France



I. INTRODUCTION&lt;br/&gt;In recent years, spin-torque transfer magnetic random access memory (STT-MRAM) has received tremendous attention due to its low switching energy, fast read/write speed, and high scalability [1]. However, thermal fluctuation and process variation have a detrimental effect on the data recovery of STT-MRAM, leading to both read errors and write errors [2]. Error correction codes (ECCs), such as the Hamming codes and Bose-Chaudhuri-Hoquenghem (BCH) codes have been employed to improve the reliability of STT-MRAM [2, 3]. Meanwhile, the channel quantizer that quantizes the signal read back from the STT-MRAM cell is critical to support the above ECCs, since high-precision analog-to-digital converters (ADCs) are not applicable for STT-MRAM. Various criteria have been proposed for designing the quantizer for STT-MRAM, including the Maximizing-Mutual-Information (MMI) criterion, the Maximizing-Cutoff-Rate (MCR) criterion, and the Optimizing- Polyanskiy-Poor-Verdu -Bound (PPVB) criterion [4]. However, the final choice of the quantization scheme is still relying on the decoding error rate performance of ECCs. It is typically evaluated by using computer simulations, which are too slow to reach the target error rate level of 10&lt;sup&gt;-12 &lt;/sup&gt;or below of data storage systems.&lt;br/&gt;&lt;br/&gt;In this work, we propose a union bound analysis which can accurately predict the word error rates (WERs) of ECCs with maximum-likelihood (ML) decoding over the quantized STT-MRAM channel. The derived bound provides a theoretical tool for comparing different quantization schemes at the targeted low error rate levels of data storage systems. Moreover, we also present a new criterion to design the quantizer by minimizing the WERs of ECC decoding that are obtained from the union bound analysis. Numerical results show that the union bound optimized quantizer can achieve better WER performance than the state-of-art quantizers for STT-MRAM.&lt;br/&gt;&lt;br/&gt;II. UNION BOUND ANALYSIS AND CHANNEL QUANTIZATION&lt;br/&gt;The union bound analysis is a theoretical technique that is used for bounding the error rate of ML decoding of ECCs. It was mainly developed for codes over symmetric channels [5]. In [6] and [7], more sophisticated union bound techniques were proposed for simple asymmetric channels, &lt;i&gt;i.e.&lt;/i&gt; the Z-channel and the general binary asymmetric channel (BAC). In this work, we consider the quantized STT-MRAM channel with both write errors and read errors, which can be modelled as a concatenation of a BAC and a binary-input multiple-output discrete memoryless channel. Our proposed union bound analysis first utilizes a function of a predefined multi-dimensional distance between two codewords to calculate the pairwise error probability (PEP) [7] of the quantized STT-MRAM channel. Note that this is different from the cases of the symmetric channels where the PEP is uniquely determined by the Hamming distance of the code. Based on the PEP, we can derive a union bound of the WER of ML decoding as a function of the weight distributions of ECC. Next, we further propose to use the WER obtained from the above union bound analysis as the criterion to design the channel quantizer of STT-MRAM. We also develop effective differential evolution algorithm to determine the optimum quantization boundaries for the multi-bit quantizers. Since our proposed union bound optimized quantizer takes into consideration the weight spectrum of ECC, it can achieve better error rate performance than the prior-art quantizers.&lt;br/&gt;&lt;br/&gt;III. NUMERICAL AND SIMULATION RESULTS&lt;br/&gt;The system parameters of STT-MRAM are taken from [2]. In the quantized STT-MRAM channel model, we assume the write error rate of 10&lt;sup&gt;-5&lt;/sup&gt;, and vary the mean normalized resistance spreads σ&lt;sub&gt;0&lt;/sub&gt;/μ&lt;sub&gt;0&lt;/sub&gt;=σ&lt;sub&gt;1&lt;/sub&gt;/μ&lt;sub&gt;1&lt;/sub&gt; of the low and high resistances to account for the influence of different process variations. The (72,64) extended Hamming code [3] is adopted to validate our analysis, and the simulated ML decoding performance is obtained using the Bahl, Cocke, Jelinek and Raviv (BCJR) decoder [5]. In Fig.1, we compare the decoding WERs provided by the union bound with those achieved by the BCJR decoder through simulations, with the proposed union bound optimized quantizer with different number of quantization bits &lt;i&gt;q&lt;/i&gt;. Observe that there is a noticeable gap between the union bound and simulated WERs for the case of &lt;i&gt;q&lt;/i&gt;=1. However, with two or more quantization bits, the performance gap becomes negligible, and when the WER is less than 10&lt;sup&gt;-6&lt;/sup&gt;, the union bound is almost identical to the simulated WER. In Fig. 2, we fix the quantization bit to be &lt;i&gt;q&lt;/i&gt;=2, and compare the WER performance with different types of quantizers. A nice agreement is observed again between the WERs predicted by the union bound and those obtained from simulations, for the various types of quantizers. Moreover, our proposed union bound optimized quantizer outperforms all the other quantizers. There is a larger performance improvement over the MMI quantizer, which is most widely adopted in the literature. This demonstrates the potential of our proposed union bound analysis in predicting the decoding error rate performance and in guiding the design of channel quantizer for STT-MRAM.&lt;br/&gt;&lt;br/&gt;Acknowledgement: This work is supported by RIE2020 Advanced Manufacturing and Engineering (AME) programmatic grant A18A6b0057 and Singapore MOE Tier 2 fund MOE2019-T2-2-123.

**References:**

[1] M. Marinella, “The future of memory,” in &lt;i&gt;Proc. Aerospace Conf, Big Sky, MN, USA&lt;/i&gt;, Mar. 2012, pp. 1–11.&lt;br/&gt;[2] K. Cai, and K. A. S. Immink, “Cascaded channel modeling, analysis, and hybrid decoding for spin-torque transfer magnetic random access memory (STT-MRAM),” &lt;i&gt;IEEE Trans. Magn.&lt;/i&gt;, vol. 53, no. 11, pp. 1–11, 2017.&lt;br/&gt;[3] K. Cai, Z. Qin, and B. Chen, “Channel capacity and soft-decision decoding of LDPC codes for spin-torque transfer magnetic random access memory (STT-MRAM),” in &lt;i&gt;Proc. Int. Conf. Comput., Netw., Commun. (ICNC), San Diego, CA, USA&lt;/i&gt;, Jan. 2013, pp. 550–554.&lt;br/&gt;[4] Z. Mei, K. Cai and L. Shi, “Information Theoretic Bounds Based Channel Quantization Design for Emerging Memories,” in &lt;i&gt;Proc. IEEE Information Theory Workshop (ITW), Guang Zhou, China&lt;/i&gt;, Nov. 2018, pp. 1–5.&lt;br/&gt;[5]W. E. Ryan and S. Lin, &lt;i&gt;Channel Codes: Classical and Modern&lt;/i&gt;, Cambridge, Cambridge University Press, 2009.&lt;br/&gt;[6] A. Barbero, P. Ellingsen, S. Spinsante, and O. Ytrehus, “Maximum likelihood decoding of codes on the Z-channel,” in &lt;i&gt;Proc. IEEE International Conference on Communications (ICC), Istanbul, Turkey&lt;/i&gt;, Jun. 2006, pp. 1–5.&lt;br/&gt;[7] G. Song, K. Cai and J. Cheng, “A union bound analysis for codes over binary asymmetric channels,” in &lt;i&gt;Proc. IEEE International Conference on Communications (ICC), Paris, France, &lt;/i&gt;May. 2017, pp. 1–5.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7687adc035569fbf9f2d0102c1f4556f.jpg)
Fig. 1. Comparison between analytical and simulation WERs, with union bound optimized quantizer with &lt;i&gt;q&lt;/i&gt;= 1,2,3.

![](https://assets.underline.io/uploads/markdown_image/1/image/ff97a646ffd6d77b07f4e57f875be682.jpg)
Fig. 2. Comparison between analytical and simulation WERs with different types of quantizers, for &lt;i&gt;q&lt;/i&gt;=2.

INTERMAG 2021

 Union Bound Analysis for Spin-Torque Transfer Magnetic Random Access Memory with Channel Quantization

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I. INTRODUCTION In recent years, spin-torque transfer magnetic random access memory (STT-MRAM) has received tremendous attention due to its low switching energy, fast read/write speed, and high scalability [1]. However, thermal fluctuation and process variation have a detrimental effect on the data recovery of STT-MRAM, leading to both read errors and write errors [2]. Error correction codes (ECCs), such as the Hamming codes and Bose-Chaudhuri-Hoquenghem (BCH) codes have been employed to improve the reliability of STT-MRAM [2, 3]. Meanwhile, the channel quantizer that quantizes the signal read back from the STT-MRAM cell is critical to support the above ECCs, since high-precision analog-to-digital converters (ADCs) are not applicable for STT-MRAM. Various criteria have been proposed for designing the quantizer for STT-MRAM, including the Maximizing-Mutual-Information (MMI) criterion, the Maximizing-Cutoff-Rate (MCR) criterion, and the Optimizing- Polyanskiy-Poor-Verdu -Bound (PPVB) criterion [4]. However, the final choice of the quantization scheme is still relying on the decoding error rate performance of ECCs. It is typically evaluated by using computer simulations, which are too slow to reach the target error rate level of 10-12 or below of data storage systems. In this work, we propose a union bound analysis which can accurately predict the word error rates (WERs) of ECCs with maximum-likelihood (ML) decoding over the quantized STT-MRAM channel. The derived bound provides a theoretical tool for comparing different quantization schemes at the targeted low error rate levels of data storage systems. Moreover, we also present a new criterion to design the quantizer by minimizing the WERs of ECC decoding that are obtained from the union bound analysis. Numerical results show that the union bound optimized quantizer can achieve better WER performance than the state-of-art quantizers for STT-MRAM. II. UNION BOUND ANALYSIS AND CHANNEL QUANTIZATION The union bound analysis is a theoretical technique that is used for bounding the error rate of ML decoding of ECCs. It was mainly developed for codes over symmetric channels [5]. In [6] and [7], more sophisticated union bound techniques were proposed for simple asymmetric channels, i.e. the Z-channel and the general binary asymmetric channel (BAC). In this work, we consider the quantized STT-MRAM channel with both write errors and read errors, which can be modelled as a concatenation of a BAC and a binary-input multiple-output discrete memoryless channel. Our proposed union bound analysis first utilizes a function of a predefined multi-dimensional distance between two codewords to calculate the pairwise error probability (PEP) [7] of the quantized STT-MRAM channel. Note that this is different from the cases of the symmetric channels where the PEP is uniquely determined by the Hamming distance of the code. Based on the PEP, we can derive a union bound of the WER of ML decoding as a function of the weight distributions of ECC. Next, we further propose to use the WER obtained from the above union bound analysis as the criterion to design the channel quantizer of STT-MRAM. We also develop effective differential evolution algorithm to determine the optimum quantization boundaries for the multi-bit quantizers. Since our proposed union bound optimized quantizer takes into consideration the weight spectrum of ECC, it can achieve better error rate performance than the prior-art quantizers. III. NUMERICAL AND SIMULATION RESULTS The system parameters of STT-MRAM are taken from [2]. In the quantized STT-MRAM channel model, we assume the write error rate of 10-5, and vary the mean normalized resistance spreads σ0/μ0=σ1/μ1 of the low and high resistances to account for the influence of different process variations. The (72,64) extended Hamming code [3] is adopted to validate our analysis, and the simulated ML decoding performance is obtained using the Bahl, Cocke, Jelinek and Raviv (BCJR) decoder [5]. In Fig.1, we compare the decoding WERs provided by the union bound with those achieved by the BCJR decoder through simulations, with the proposed union bound optimized quantizer with different number of quantization bits q. Observe that there is a noticeable gap between the union bound and simulated WERs for the case of q=1. However, with two or more quantization bits, the performance gap becomes negligible, and when the WER is less than 10-6, the union bound is almost identical to the simulated WER. In Fig. 2, we fix the quantization bit to be q=2, and compare the WER performance with different types of quantizers. A nice agreement is observed again between the WERs predicted by the union bound and those obtained from simulations, for the various types of quantizers. Moreover, our proposed union bound optimized quantizer outperforms all the other quantizers. There is a larger performance improvement over the MMI quantizer, which is most widely adopted in the literature. This demonstrates the potential of our proposed union bound analysis in predicting the decoding error rate performance and in guiding the design of channel quantizer for STT-MRAM. Acknowledgement: This work is supported by RIE2020 Advanced Manufacturing and Engineering (AME) programmatic grant A18A6b0057 and Singapore MOE Tier 2 fund MOE2019-T2-2-123.

**References:**

[1] M. Marinella, “The future of memory,” in Proc. Aerospace Conf, Big Sky, MN, USA, Mar. 2012, pp. 1–11. [2] K. Cai, and K. A. S. Immink, “Cascaded channel modeling, analysis, and hybrid decoding for spin-torque transfer magnetic random access memory (STT-MRAM),” IEEE Trans. Magn., vol. 53, no. 11, pp. 1–11, 2017. [3] K. Cai, Z. Qin, and B. Chen, “Channel capacity and soft-decision decoding of LDPC codes for spin-torque transfer magnetic random access memory (STT-MRAM),” in Proc. Int. Conf. Comput., Netw., Commun. (ICNC), San Diego, CA, USA, Jan. 2013, pp. 550–554. [4] Z. Mei, K. Cai and L. Shi, “Information Theoretic Bounds Based Channel Quantization Design for Emerging Memories,” in Proc. IEEE Information Theory Workshop (ITW), Guang Zhou, China, Nov. 2018, pp. 1–5. [5]W. E. Ryan and S. Lin, Channel Codes: Classical and Modern, Cambridge, Cambridge University Press, 2009. [6] A. Barbero, P. Ellingsen, S. Spinsante, and O. Ytrehus, “Maximum likelihood decoding of codes on the Z-channel,” in Proc. IEEE International Conference on Communications (ICC), Istanbul, Turkey, Jun. 2006, pp. 1–5. [7] G. Song, K. Cai and J. Cheng, “A union bound analysis for codes over binary asymmetric channels,” in Proc. IEEE International Conference on Communications (ICC), Paris, France, May. 2017, pp. 1–5.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7687adc035569fbf9f2d0102c1f4556f.jpg)
Fig. 1. Comparison between analytical and simulation WERs, with union bound optimized quantizer with q= 1,2,3.

![](https://assets.underline.io/uploads/markdown_image/1/image/ff97a646ffd6d77b07f4e57f875be682.jpg)
Fig. 2. Comparison between analytical and simulation WERs with different types of quantizers, for q=2.

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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

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.

 Second harmonic detection of chiral spin structures in a magnetic multilayer

In the chiral ferromagnets with the Dzyaloshinskii-Moriya (DM) spin-orbit interaction, vortex spin microstructures arise, which are considered as a unit of information for computer memory. In strongly correlated itinerant ferromagnets, these spin structures arise due to the first-order thermodynamic phase transitions, in which an intermediate phase of the chiral spin liquid is realized. However, the experiments indicate the realization of a more complex and still unexplained picture of spin microstructures, which depends not only on temperature and external magnetic field, but also on pressure. In this communication, we investigate the conditions for the occurrence of such spin microstructures in the framework of the spin-fluctuation theory of strongly correlated band ferromagnets with the DM-interaction. We consider the Hubbard model and DM-interaction, which, due to the small relativism, is considered in the mean-field approximation. The resulting expression for the free energy functional contains pressure-dependent magnetic, electronic, and spin-fluctuation terms, which are self-consistent with each other. The conditions for the minimum of free energy and electroneutrality are considered at various pressures in the model of the electronic structure of MnSi obtained from the LDA+U+SO calculations. The effect of pressure on the DOS determined by LDA+U+SO-calculation will be considered in the model of corresponding states. The initial state is chiral ferromagnetic helicoidal spin spiral (CFH) phase. An increase in temperature and pressure leads to thermodynamic and quantum transitions, which are responsible for the formation of the MnSi phase diagram (see fig.). A crossover of a first-order thermodynamic and quantum phase transition is obtained, which leads to the appearance of a chiral spin fluid (CSF). An increase in pressure suppresses local DOS minimum and zero-point spin fluctuations. This leads to the disappearance of the crossover of quantum and thermodynamic phase transitions at the multicritical boundary point (T0, p0). Below the temperature T0, at p=p0, a quantum transition to the state of phase segregation (SP) occurs. This transition is accompanied by an increase in the DOS at the Fermi level and leads to the appearance of a ferromagnetic state. In this state, due to the DM-interaction and fluctuating exchange fields, skyrmions of the left and right spin chirality appear. Moreover, in the SP-region baric magneto-volumetric effect is realized, which is observed experimentally. An increase in pressure at T&lt;T0 and p&gt;p0 leads to the crossing of the DOS maximum by the chemical potential and a quantum transition to a paramagnetic state with a nonzero local magnetization (CPM). In the CPM-region (p&gt;pS(T)), the spin chirality fluctuations are realized. An increase in the amplitude of spin fluctuations with increasing pressure in this phase ensures the disappearance of local magnetization. As a result, at p=pC(T), a second-order phase transition to the paramagnetic phase (PM) without spin chirality occurs. The (p-T)-diagram of MnSi, calculated in the framework of the developed spin-fluctuation approach using the LDA+U+SO spectrum, is consistent with experiment.

**References:**



**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/a75fc5d7b43c1a1ef3f59af9aaec7ab5.jpg)

 Quantum phase transitions under pressure in a chiral itinerant ferromagnet MnSi

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

 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

Recently, the problem of information storage becomes more serious due to the information explosion. It is widely known that future advances in the areal density of hard disk drives (HDDs) are essentially limited by the magnetic recording trilemma arising from the conflicting requirements between signal-to-noise ratio, writeability, and stability[1]. Energy-assisted magnetic recording (EAMR) such as heat-assisted magnetic recording[2] and microwave-assisted magnetic recording[3] is thought to solve the trilemma by improving writability and enhancing areal recording density. Two-dimensional magnetic recording with shingled writing[4] has also been proposed for increasing track density, which is enabled by partial overlapping of recorded tracks. EAMR combined with SMR is expected to be achieved for the next generation recording schemes of HDDs. On the other hand, servo signal writing is one of serious problems for high areal recording density in hard disks. The self-servo track writing technology, which regenerated from previous written servo patterns, has been adopted during head-disk assembly. However, since the servo signal writing in the present hard disks takes more than a few days per one drives on manufacturing process, and the supply of HDDs can not keep up with the social demand. Magnetic printing with perpendicularly magnetized patterns is a strong candidate for servo track writing with extremely high speed and low cost[5]. However, the recording field strength by utilizing the conventional magnetic printing master medium is not enough to write servo-signals on HDDs with large coercivity for EAMR, and it becomes much weaker as the pattern size or the track pitch decreases. The decrease in the recording field or the field gradient deteriorates the signal-to-noise ratio of servo-signals, and this problem becomes more serious in HDDs which has narrower track width. In order to accurately write servo-signals on HDDs for EAMR combined with SMR, it is essential to enhance the recording field. I propose a new master structure, herein called double magnet master (DMM) medium[6], for magnetic printing technique to improve recording characteristics. Figure 1 shows the schematic illustration of double magnet master medium and recording layer. The DMM medium consists of two magnetic materials with different coercivity. Both materials have perpendicular magnetic anisotropy but exchange interaction between both materials does not act. One material has a coercivity higher than the coercivity of the recording layer, while the other has a lower coercivity. During application of printing field Hp, the magnetization of low coercivity parts (magnet 1) orients to the direction of printing field while high coercivity parts (magnet 2) is not subject to magnetization reversal. As a result, the recording field can be enhanced. The recording field distributions of L/S patterns with pattern length of 10 nm, which corresponds to bit length, were calculated by micromagnetic simulation for conventional perpendicular master (CPM)[5] and DMM media. Tha magnetic spacing is 2 nm in this case. The printing field Hp was varied from 5 kOe to 50 kOe. In CPM, the recording field Hr consists of the magnetostatic field generated from only the magnet 1, while in DMM, the recording field Hr consists of the magnetostatic fields generated from both magnets 1 and 2. The recording field difference ΔHr of about 11 kOe for DMM is twice as that for CPM in various printing field. It was clearly found that the recording field can be enhanced by DMM. Since the magnetostatic field generated from double magnets is effective when the magnetization in the harder magnet does not reverse, the printing field is less than the coercivity of the harder magnet, which is about 73kOe by using L10-FePt patterms as magnet 2. In order to evaluate printing characteristics of CPM and DMM, the printed magnetization distributions of ECC-type EAMR media [7] with the coercivities of 15kOe and 22kOe were calculated. Figure 2 shows printing performance [8] as a function of printing field Hp. The printing performance of 1 means the perfect printing. The optimum printing fields are slightly less than those of the coercivity of recording layer. The printing performances of DMM were dramatically improved comparing with CPM printing characteristics. It was clearly confirmed that the printing performances of DMM are superior to that of CPM. Therefore, the magnetic printing by utilizing DMM is a promising way to realized high-speed servo-track writing for EAMR combined with SMR.

**References:**

[1] H. J. Richter, A. Y. Dobin, J. Magn. Magn. Mater. 287 (2005) 4150. [2] M. H. Kryder, E. C. Gage, T. W. McDaniel, W. A. Challener, R. E. Rottmayer, G. Ju, Y. T. Hsia, M. F. Erden, Proc. IEEE 96 (2008) 1810-1835. [3] J.-G. Zhu, X. Zhu, Y. Tang, IEEE Trans. Magn. 44 Iss. 1 (2008) 125-131. [4] A. Amer, J. Holliday, D. D. E. Long, E. L. Miller, J.-F. Paris, T. Schwarz, IEEE Trans. Magn. 47, Iss. 10 (2011) 3691-3697. [5] N. Sheeda, M. Nakazawa, H. Konishi, T. Komine, and R. Sugita, IEEE Trans. Magn. 45 (2009) 3676-3678. [6] T. Komine, submitted to J. Magn. Magn. Mater. 2021. [7] T. Tanaka, S. Kashiwagi, Y. Kanai, K. Matsuyama, J. Magn. Magn. Mater. 416 (2016) 188-193. [8] T. Komine, T. Murata, Y. Sakaguchi, R. Sugita, IEEE Trans. Magn. 44 Iss. 11 (2008) 3416-3418.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d8f8ad00cc0a9b7cd3f35f27da53b2ef.jpg)
Fig.1 Schematic illustration of magnetic printing process by utilizing double magnet master medium.

![](https://assets.underline.io/uploads/markdown_image/1/image/dda58c7ac541a3940c784a3e79047b17.jpg)
Fig.2 Printed magnetization distribution of recording layer and printing performance as a function of printing field.

 Double magnet master media for magnetic printing onto energy-assisted magnetic recording media

The ability to engineer magnetic phase distributions over nanometer length scales is invaluable in the development of spintronic and data storage technologies with enhanced efficiency. Equiatomic FeRh with B2 crystallography is a useful material for local manipulation of magnetic phases, owing to its unusual first-order metamagnetic phase transition (MPT) from antiferromagnetic (AF) to ferromagnetic (FM) ordering, which occurs at ~100 °C [1]. It has been shown that in addition to the thermally driven MPT, there is a disorder-induced MPT from AF to FM ordering, and that with further disordering a second MPT to PM behaviour can be induced [2]. Bennett et al. [3] have demonstrated that by irradiating FeRh with noble gas ions, the disorder-induced MPTs can be exploited to produce a rich variation in the magnetism as a function of depth. In the present work [4], we combine several non-destructive experimental techniques with extensive simulations to identify a direct correspondence between the nature and concentration of induced defects and the resulting modification of the magnetic depth profile in 40 nm FeRh thin films as the fluence of 25 keV Ne+ ions increases. Polarised neutron reflectometry (PNR) was used to determine the magnetisation depth profiles for a range of sample temperatures under an applied field of 1 kOe. The data were fitted to dynamical simulations of the nuclear and magnetic scattering length density (n/mSLD) profiles (Figure 1), which are proportional to the density and in-plane magnetisation, Ms, respectively. The data demonstrate that it is possible to produce a uniform and thermally stable magnetisation with a single-step irradiation, as in Figure 1(b), or to produce a simple bilayer phase structures, as in Figure 1(c) and (d). To elucidate the interplay between the magnetisation depth distributions, the coexistence of magnetic phases, and the ion-induced damage profiles, conversion electron Mössbauer spectroscopy (CEMS) and Doppler broadening variable energy positron annihilation spectroscopy (DB-VEPAS) were performed on each sample. DB-VEPAS allows relative changes of defect concentration to be determined via the line broadening of gamma photons produced from the annihilation of positrons with varying energy Ep. By fitting the DB-VEPAS data, the mean positron diffusion length as a function of increasing fluence is obtained, which is inversely proportional to the concentration of open volume defects. Thus, we demonstrate that the defect concentration monotonically increases with the irradiating fluence. To complement the DB-VEPAS measurements, variable energy positron annihilation lifetime spectroscopy (VEPALS) measurements were also performed on each sample. Here, variations in positron lifetime with increasing fluence signify changes to the size of open volumes; thus, changes in the concentration and the type of defects are determined. By comparing the measured data to theoretical lifetime values calculated using density functional theory (DFT), we identify the predominant defect type for each irradiation condition with depth sensitivity. The PNR, CEMS, and positron data allow the effect of increasing fluence to be explained as shown schematically in Figure 2, wherein the onset of FM and PM phases are associated with the emergence of static disorder and monovacancy defects respectively. In the as-grown case, there is a substantial concentration of monovacancy defects. At the lowest value of the fluence (7.5 × 1013 ions cm-2), the defect concentration increases and sub-unit cell atomic displacements begin to dominate. The generation of static disorder continues as the fluence is increased to 3 × 1014 ions cm-2, with no change to the predominant defect type. On increasing the fluence to 6 × 1014 ions cm-2, atomic displacements give way to monovacancies, with some static disorder remaining at the substrate interface. For all samples, our data are well described assuming AF, FM, and PM phases only, and do not present evidence for states such as spin glasses or magnetic dead layers. This feature, coupled to the fact that the mSLD profiles of the irradiated samples comprise relatively few sublayers with discrete magnetisations which are stable with respect to increasing temperature, highlights the effectiveness of sensitive and controlled disordering of ordered alloy precursors for producing technologically useful magnetic configurations. Our results provide the basis for the patterning of single layer or multilayer structures that incorporate FeRh to create devices on sub-100 nm length scales with AF and FM order in close physical proximity.

**References:**

[1] M. Fallot, Ann. Phys., vol. 10, pp. 291–332 (1938) [2] A. Iwase et al., Nucl. Instrum. Methods Phys. Res., vol. 256, pp. 429–433 (2007) [3] S. P. Bennett et al., Mater. Res. Lett., vol. 6, pp. 106–112 (2018) [4] W. Griggs et al., APL Mater., vol. 8, p. 121103 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/69319fe9359c4803c0c33b75650a7a95.jpg)
Fig. 1. (a)-(d) Magnetic (coloured) and nuclear (grey) scattering length density (mSLD and nSLD) profiles corresponding to the fitted PNR data for each sample across the range of measured temperatures.

![](https://assets.underline.io/uploads/markdown_image/1/image/85121693078388560eeaf904d351ce49.png)
Fig. 2. The effect of increasing the irradiating fluence on the depth-dependent magnetisation, magnetic phase fractions, and concentration and type of structural defects is illustrated.

Union Bound Analysis for Spin-Torque Transfer Magnetic Random Access Memory with Channel Quantization

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Union Bound Analysis for Spin-Torque Transfer Magnetic Random Access Memory with Channel Quantization

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