpi and noise power for 1600 kbpi, where a 30-nm-wide single-pole-type head with a spin torque oscillator with the frequency at 22 GHz [3] was assumed in recording process. Fig. 1(b) presents the MAS plots where Aint12, Aint23, and Aint34 are 0.5, 0.5, and 0.7 μerg/cm and Hk4 is 35 kOe, which were estimated under the continuous AC with pulse DC fields. The figure shows that higher switchable max. averaged Hk (Hk,ave) is obtained when only one of the three layers has large Hk and the other two layers have low Hk. This study found that the MAS properties are sensitive to field durations of DC and AC. In the case when a continuous DC and pulse AC fields, higher switchable max. Hk,ave was obtained only when media structure has large Hk1, low Hk2 and low Hk3. The calculation also indicated that relatively high Aint34 (&gt; 0.5 μerg/cm) was preferable. Fig. 2 compares SNRs for three the type of Hk layer structure media where Aint12, Aint23, and Aint34 were fixed to 0.5, 0.5, and 0.7 μerg/cm. The figure suggests that SNR for large Hk1 structure media reaches 16.7 dB around 25 kOe of Hk,ave. On the other hand, SNRs for large Hk2 and large Hk3 structure media decrease with increase in Hk,ave. The poor SNRs attribute to low signal power for 200kbpi. This may be translated by the MAS properties for multilayer films sensitive to the field durations.

![](https://assets.underline.io/uploads/markdown_image/1/image/f69d815b065f51bebe061d7e60d1fa7e.jpg)

Fig.1. (a) Schematic of medium layer structure and anisotropy fields and (b) MAS plots for a multilayer.

![](https://assets.underline.io/uploads/markdown_image/1/image/fd78099fea71c0873d44bece0fddeedf.jpg)

Fig.2. SNRs comparison among three type of layer Hk structure media with r = 0.4.

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

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