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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].&lt;br/&gt;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 &lt;i&gt;x&lt;/i&gt; 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, &lt;i&gt;J&lt;/i&gt;&lt;sub&gt;ex&lt;/sub&gt;, was assumed between FGL1 and FGL2.&lt;br/&gt;Fig. 2 shows the in-plane component of the magnetization of FGL1 and FGL2 (&lt;i&gt;M&lt;/i&gt;&lt;sub&gt;in-plane&lt;/sub&gt;/&lt;i&gt;M&lt;/i&gt;&lt;sub&gt;s&lt;/sub&gt;) averaged over the FGL volume, relative to a unit vector lying in the plane of the FGL. When &lt;i&gt;M&lt;/i&gt;&lt;sub&gt;in-plane&lt;/sub&gt;/&lt;i&gt;M&lt;/i&gt;&lt;sub&gt;s&lt;/sub&gt; = 1 the FGL rotates perfectly in the FGL plane. Fig. 2(a) shows &lt;i&gt;M&lt;/i&gt;&lt;sub&gt;in-plane&lt;/sub&gt;/&lt;i&gt;M&lt;/i&gt;&lt;sub&gt;s&lt;/sub&gt; for &lt;i&gt;J&lt;/i&gt;&lt;sub&gt;ex&lt;/sub&gt; = -1.33 erg/cm&lt;sup&gt;2&lt;/sup&gt; [2] and (b) &lt;i&gt;J&lt;/i&gt;&lt;sub&gt;ex&lt;/sub&gt; = -10 erg/cm&lt;sup&gt;2&lt;/sup&gt;. The average values of &lt;i&gt;M&lt;/i&gt;&lt;sub&gt;in-plane&lt;/sub&gt;/&lt;i&gt;M&lt;/i&gt;&lt;sub&gt;s&lt;/sub&gt; 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 &lt;i&gt;J&lt;/i&gt;&lt;sub&gt;ex&lt;/sub&gt; = +1.33, 0, -5, -50, and -100 erg/cm&lt;sup&gt;2&lt;/sup&gt;. We confirmed that &lt;i&gt;J&lt;/i&gt;&lt;sub&gt;ex&lt;/sub&gt; = +1.33 did not work; there was a significant improvement between &lt;i&gt;J&lt;/i&gt;&lt;sub&gt;ex&lt;/sub&gt; = 0 and -10; while there was little change from -10 erg/cm&lt;sup&gt;2&lt;/sup&gt; for &lt;i&gt;J&lt;/i&gt;&lt;sub&gt;ex&lt;/sub&gt; = -50 and -100 erg/cm&lt;sup&gt;2&lt;/sup&gt;. 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.&lt;br/&gt;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.

2022 Joint MMM-INTERMAG

Micromagnetic Analysis of Dual FGL STO

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INTERMAG 2021

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.

poster

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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.

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

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

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