France



&lt;br/&gt;Magnetocapacitance (MC) is the change in capacitance of a solid with applied magnetic field. This change in capacitance is due to spin accumulation which coexists with geometric capacitance [1]. The effect is explained using the formation of spin channels in solids due to selective spin scattering either at interface [2] or by magnetic ions [3]. The MC is observed in magnetoelectrically coupled solids or interfaces as well. It is used as a tool to evaluate the magnetoelectric effect [4]. Spin accumulation in 2-D materials is one of interesting areas of investigation because of its device applications, studied with the help of magnetocapacitance effect. Hence Graphene and related materials have been studied extensively for spin transport [5]. Graphene oxide (GO) is a unique material and has a monomolecular layered structure similar to that of graphene with various oxygen containing functional groups which make the atomic thick layers’ defect reach [6]. Delocalized π electrons in aromatic ring structures induce diamagnetism in Graphene like compounds, but in GO π electrons develop ferromagnetism due to its unique structural properties. It has been proven that GO exhibits ferromagnetism with the presence of epoxy groups and it is a soft ferromagnetic material with shapedM-H curve [7]. In the present work we investigate spin dependent capacitance in GO with different frequencies and estimate the spin accumulation length. The magnetic interaction and the origin of magnetocapacitance in these was interpreted with sign of MC and AC magnetoresistance study. The results of GO are compared to reduced GO (rGO) to depict the spin accumulation capacity.&lt;br/&gt;&lt;br/&gt;For spin accumulation study GO was prepared by modified Hummer’s method from graphitic powder and it was further reduced to form rGO using hydrazine hydrate. This method is expected to yield GO with large defect densities. The formed GO was subjected to AFM (inset of Figure 1) and Raman spectroscopic study (Figure 1) confirming the formation of flakes like GO. The magnetism in GO was investigated using a vibrating sample magnetometer showing soft ferromagnetism at room temperature. The GO and rGO were compressed to form parallel plate capacitors to study spin accumulation in them. The AC magnetoresistance (MR) and MC were studied from 100 Hz to 5 MHz at room temperature (Figure 2a). The change in capacitance with frequency was modelled with the Havriliak-Negami relaxation process (inset of Figure 2a). The change in asymmetry parameter showed contribution of defects in the relaxation process. The relaxation time was found to change with applied magnetic field supporting strong magnetic field dependent scattering in GO. We observed positive MC of 20 % at around 200 Hz and 100 Oe, where capacitive coupling is higher and 2.2 % at 1.1 MHz (Figure 2b) which is due to predominant inductive coupling. The positive sign of MC indicates the interface mediated spin accumulation process in GO [1]. This low field MC is very important for device development The spin diffusion length calculated, equivalent to capacitance change was of the order of 56nm. MR in GO was observed to change sign from negative to positive indicating the increase of interface mediated effects with higher magnetic fields. In contrast to GO, rGO showed no spin accumulation effects except significant change in impedance, attributed to surface permeability change. The report projects GO as a significant spin accumulation material useful in different spintronic device applications.

**References:**

1. Y. H. Zhu, X. X. Zhang, J. Liu and P. S. He &lt;i&gt;J. Appl. Phys.&lt;/i&gt; Vol. 122, p. 043902 (2017).&lt;br/&gt;2. G. Catalan ,&lt;i&gt;Appl. Phys. Lett.&lt;/i&gt; Vol. 88, p.1–4(2006).&lt;br/&gt;3. K. T. McCarthy, A. F. Hebard and S. B. Arnason &lt;i&gt;Phys. Rev. Lett.&lt;/i&gt; Vol. 90(11), p.1172015–8 (2002).&lt;br/&gt;4. K. Manna, R. S. Joshi, S. Elizabeth and P. S. Anil Kumar, Appl. Phys. Lett. Vol.104 p.2–6(2014).&lt;br/&gt;5. E. C Ahn npj 2D Materials and Applications Vol. 4, p.17 (2020).&lt;br/&gt;6. A. Stergiou,., et al.. J. Nanotechnol. 5, p.1580-1589 (2014).&lt;br/&gt;7. L. Dongwook, S.Jiwon, Appl. Phys. Lett. 106, p.172402 (2015).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/1f85cb12ca24b6e94fac2e26df228ffc.jpg)
Figure 1: Raman spctra for GO (Inset shows AFM impage of as prepared GO  grafted on SiO&lt;sub&gt;2&lt;/sub&gt; ) substrate

![](https://assets.underline.io/uploads/markdown_image/1/image/fdb7c4d2f9cacda291b12cb7af91696b.jpg)
Figure 2 (a). Variation of capacitance with respect to frequency for different applied magnetic fields in GO (inset shows fit of Havriliak-Negami relaxation model to the data) (b) MC in graphene with applied magnetic field for different frequencies.

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 Large Low Magnetic Field Magnetocapacitance Effect and Spin Accumulation in Graphene Oxide

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Magnetocapacitance (MC) is the change in capacitance of a solid with applied magnetic field. This change in capacitance is due to spin accumulation which coexists with geometric capacitance [1]. The effect is explained using the formation of spin channels in solids due to selective spin scattering either at interface [2] or by magnetic ions [3]. The MC is observed in magnetoelectrically coupled solids or interfaces as well. It is used as a tool to evaluate the magnetoelectric effect [4]. Spin accumulation in 2-D materials is one of interesting areas of investigation because of its device applications, studied with the help of magnetocapacitance effect. Hence Graphene and related materials have been studied extensively for spin transport [5]. Graphene oxide (GO) is a unique material and has a monomolecular layered structure similar to that of graphene with various oxygen containing functional groups which make the atomic thick layers’ defect reach [6]. Delocalized π electrons in aromatic ring structures induce diamagnetism in Graphene like compounds, but in GO π electrons develop ferromagnetism due to its unique structural properties. It has been proven that GO exhibits ferromagnetism with the presence of epoxy groups and it is a soft ferromagnetic material with shapedM-H curve [7]. In the present work we investigate spin dependent capacitance in GO with different frequencies and estimate the spin accumulation length. The magnetic interaction and the origin of magnetocapacitance in these was interpreted with sign of MC and AC magnetoresistance study. The results of GO are compared to reduced GO (rGO) to depict the spin accumulation capacity. For spin accumulation study GO was prepared by modified Hummer’s method from graphitic powder and it was further reduced to form rGO using hydrazine hydrate. This method is expected to yield GO with large defect densities. The formed GO was subjected to AFM (inset of Figure 1) and Raman spectroscopic study (Figure 1) confirming the formation of flakes like GO. The magnetism in GO was investigated using a vibrating sample magnetometer showing soft ferromagnetism at room temperature. The GO and rGO were compressed to form parallel plate capacitors to study spin accumulation in them. The AC magnetoresistance (MR) and MC were studied from 100 Hz to 5 MHz at room temperature (Figure 2a). The change in capacitance with frequency was modelled with the Havriliak-Negami relaxation process (inset of Figure 2a). The change in asymmetry parameter showed contribution of defects in the relaxation process. The relaxation time was found to change with applied magnetic field supporting strong magnetic field dependent scattering in GO. We observed positive MC of 20 % at around 200 Hz and 100 Oe, where capacitive coupling is higher and 2.2 % at 1.1 MHz (Figure 2b) which is due to predominant inductive coupling. The positive sign of MC indicates the interface mediated spin accumulation process in GO [1]. This low field MC is very important for device development The spin diffusion length calculated, equivalent to capacitance change was of the order of 56nm. MR in GO was observed to change sign from negative to positive indicating the increase of interface mediated effects with higher magnetic fields. In contrast to GO, rGO showed no spin accumulation effects except significant change in impedance, attributed to surface permeability change. The report projects GO as a significant spin accumulation material useful in different spintronic device applications.

**References:**

1. Y. H. Zhu, X. X. Zhang, J. Liu and P. S. He J. Appl. Phys. Vol. 122, p. 043902 (2017). 2. G. Catalan ,Appl. Phys. Lett. Vol. 88, p.1–4(2006). 3. K. T. McCarthy, A. F. Hebard and S. B. Arnason Phys. Rev. Lett. Vol. 90(11), p.1172015–8 (2002). 4. K. Manna, R. S. Joshi, S. Elizabeth and P. S. Anil Kumar, Appl. Phys. Lett. Vol.104 p.2–6(2014). 5. E. C Ahn npj 2D Materials and Applications Vol. 4, p.17 (2020). 6. A. Stergiou,., et al.. J. Nanotechnol. 5, p.1580-1589 (2014). 7. L. Dongwook, S.Jiwon, Appl. Phys. Lett. 106, p.172402 (2015).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/1f85cb12ca24b6e94fac2e26df228ffc.jpg)
Figure 1: Raman spctra for GO (Inset shows AFM impage of as prepared GO grafted on SiO2 ) substrate

![](https://assets.underline.io/uploads/markdown_image/1/image/fdb7c4d2f9cacda291b12cb7af91696b.jpg)
Figure 2 (a). Variation of capacitance with respect to frequency for different applied magnetic fields in GO (inset shows fit of Havriliak-Negami relaxation model to the data) (b) MC in graphene with applied magnetic field for different frequencies.

poster

Introduction: Double CoFeB/MgO interface perpendicular MTJ has become the de facto standard technology of current STT-MRAM. However, there are still issues for high density memory application and high reliable application including Automobile applications. Furthermore, achievement of high speed and high endurance features, in the other words, high speed SRAM replacement is still challenging for STT-MRAM, because of its tradeoff relationship among the retention, endurance, and operational speed. Recently, Spin Orbit Torque (SOT) devices have been intensively researched and developed because SOT devices have the potential to achieve high speed and high endurance. To realize the practical use of SOT-MRAM for LSI applications, these issues must be addressed. This invited paper reviews our recent progresses in STT-MTAM, SOT-MRAM and Nonvolatile AI processors with CMOS/MTJ hybrid technology. The STT-MRAM and SOT-MRAM as NV working memory and AI processors as these applications are particularly suitable for future IoT / AI systems that require ultra-low-power and high-performance computing at the same time. 128Mb STT-MRAM and Quad interface MTJ technology: We developed novel damage control integration process technologies including new low-damage MgO deposition process, low-damage RIE process, and low temperature SiN-cap process. By applying the developed damage control integration process technologies to double interface p-MTJ fabrication, TMR ratio, thermal stability factor, and switching efficiency of Double p-MTJ were successfully improved. Moreover, it was shown that despite the significant increase in thermal stability factor, the endurance of the fabricated Double p MTJs was over 1010. Finally, with our double-interface p-MTJ technology and novel damage control integration process technologies, fabricated 128Mb STT-MRAMs successfully achieved 14ns/7ns write speed at Vdd of 1.2V/1.8V, respectively. Next, for further scaling of STT-MRAM, we proposed novel Quad-interface p-MTJ technology which brings forth an increase of thermal stability factor compared with conventional Double-interface p-MTJ technology. We successfully fabricated the quad-interface MTJ using 300mm process based on the damage control integration process. The fabricated Quad p-MTJs achieved an enhancement of switching efficiency in addition to an approximately two times larger thermal stability factor without degradation of TMR ratio. The developed Quad p-MTJ technology will become an essential technology for the scaling of the STT-MRAM beyond 20nm without changing material and process sets from mass-production STT-MRAM. Moreover, the high reliable Quad p-MTJ technology with enough thermal stability factor is suitable for Automobile applications that require high temperature operation such as 150○C. Field-free canted SOT-Device with 350ps write speed and its SOT-MRAM: To realize practical use of SOT-MRAM for LSI applications, we demonstrated 55 nm-CMOS/SOT-device hybrid MRAM cell with magnetic field free writing for the first time. For field free writing, we developed canted SOT device under 300 mm BEOL process with 400°C thermal tolerance. In addition, we developed its advanced process as follows; PVD process of SOT channel layer for high spin Hall angle under 400°C thermal tolerance, low damage RIE technology for high TMR/thermal stability factor, and ultra-smooth surface metal via process under SOT device to reduce contact resistance. By using the developed technologies, our canted SOT devices achieved fast write speed of 0.35 ns without magnetic field, an enough thermal stability factor of 70 for non-volatile memory (over 10 years retention), and a high TMR ratio of 167%, simultaneously. Moreover, we fabricated a field free canted SOT-MRAM cell with 55 nm CMOS technology and demonstrated its write/read performance. These technologies will open to high speed write non-volatile memory such as 1 level cache application of many kinds of application processors. CMOS/MTJ hybrid AI Processor for low power application: Focusing on the indispensable nearest neighbor search (NNS) function of the brain, we developed a nonvolatile object recognition processor with NSS full-adaptive to any data format, employing nonvolatile memories base on our p-MTJ. The developed object recognition processor was fabricated under 90nm-CMOS/70nm-MTJ hybrid process on 300mm-wafer. The 4-Transistors & 2-MTJs type memory cells were adopted to completely eliminate standby power. Moreover, we successfully developed a self-directed power-gating technique leveraging the non-volatility, high access speed and unlimited endurance features of the p-MTJs. This developed self directed power-gating technique was employed to shut down idle circuit blocks during not only the standby periods but also the full operation periods. The measured peak operation power consumption of the prototype chip was only 130μW and can be further optimized corresponding to the format of reference data. Compared to the latest conventional researches, the significant improvements of both power performance (about 2-3 order reduction) and circuit density (about 2 order improvement) were achieved.

**References:**

[1] S. Ikeda et al., Nat. Mater. 9, 721 (2010). [2] T. Endoh et al., 2011 International Electron Device Meeting (IEDM), 2011. 34 [3] H. Sato et al., Appl. Phys. Lett. 101, 022414 (2014). [4] T. Saito et al., IEEE T-Magn. 54, 3400505 (2018). [5] H. Sato et al., IEDM 2018, p. 608. [6] K. Lee et al., IEDM 2018, p. 604. [7] Y. J. Song et al., IEDM 2018, p. 416. [8] C. Park et al., VLSI 2018, p. 185. [9] O. Golonzka et al., IEDM 2018, p. 412. 6 [10] C. Park et al., VLSI 2018, p. 185. [11] T. Endoh, IEEE ISCASS2018. (Invited) [12] K. Nishioka et al., VLSI 2019, T11-4; IEEE T-ED 67, 995 (2020). [13] S. Fukami et al., VLSI 2016, T61. 10 [14] H. Honjo et al, IEDM 2019, 28.5.1. [15] T. Endoh, MRAM Development Day 2019 (Invited) [16] T. Endoh, 2020 International Solid-State Circuits Conference (ISSCC) (Invited) [17] T. Endoh et al, VLSI Symp. 2020 (Invited) [18] T. Endoh, Tutorial of IEDM2020 (Invited)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/4ab86aad73439f39476f104f23a8eb59.jpg)
Fig. 1: (a) developed 128Mb STT-MRAM. (b) photograph of its subarray. (c) Shmoo plot of subarray in 128Mb STT-MRAM. (d) Switching efficiency as a function of CD for the Quad- and Double-interface MTJs.

![](https://assets.underline.io/uploads/markdown_image/1/image/b8abc4d21c732c328f6998e190bbd9cb.jpg)
Fig. 2: (a) Developed AI object recognition processor with 90nm-CMOS/70nm-MTJ. (b) Further performance comparisons of the operation power efficiency and circuit density.

 Advanced MTJ and SOT Technology for AI and Automobile applications INVITED

As the crucial device element for next-generation cache and edge computing technologies, the magnetic tunnel junction (MTJ) has been widely studied with recent focus towards using spin-orbit torque (SOT) to switch the free layer magnetization. Writing the MTJ using SOT approach is attractive, because it can increase the junction endurance as switching current path does not go through the tunneling barrier. In this work, we aim to enhance the SOT switching efficiency in the MTJ through various interlayer insertion designs in the free layer. Our study is carried out using in-plane MTJ (IP-MTJ), which has been demonstrated that no external magnetic field is required for SOT switching [1]. We fabricated W underlayer-based IP-MTJs, and investigate how the insertion layer between the W and CoFeB free layer affects the switching current density of the MTJ. Interestingly, with the insertion of ultra-thin layers such as MgO, Ru and TiW, the switching current drops significantly with the MgO insertion offering ~70% of reduction. Here, standalone 3-terminal IP-MTJ devices of elliptical sizes 300 nm x 600 nm are fabricated using Canon X5 stepper lithography on 8 inch Si wafers. We measure the pulse current switching loops with our home-built probe station [2]. Without insertion layer, the switching current is ~11.9 mA, equivalent to ~1.9x1011 A/cm2. The large value of switching current density is because of the dramatic reduction in the spin Hall angle (SHA) of W underlayer after annealing at 330°C, which also leads to a structural phase change to α-W. While the introduction of an insertion layer universally reduces the current density. The MgO interlayer reduces it by more than 3 times making it superrior for insertion layer selection. To understand it better, we performed spin-torque ferromagnetic resonance (ST-FMR) measurement [3] to determine the damping constant, spin Hall angle, and effective saturation magnetization (Ms_eff). A representative example of ST-FMR spectra for MgO insertion layer is shown in Fig. 1. We found the thin insertion layer has little influence on the damping constant, but some modifications in Ms_eff and SHA are found, which can be correlated with the switching current reduction results. Further investigations on the role of insertion layer thickness are ongoing and are expected to reveal the optimal insertion layer for SOT current reduction.

**References:**

[1] L. Liu, C.F. Pai, Y. Li et al., Science, Vol. 336, p.555 (2012) [2] J. Lourembam, J. Huang, S.T. Lim et al., AIP Adv., Vol. 8, p.055915 (2017) [3] L.S. Huang, S.K. He, Q.J. Yap et al., Appl. Phys. Lett., Vol. 113, p.022404 (2018)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/ad011acedbe5cfe942648e452a3ab9a6.jpg)
Fig. 1 Field sweep ST-FMR spectra of W/MgO/CoFeB/MgO film. The RF frequency varies from 7 to 20 GHz.

 Effect of insertion layer on reducing switching current density in spin-orbit torque magnetic tunnel junction

Spin-orbit torque (SOT) induced magnetic switching has attracted an extensive interest due to its potential high efficiency in terms of switching time and power consumption [1]. Chiral domain wall (DW) motion driven by SOT in nanowires in the presence of Dzyaloshinskii–Moriya Interaction (DMI) at interfaces between HM and FM layers has shown a great potential for applications to racetrack memory devices [2]. One potential advantage of DW driven switching is that multi-level states can be obtained by controlling the position of the DW across a nanostructure such as wires or dots. As such, domain wall Memristors and the possibility to use DWs motion as synapses [3] have been recently demonstrated. Here, we study SOT driven multi-level states in Hall bars based on W-CoFeB-MgO structures by reducing locally the perpendicular magnetic anisotropy through He+ ion irradiation across a mask [4]. The structure of the investigated samples is W (4 nm)/Co20Fe60B20 (0.6 nm)/MgO (2 nm)/Ta (3 nm). The films were then patterned into Hall bars consisting in 20 μm wide and 140 μm long wires including 2 Hall crosses with 5 μm wide pads. As shown in previous studies [6], light He+ ion irradiation induced interface intermixing can be used to finely tune Ms, interface anisotropy and DMI in CoFeB-MgO structures. One of the Hall cross was then subsequently irradiated through a mask by He+ ions with an energy of 15 keV at a fluence of 1×1019 ions/m2 to reduce locally the perpendicular anisotropy Keff. Anomalous Hall Effect measurements combined with Kerr microscopy indicate that the SOT switching process is dominated by domain wall nucleation in the irradiated region followed by rapid domain propagation to switch the irradiated area (figure 1). The reduction of Keff by 20% under ion irradiation is accompanied by a large reduction of the switching current from 4.07 to 1.68 MA/cm2 under an in-plane 50 mT field; The switching current is determined by the threshold current to induce DW nucleation as checked by Kerr microscopy. As Keff is lowered in the irradiated Hall bar, the energy barrier to induce DW nucleation by SOT is also reduced. Due to the strong pinning of the DW at the transition between the irradiated and the non-irradiated region as evidenced by Kerr microscopy, we observe an intermediate Hall resistance state between the 2 saturated states (Figure 2). The possibility to increase the number of intermediate states by designing different value of perpendicular anisotropy through He+ ion irradiation paves the way towards efficient neuromorphic and memristor devices where the nucleation, motion and pinning can be tailored.

**References:**

[1] M. Cubukcu, et al, IEEE Transactions on Magnetics 54, 9300204 (2018) [2] K. S. Ryu et al, Nat. Nanotechnol. 8, 527 (2013) [3] S. Lequeux, et al, Scientific Reports 6, 31510 (2016) [4] X.Zhao, Appl. Phys. Lett 116, 242401 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/15951bba9072bcb7b7faba4b2ff3dd61.jpg)
 
Figure 1. Kerr images showing the SOT induced magnetization switching process of the Hall bar for an increasing current under a 50 mT in-plane magnetic field. The white dashed lines indicate the longitudinal wire and the transverse Hall voltage pads. The gray box corresponds to the irradiated area. The yellow arrow in (iv) shows the direction of domain wall motion.

![](https://assets.underline.io/uploads/markdown_image/1/image/50371af00eff66ca4a5d33d727cefb07.jpg)
Figure 2. SOT switching curve for an irradiated Hall cross measured under 50 mT. States 1, 2, 3 correspond to down magnetization, intermediate state and up magnetization respectively

 Spin orbit torque driven multi-level switching in He+ irradiated W-CoFeB-MgO Hall bars with perpendicular anisotropy

In the last decade, spin-orbit torque (SOT) [1-3] has enabled us to electrically control spins of magnetic materials, which provides an idea manner to develop magnetic random-access memory (SOT-MRAM). One crucial step toward the SOT-MRAM is realization of magnetic tunnel junctions (MTJ) whose magnetic states can be switched by SOT at zero applied field. In our previous paper [4], we have realized field-free SOT switching of a perpendicular CoFeB film which coupled with another CoFeB film with in-plane magnetization via a Ta spacer. This structure composed by the perpendicular CoFeB, the in-plane CoFeB and the spacer Ta is entitled as the“T-Type” magnetic structure. Here, by using an in-plane CoFeB/Ta/perpendicular CoFeB T-type magnetic structure as a free layer, we have realized a MTJ switchable by SOT at the field-free condition (Fig. 1) [5]. The spacer layer Ta plays three roles. Besides of absorbing the B element during annealing process to improve TMR ratio, it also mediates exchange coupling and finally generates large enough SOT to drive sharp switching between the parallel and antiparallel states of the MTJ without any applied fields. This work could provide a useful stack with the T-type structure to realize field-free SOT-MRAM or logic devices.

**References:**

[1] I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V. Costache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl, and P. Gambardella, Nature, 476, 189 (2011). [2] L. Liu, C. F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science, 336, 555–558 (2012) [3] L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 109, 096602 (2012) [4] W. J. Kong, C. H. Wan, X. Wang, B. S. Tao, L. Huang, C. Fang, C. Y. Guo, Y. Guang, M. Irfan, and X. F. Han, Nat. Commun. 10, 233 (2019) [5] W. J. Kong, C. H. Wan, C. Y. Guo, C. Fang, B. S. Tao, X. Wang, and X. F. Han, Appl. Phys. Lett. 116, 162401 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/ab60392e9068d8cdd9d3b5c29751a49c.jpg)
Fig.1 (a) Schematics of a MTJ with the T-type magnetic structure as its synthetic free layer. (b) Out-of-plane and (c) in-plane hysteresis loops of the stack Sub//Ta(8)/ Co40Fe40B20(2)/Ta(1.3)/Co40Fe40B20(0.6)/MgO(2.5)/Co40Fe40B20(1.2)/Ta(5)/Ru(5 nm) to show a perpendicular layer and an in-plane layer in it. (d) The TMR ratio as a function of the Hz field and (e) Current-driven magnetization switching at +/-300 Oe and 0 field. The switching direction at zero field depends on the history of reducing field.

 All-electrical manipulation of magnetization in magnetic tunnel junction via spin–orbit torque

Spin-orbit torque magnetic RAM (SOT-MRAM) is affected by read disturbances due to spin current in the heavy metal layer that arises from spin-orbit interaction at the time of reading [1]. To resolve this issue, we have proposed a stabilized structure with a bi-directional read path, based on the idea that spin currents flowing in opposite directions on either side of the heavy metal layer cancel each other’s effects, that improves the read reliability of SOT-MRAM. We have already reported that this structure reduces the read disturbance [2]. In this study, we evaluated the movement of the magnetization of the free layer at the time of reading in a bi-directional read SOT-MRAM by using an LLG simulator. It was found that the read disturbance caused by SOT is removed in the free layer, as shown in Fig. 1 (a). In other words, the spin currents generated in opposite directions in the heavy metal layer directly affect the magnetization of the free layer, and the free layer functions as if it were two ferromagnets, whose balanced magnetizations cancel the read disturbance. To clarify this mechanism, Fig. 1 (b) shows the simulated magnetization for only the free layer at extremely large currents. The free layer is a ferromagnet magnetized in two opposite directions due to the central vortex magnetization, and the magnetization does not reverse to either side. (That is, it is not a mechanism in which the spin currents cancel themselves in both directions in the heavy metal layer.) Toward the realization of this bidirectional read SOT-MRAM structure, we evaluated the effect on the read disturbance of the size of the direction in which current flows through the heavy metal layer of an SOT structure. The device structure had an magnetic tunnel junction (MTJ) with two ferromagnets (cobalt), in which a pinned layer with a thickness of 3.5 nm and a free layer with a thickness of 1.5 nm sandwiched an insulating layer with a thickness of 1.0 nm, and a heavy metal layer (tungsten) with a thickness of 2.0 nm under the free layer. The magnetization of the two ferromagnetic layers directed toward the short side direction of the device, and the current passed from the pinned layer to both ends in the long side direction of the heavy metal layer. The width in the short side direction was uniformly 30 nm, and there were four device widths for the long side direction, i.e., 30 nm, 60 nm, 120 nm, and 180 nm. We evaluated the magnetization reversal probability of a conventional SOT-MRAM structure and the stabilized SOT-MRAM structure for a read time of 5 ns. Fig. 2 shows that the stabilized SOT-MRAM was about 6 to 10 times more effective than the conventional structure. In the conventional SOT-MRAM, the threshold current was proportional to the width of the device, whereas in the stabilized SOT-MRAM, the threshold current decreased as the width of device increased. The magnetization reversal of the conventional SOT-MRAM was strongly affected by the SOT disturbance due to the spin Hall effect, and the difference in thermal stability depending on the device size also had an influence. On the other hand, in the stabilized SOT-MRAM, the disturbance was canceled out at the free layer and only spin transfer torque in MTJ affected the magnetization reversal. Thus, we expected that the relationship between thermal stability and device size would have a large effect on magnetization reversal, but the result was contrary to that. These results indicate that the force exerted by the spin Hall effect on the magnetization of the free layer is relatively large and easy to act when the device is small. Therefore, a small device of the conventional SOT-MRAM tend to easily occur magnetization reversal, whereas in a smaller device of the stabilized SOT-MRAM, the effect on the magnetization of the free layer is strong enough to cancel the SOT disturbance and the magnetization reversal becomes more difficult. Acknowledgments This work was partly supported by a Grant-in-Aid for Scientific Research (19K04536) from Japan Society for the Promotion of Science.

**References:**

[1] H. Kazama and T. Kawahara, “Spin-Orbit Torque MRAM Read Reliability”, 2017 IEEE International Magnetics Conference (Intermag2017), DOI: 10.1109/INTMAG.2017.8007571, 2017. [2] Y. Kishi, K. Tabata, M. Ke, and T. Kawahara, "Evaluation of Read Disturbance Reduction Effect by Bi-directional Read on Ferromagnetic Material Properties of SOT-MRAM", ABSTRACT of IEEE/AIP 2020 Magnetism and Magnetic Materials Conference (MMM2020), O2-14, pp.434-435, 2020.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/3e0ae5fa7997c288903464f85d3c9b8c.jpg)
Fig. 1 (a) Stabilized structure (bi-directional read) of SOT-MRAM device [1]. The blue arrows indicate the current directions. Two red circles in the free layer canceled the SOT disturbance. (b) Example of simulation results showing the movement of the magnetization in the free layer of the stabilized structure. We simulated with a two-layer structure consisting of a free layer and a heavy metal layer. A vortex magnetization is created in the middle, dividing the free layer. The two split ferromagnets are magnetized in opposite directions.

![](https://assets.underline.io/uploads/markdown_image/1/image/7e606b44e402730cd84b8ea8cae483f6.jpg)
Fig. 2 Comparison of conventional and stabilized (bi-directional read) SOT-MRAM threshold currents for different X sizes of SOT structure for a read times of 5ns. In the SOT structure, two ferromagnetic layers (cobalt) sandwich an insulator, and a heavy metal layer (tungsten) is under the free layer; the current flows from the pinned layer to the long side of the heavy metal layer. The magnetization of the ferromagnetic layers are basically in the short-side direction, but the magnetization of the free layer is in any direction.

 Evaluation of Read Disturbance Reduction Effect by SOT-MRAM Bi-directional Read on Device Size Dependence

Spin-orbit torque (SOT) devices1 are promising candidates for the future magnetic memory landscape as they promise a low read disturbance, high endurance and low read error, in comparison to the spin-transfer torque devices. However, SOT memories are area intensive due to the requirement of two access transistors per bit2. Here, we report a SOT memory cell which has multiple bits sharing a common write channel. This enables twice the memory density compared to the conventional SOT-MRAM design. The shared write channel (SWC) design is shown in Fig. 1. Multiple bits share a common write path thereby reducing the need for two access transistors for every bit. With a single pulse of current, different combination of digital information can be written on all the bits on the shared path. SPICE circuit simulations were performed and the area per bit of the 32x32 SWC memory is found to be 0.0414 µm2 while that of the normal 2 transistor (T)-SOT memory is around 93% larger at 0.0801 mm2. However, an essential design requirement for SWC memory is programmable SOT device for which the direction of SOT can be toggled dynamically. With different polarities of SOTs for individual bits on a shared path, information such as ‘0101..’ and ‘1001..’ can be written. A SWC device with 2-bits is demonstrated using a Pt/Co device with a GdOx gate oxide (Fig. 2(a)). The direction/polarity of SOT in such a device can be toggled by modulating the oxygen content at the Pt/Co interface3, 4. This can be achieved by the application of gate voltage through the GdOx gate. As shown in Fig. 2(b), the multi-bit device can be programmed in different states so that all combination of digital information can be written using an unidirectional current pulse. Our prototype device paves a way towards a high density SOT MRAM architecture.

**References:**

 1. I. Mihai Miron, G. Gaudin, S. Auffret, et al. Nature Materials Vol. 9, p. 230 (2010). 2. R. Bishnoi, M. Ebrahimi, F. Oboril et al. in 2014 19th Asia and South Pacific Design Automation Conference (ASP-DAC) p. 700 (2014). 3. R. Mishra, F. Mahfouzi, D. Kumar et al. Nature Communications Vol. 10, p. 248 (2019). 4. X. Qiu, K. Narayanapillai, Y. Wu, et al. Spin–orbit-torque engineering via oxygen manipulation. Nature Nanotechnology Vol. 10, p. 333 (2015).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/8c4ae1e6e475f27a9ef80e64dedab035.jpg)
Figure 1. A shared write channel based SOT design. Multiple bits share a common write channel, thereby reducing the need for dedicate write transistor for individual bits.

![](https://assets.underline.io/uploads/markdown_image/1/image/d648fbc35ed503d00c3ea8b6a917580f.jpg)
Figure 2. (a) A Pt/Co/GdOx based 2-bit memory cell in which SOT polarity of individual bit can be programmed independently. (b) The two bits can be switched independently in different directions depending on the programmed SOT state. The blue switching curve represents normal SOT switching for a Pt device. The red switching curve is obtained after the application of negative gate voltage on the bit, which increases the oxygen content at the Pt/Co interface thereby changing SOT polarity to negative.

 Multi-bit Spin-orbit Torque Device for High Density MRAM

Magnetic random access memory (MRAM) with its inherent non-volatility is believed to address the large stand-by energy issues in the present memory hierarchy [1]. In recent years, the spin-transfer torque (STT)-MRAM has gradually matured and started to appear in the market. Typically, STT writing of perpendicular magnetic tunnel junction (pMTJ) is limited to a few nanoseconds. This is because STT is collinear and competing with the intrinsic damping of the free-layer (FL), resulting in a long incubation delay [2] and large energy consumption. To mitigate STT limitations, spin-orbit torque (SOT) and voltage control of magnetic anisotropy (VCMA) were proposed as alternative MRAM writing mechanisms. SOT is able to switch the FL of a pMTJ by injecting an in-plane current in an adjacent SOT layer, which enables a three-terminal cell structure with energy-efficient and reliable sub-ns writing capabilities [3]. On the other hand, VCMA promises significant advances toward low-power MRAM. The electronic-based VCMA effect can instantly modify the perpendicular magnetic anisotropy (PMA) of the FL to induce precessional switching at GHz rates and ultra-low energy consumption (fJ) [4]. However, both mechanisms also present technological challenges. The challenges for SOT are currently related to the density and write efficiency: the 2-transistor 1-MTJ cell structure limits the array density, and the SOT write current remains larger than for STT-MRAM, which imposes a large selector transistor to accommodate its write current. For VCMA, the write margin is small, and is subject to the variations in VCMA coefficient and field amplitudes, making write control difficult in dense arrays. Further, a large VCMA effect (&gt;1000fJ/Vm) is mandatory to avoid compromising retention in sub-30nm pMTJ, which remains as a major challenge as typical values in pMTJs are 30-60 fJ/Vm at device level [5]. To overcome the above-cited limitations and to combine the best of these two approaches, voltage-gate assisted spin-orbit torque (VGSOT)-MRAM concept [Fig. 1(a)] has been proposed [6]. In this work, we report on VGSOT switching properties in pMTJ, integrated with our 300mm SOT-MRAM platform [3]. The stack is composed of W(3.5nm)/CoFeB(1nm)/MgO(1.7nm, RA ~ 5kΩ.μm2)/RL/SAF, where the thick MgO ensures pure VCMA gate control without STT contribution. We use an electrical scheme that allows for individual control over SOT/gate pulse amplitudes (VSOT/Vg), and variable pulse duration (tp). Fig. 1(b) shows exemplary SOT switching probability (Psw) curves under different Vg, at tp = 0.4ns. We observe a clear decrease (increase) in VSOT under Vg of 1V (-1V) for both AP-P and P-AP transitions. The critical switching voltage, defined at Psw = 50%, is converted into the critical switching current (Ic) and plotted as a function of 1/tp in Fig. 1(c). It shows a typical linear scaling, i.e. Ic = Ic0 + q/tp in the sub-ns regime for all gate values [7], where Ic0 is the intrinsic critical current and q indicates the nucleation energy for magnetization reversal. The linear dependences of Ic0 and q on Vg [Fig. 1(d)] reflect the direct modification of PMA upon Vg applications and the change of nucleation energy, respectively. We report here a 25% reduction in Ic under Vg = 1V, which corresponds to a 45% reduction in total switching energy to 30fJ at 0.4ns. We further demonstrate that VGSOT offers reliable sub-ns switching with a low write error rate (&lt;10-5) and that VGOST-pMTJ is highly durable against large read/write stresses (&gt;1012) Finally, we benchmark the VGSOT performance at the 5nm technology node against other embedded memory technologies to highlight the VGSOT benefits. It reveals that VGSOT-4MTJ design can reduce the effective cell area to &lt;0.5x of the high-density SRAM at minimal performance degradation. Meanwhile, the constrain of achieving challenging material parameters for VCMA-MTJ (ξ &gt; 1000fJ/Vm) and SOT-MTJ (θSH &gt; 1.5) can be relaxed to reasonable values: ξ = 300fJ/Vm and θSH = 0.45. Our study shows the great potential of this VGSOT approach for fast, dense and low power embedded memory applications.

**References:**

[1] B. Dieny et al., Nat. Electron. 3, pp. 446–459 (2020) [2] T. Devolder et al., Phys. Rev. Lett. 100, 057206 (2008) [3] K. Garello et al., IEEE VLSI Circ., pp. 81-82 (2018) [4] Y. C. Wu et al., IEEE VLSI Tech., TMFS.4 (2020) [5] C. Grezes et al., IEEE Magn. Lett. 8, 3102705 (2017) [6] H. Yoda et al., IEEE IEDM, pp. 27.6.1-27.6.4 (2016) [7] K. Garello et al., Appl. Phys. Lett. 105, 212402 (2014)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/704e6e8a749f9a7c8eea400c0ddf393c.jpg)
Fig. 1. (a) Illustration of the VGSOT writing scheme. (b) SOT switching probability (Psw) as a function of VSOT under different Vg at tp = 0.4ns. (b) Critical SOT switching current and current density at Psw = 50% as a function of 1/ tp, showing typical linear scaling for all Vg. (c) Ic0 and q as a function of Vg.

 Voltage-Gate Assisted Spin-Orbit Torque MRAM for High-Density and Low Power Embedded Applications INVITED

Demand for Artificial Intelligence (AI) has been growing due to their usefulness in commercial electronics as well as tackling global issues, such as a pandemic [1]. Current devices that perform AI rely on the conventional von Neumann architecture, which requires extensive programming and power [2]. Therefore, significant attention is dedicated to the brain-inspired neuromorphic computing (NC) as an alternative that is more power-efficient and a more suitable platform for AI [3]. Neuromorphic architecture consists of a network of interconnected synapses and neurons that mimic their biological counterparts [4]. Sengupta et al. have proposed a neuron design based on a domain wall device that uses a magnetic tunnel junction (MTJ) [5] at one end. The tunneling magnetoresistance (TMR) depends on the position of the DW. A current is required to move the domain wall forward or backward (integrate or leak functions respectively). In this work, we place emphasis on designing a domain wall (DW) based artificial neuron, which displays the leaky function without the requirement for a driving current. To achieve leaky function and self-reset process of neuron, we propose a domain wall device, which has an anisotropy field (Hk) gradient in the region of interest (ROI), as shown in Fig. 1(a). This anisotropy gradient is crucial in inducing the automatic returning motion of DW in the absence of current. The design is simulated using Mumax3, and the DW in the free layer nanowire (NW) is driven by spin-transfer-torque (STT). As shown in Fig. 1, current is first applied to propagate the DW towards the threshold position to “fire” the neuron device; current is subsequently removed and DW motion is observed. The micromagnetic parameters used here represent the magnetic properties of CoFeB film with a perpendicular magnetic anisotropy. The magnetic anisotropy constant (Ku) and saturation magnetization (Ms) are graded in ROI, so that Hk gradient is achieved. Some results are presented in Fig. 2(b), showing DW position vs time graph of the reference NW, without any gradient. One may notice that the DW does not return towards the initial position in these devices. Fig. 2(c) corresponds to a device with an Hk gradient. We can notice that the device enables natural return of the domain wall towards the starting position in the absence of current. This suggests that the device displays the leakage function, as well as the ability to reset after the neuron has fired. Fig. 2(d) corresponds to the same device as that of Fig. 2(c), except in this case, three current pulses are consecutively applied with a time interval of 60 ns. These results indicate that the neurons can function with a reset time of about 60 ns. The proposed design has potential applications for DW-based artificial neuron since it displays the integration and leakage functions, while the firing function can be achieved with the utilization of MTJ during the fabrication process.

**References:**

[1] Z. Hu, Q. Ge, S. Li, L. Jin, and M. Xiong, “Artificial Intelligence Forecasting of Covid-19 in China,” Feb. 2020, Accessed: Jun. 25, 2020. [Online]. Available: http://arxiv.org/abs/2002.07112. [2] R. A. Nawrocki, R. M. Voyles, and S. E. Shaheen, “A Mini Review of Neuromorphic Architectures and Implementations,” IEEE Trans. Electron Devices, vol. 63, no. 10, pp. 3819–3829, 2016, doi: 10.1109/TED.2016.2598413. [3] H. Farkhani, M. Tohidi, S. Farkhani, J. K. Madsen, and F. Moradi, “A Low-Power High-Speed Spintronics-Based Neuromorphic Computing System Using Real-Time Tracking Method,” IEEE J. Emerg. Sel. Top. Circuits Syst., vol. 8, no. 3, pp. 627–638, Sep. 2018, doi: 10.1109/JETCAS.2018.2813389. [4] J. Grollier, D. Querlioz, K. Y. Camsari, K. Everschor-Sitte, S. Fukami, and M. D. Stiles, “Neuromorphic spintronics,” Nat. Electron., 2020, doi: 10.1038/s41928-019-0360-9. [5] A. Sengupta, P. Panda, P. Wijesinghe, Y. Kim, and K. Roy, “Magnetic tunnel junction mimics stochastic cortical spiking neurons,” Sci. Rep., vol. 6, no. July, pp. 1–9, 2016, doi: 10.1038/srep30039.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/e015896fad0991ac5a8452eee8bf2d23.jpg)
The proposed 3T-MTJ device. Magnetizations of the pinned layers are fixed while that of the MTJ free layer NW can be manipulated by applying electrical current. Conduction electrons are polarized by a pinned layer and push the DW by STT. Neuronal firing happens when the DW propagates past the MTJ device, i.e. when parallel configuration of the MTJ is achieved.

![](https://assets.underline.io/uploads/markdown_image/1/image/e66f29e5f0d82f3c0bbcc2c2314dc9be.jpg)
(a) Schematic of the NW with the ROI labelled. (b and c) DW position vs time graphs for the reference NW and the result where DWs returned and stabilized around the initial position after neuronal firing. (d) is the case with three pulses sequentially applied. The interval is the same between each pulse, sufficient for DWs to stabilize around the initial position.

 Domain wall device with a graded anisotropy field as artificial neuron for neuromorphic computing

Topological insulator is a material that is conductive on the surface and internally insulated [1].The unique topological protection characteristics of topological insulators showing robustness to inherent and introduced disorder have attracted widespread attention. Inspired by topological insulators, scientists have tried to extend the magical properties of topological insulators to optical systems, and proposed different systems of optical topological insulators in succession. Recently, Two-dimensional coupled resonant optical waveguide (CROW) has become an advantageous tool for studying optical topological insulators due to their unique structure and compatibility with traditional photonic integrated circuits [2]. This paper mainly studies the coupling strength k range between adjacent rings of two-dimensional micro-ring arrays (as shown in Fig.1) forming optical topological insulators. First, using the transfer matrix method analyzed the transmission relationship and energy band of the micro-ring array to determine the range of appropriate coupling strength. Then we simulated the energy band diagram of the proposed structure under different coupling strengths to verify its topological state. Finally, we explored the influence of the coupling strength on the spectral transmission in the topological state. Based on the above analysis, we obtain the critical value of the coupling strength k to form the optical topological insulator. When the coupling strength is greater than 0.649, the two-dimensional micro-ring topological array becomes an optical topological insulator. Within this coupling strength range, the change of coupling strength will also affect the spectral transmission, as shown in Fig.2. When the k satisfies the condition of the edge state, the spectral transmission is maintained at a relatively high level. In addition, when k is 0.775 in Fig.2(c), the spectral transmission of the system behaves more stable and power value is maintained above 0.9 during the spectrum of 1.557μm-1.562μm. Whereas, in other condition such as shown in Fig 7(a), (b) and (d), the spectral transmission fluctuates at a certain value. The coupling coefficient k is farther away from the value of 0.775, the fluctuations are even stronger. The simulation results reveal that there is the optimal coupling coefficient to ensure a flat spectral response in the topological edge state. This research will be helpful for investigating the application of photonic communications based on optical topological insulators in the future works.

**References:**

[1] M. Hasan, and C. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82(4) (2010): 3045-3067. [2] M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, Robust optical delay lines with topological protection, Nature Physics 7(11) (2011): 907-912.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/3b5bba06e2939c1a29625d319302ee34.jpg)
Fig. 1 Schematic diagram of two-dimensional resonators topological insulator.

![](https://assets.underline.io/uploads/markdown_image/1/image/38d3d818d0b5512e54f28e89a4eb4904.jpg)
Fig. 2 Transmission characteristic curve of a 5× 5 site ring array system, during different coupling coefficient k.

 Study on the Critical State of Two-Dimensional Resonators Topological Insulator

The utilization of current-induced spin-orbit torques (SOTs) in magnetic heterostructures has been devoted lots of research attention as it shows promise for future ultrafast and power conservation magnetic memories [1]. Features of three-terminal non-volatile magnetic memories utilizing SOT-induced magnetization switching promote us to develop kinds of materials in which the SOTs with desiring polarization and orientation are able to be generated efficiently [2]. Comparing to conventional heavy metals like Pt, W, Ta, etc., Mn-based L12 type antiferromagnets (AFM) XMn3 (X=Ir, Pt, etc.) are of significant interest due to the high charge current to spin current conversion efficiency, i.e., large spin Hall angle (SHA) they show [3], as well as the potential of controlling SHA and SOTs orientation by tuning the spin texture [4], crystal structure [5], facet orientation [6], etc. they possess. Since recent research showed that the generation of spin current in Mn-based AFM strongly related to magnetic structure [7], it is necessary to fabricate high quality Mn-based AFM single crystal films with known magnetic structure to analysis the SHA features. In this work, we report the fabrication of L12 PtMn3 single crystal epitaxial films which the magnetic structure is known as kagome-type lattice on a (111) plane, and the PtMn3 thickness dependence of effective SHA θeffSH. All the samples were prepared by magnetron sputtering system. The stack structure was MgO(001) sub. / PtMn3 (t) / Py (4) / MgAl2O4 (2) / Pt (3) in nm. PtMn3 layer was deposited at substrate temperature (Ts) of 450oC, while other layers were prepared at Ts = R.T. θeffSH were estimated by Spin-Torque ferromagnetic resonance (ST-FMR) measurement with lock-in amplifier. Fig. 1 shows lattice constant of a-axis and c-axis of PtMn3 films as function of thickness. Since the lattice constant of MgO substrate and L12 PtMn3 is 4.22Å, 3.833Å respectively, PtMn3 epitaxial films thus under tensile strain which is induced by MgO substrate. With the decreasing of PtMn3 thickness, a-axis became larger while c-axis became smaller, which corresponds to the epitaxial relation. We note that a similar phenomenon also had been observed in previous work [8]. Fig. 2 shows the thickness dependence of measured θeffSH. We found that the calculated θeffSH exhibits little variation as a function of the RF excitation frequency, therefore we plotted the θeffSH as a function of PtMn3 thickness at 6 GHz. θeffSH of 15nm and 30nm PtMn3 were estimated as 0.27 and 0.19 respectively, which are comparable to that found in IrMn3 single crystal film. We observed a significant increasement of θeffSH, reaches 0.55, for 20nm PtMn3. We speculate that such a dramatic variation of θeffSH originates from the change of magnetic structure of PtMn3, which a similar tendency of anomalous Hall conductivity v.s. thickness also reported by previous work [8]. We will show more detail in the conference. This work was supported in part by GP-spin program of Tohoku University.

**References:**

[1] R. Ramaswamy, J. M. Lee, K. Cai, and H. Yang, Appl. Phys. Rev. 5, 031107 (2018). [2] A. Manchon, J. Zelezny, I. M. Miron, T. Jungwirth, J. Sinova, A. Thiaville, K. Garello, and P. Gambardella, Rev. Mod. Phys. 91 (2019). [3] W. Zhang, M. B. Jungfleisch, W. Jiang, J. E. Pearson, A. Hoffmann, F. Freimuth, and Y. Mokrousov, Phys. Rev. Lett. 113, 196602 (2014). [4] J. Zhou et al., Sci. Adv. 5, eaau6696 (2019). [5] W. Zhang, M. B. Jungfleisch, F. Freimuth, W. Jiang, J. Sklenar, J. E. Pearson, J. B. Ketterson, Y. Mokrousov, and A. Hoffmann, Phys. Rev. B 92, 144405 (2015). [6] W. Zhang, W. Han, S.-H. Yang, Y. Sun, Y. Zhang, B. Yan, and S. S. Parkin, Sci. Adv. 2, e1600759 (2016). [7] M. Kimata et al., Nature 565, 627 (2019). [8] Z. Liu et al., Nat. Electron. 1, 172 (2018).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/fb1c8f842dcba766d7bbd0ed4f51ab6e.jpg)
Lattice constants of a, c-axis as functions of thickness. Dotted line indicates the bulk value of lattice constants for both a-axis and c-axis

![](https://assets.underline.io/uploads/markdown_image/1/image/1311eb90b4623c63e35c7682ada12d79.jpg)
Thickness dependence of effective SHA θeffSH

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