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Reliable simulation tools, capable of handling complex geometries composed of multiple materials, provide valuable help in improving the design of MRAM cells. In this work, we describe a solver based on the finite element implementation of the Landau\u2013Lifshitz\u2013Gilbert equation coupled to the spin and charge drift-diffusion formalism. The torque acting in all layers from different contributions is computed from a unified expression. In consequence of the versatility of the finite element implementation, the solver is applied to switching simulations of recently proposed structures based on spin-transfer torque, with a double reference layer or an elongated and composite free layer, and of a structure combining spin-transfer and spin-orbit torques.<\/jats:p>","DOI":"10.3390\/mi14050898","type":"journal-article","created":{"date-parts":[[2023,4,24]],"date-time":"2023-04-24T04:01:23Z","timestamp":1682308883000},"page":"898","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Finite Element Approach for the Simulation of Modern MRAM Devices"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2982-7629","authenticated-orcid":false,"given":"Simone","family":"Fiorentini","sequence":"first","affiliation":[{"name":"Christian Doppler Laboratory for Nonvolatile Magnetoresistive Memory and Logic at the Institute for Microelectronics, TU Wien, Gu\u00dfhausstra\u00dfe 27-29\/E360, 1040 Vienna, Austria"},{"name":"Institute for Microelectronics, TU Wien, Gu\u00dfhausstra\u00dfe 27-29\/E360, 1040 Vienna, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8499-1563","authenticated-orcid":false,"given":"Nils Petter","family":"J\u00f8rstad","sequence":"additional","affiliation":[{"name":"Christian Doppler Laboratory for Nonvolatile Magnetoresistive Memory and Logic at the Institute for Microelectronics, TU Wien, Gu\u00dfhausstra\u00dfe 27-29\/E360, 1040 Vienna, Austria"},{"name":"Institute for Microelectronics, TU Wien, Gu\u00dfhausstra\u00dfe 27-29\/E360, 1040 Vienna, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0972-0875","authenticated-orcid":false,"given":"Johannes","family":"Ender","sequence":"additional","affiliation":[{"name":"Christian Doppler Laboratory for Nonvolatile Magnetoresistive Memory and Logic at the Institute for Microelectronics, TU Wien, Gu\u00dfhausstra\u00dfe 27-29\/E360, 1040 Vienna, Austria"},{"name":"Institute for Microelectronics, TU Wien, Gu\u00dfhausstra\u00dfe 27-29\/E360, 1040 Vienna, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9019-4835","authenticated-orcid":false,"given":"Roberto Lacerda","family":"de Orio","sequence":"additional","affiliation":[{"name":"Institute for Microelectronics, TU Wien, Gu\u00dfhausstra\u00dfe 27-29\/E360, 1040 Vienna, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5583-6177","authenticated-orcid":false,"given":"Siegfried","family":"Selberherr","sequence":"additional","affiliation":[{"name":"Institute for Microelectronics, TU Wien, Gu\u00dfhausstra\u00dfe 27-29\/E360, 1040 Vienna, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5994-2682","authenticated-orcid":false,"given":"Mario","family":"Bendra","sequence":"additional","affiliation":[{"name":"Christian Doppler Laboratory for Nonvolatile Magnetoresistive Memory and Logic at the Institute for Microelectronics, TU Wien, Gu\u00dfhausstra\u00dfe 27-29\/E360, 1040 Vienna, Austria"},{"name":"Institute for Microelectronics, TU Wien, Gu\u00dfhausstra\u00dfe 27-29\/E360, 1040 Vienna, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wolfgang","family":"Goes","sequence":"additional","affiliation":[{"name":"Silvaco Europe Ltd., Cambridge PE27 5JL, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1736-6976","authenticated-orcid":false,"given":"Viktor","family":"Sverdlov","sequence":"additional","affiliation":[{"name":"Christian Doppler Laboratory for Nonvolatile Magnetoresistive Memory and Logic at the Institute for Microelectronics, TU Wien, Gu\u00dfhausstra\u00dfe 27-29\/E360, 1040 Vienna, Austria"},{"name":"Institute for Microelectronics, TU Wien, Gu\u00dfhausstra\u00dfe 27-29\/E360, 1040 Vienna, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1844","DOI":"10.1109\/JPROC.2016.2574939","article-title":"Standby-Power-Free Integrated Circuits Using MTJ-based VLSI Computing","volume":"104","author":"Hanyu","year":"2016","journal-title":"Proc. IEEE"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Gallagher, W.J., Chien, E., Chiang, T., Huang, J., Shih, M., Wang, C.Y., Weng, C., Chen, S., Bair, C., and Lee, G. (2019, January 7\u201311). 22nm STT-MRAM for Reflow and Automotive Uses with High Yield, Reliability, and Magnetic Immunity and with Performance and Shielding Options. Proceedings of the 2019 IEEE International Electron Devices Meeting (IEDM), San Francisco, CA, USA.","DOI":"10.1109\/IEDM19573.2019.8993469"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Han, S.H., Lee, J.M., Shin, H.M., Lee, J.H., Suh, K.S., Nam, K.T., Kwon, B.S., Cho, M.K., Lee, J., and Jeong, J.H. (2020, January 12\u201318). 28-nm 0.08 mm2\/Mb Embedded MRAM for Frame Buffer Memory. Proceedings of the IEDM Conference, San Francisco, CA, USA.","DOI":"10.1109\/IEDM13553.2020.9372040"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Shih, Y.C., Lee, C.F., Chang, Y.A., Lee, P.H., Lin, H.J., Chen, Y.L., Lo, C.P., Lin, K.F., Chiang, T.W., and Lee, Y.J. (2020, January 12\u201318). A Reflow-Capable, Embedded 8Mb STT-MRAM Macro with 9ns Read Access Time in 16nm FinFET Logic CMOS Process. Proceedings of the IEDM Conference, San Francisco, CA, USA.","DOI":"10.1109\/IEDM13553.2020.9372115"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Naik, V.B., Yamane, K., Lee, T., Kwon, J., Chao, R., Lim, J., Chung, N., Behin-Aein, B., Hau, L., and Zeng, D. (2020, January 12\u201318). JEDEC-Qualified Highly Reliable 22nm FD-SOI Embedded MRAM For Low-Power Industrial-Grade, and Extended Performance Towards Automotive-Grade-1 Applications. Proceedings of the IEDM Conference, San Francisco, CA, USA.","DOI":"10.1109\/IEDM13553.2020.9371935"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Yau, J.B., Fung, Y.K.K., and Gibson, G.W. (2017, January 8\u20139). Hybrid Cryogenic Memory Cells for Superconducting Computing Applications. Proceedings of the ICRC Conference, Washington, DC, USA.","DOI":"10.1109\/ICRC.2017.8123684"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1038\/s41598-018-37204-3","article-title":"A Cryogenic Spin-Torque Memory Element with Precessional Magnetization Dynamics","volume":"9","author":"Rowlands","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"022409","DOI":"10.1063\/1.5129553","article-title":"A Low Temperature Functioning CoFeB\/MgO-Based Perpendicular Magnetic Tunnel Junction for Cryogenic Nonvolatile Random Access Memory","volume":"116","author":"Lang","year":"2020","journal-title":"Appl. Phys. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"L1","DOI":"10.1016\/0304-8853(96)00062-5","article-title":"Current-Driven Excitation of Magnetic Multilayers","volume":"159","author":"Slonczewski","year":"1996","journal-title":"J. Magn. Magn. Mater."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"9353","DOI":"10.1103\/PhysRevB.54.9353","article-title":"Emission of Spin Waves by a Magnetic Multilayer Traversed by a Current","volume":"54","author":"Berger","year":"1996","journal-title":"Phys. Rev. B"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"024411","DOI":"10.1103\/PhysRevB.71.024411","article-title":"Currents, Torques, and Polarization Factors in Magnetic Tunnel Junctions","volume":"71","author":"Slonczewski","year":"2005","journal-title":"Phys. Rev. B"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1016\/0375-9601(71)90196-4","article-title":"Current-Induced Spin Orientation of Electrons in Semiconductors","volume":"35","author":"Dyakonov","year":"1971","journal-title":"Phys. Lett. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"036601","DOI":"10.1103\/PhysRevLett.101.036601","article-title":"Electric Manipulation of Spin Relaxation Using the Spin Hall Effect","volume":"101","author":"Ando","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"14855","DOI":"10.1038\/srep14855","article-title":"A Three-Dimensional Spin-Diffusion Model for Micromagnetics","volume":"5","author":"Abert","year":"2015","journal-title":"Sci. Rep."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1038\/s41598-016-0019-y","article-title":"A Self-Consistent Spin-Diffusion Model for Micromagnetics","volume":"6","author":"Abert","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"12937","DOI":"10.1038\/s41598-017-13181-x","article-title":"Unified Treatment of Spin Torques Using a Coupled Magnetisation Dynamics and Three-Dimensional Spin Current Solver","volume":"7","author":"Lepadatu","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.camwa.2020.06.009","article-title":"MFEM: A Modular Finite Element Library","volume":"81","author":"Anderson","year":"2020","journal-title":"Comp. Math. Appl."},{"key":"ref_18","unstructured":"(2023, April 19). MFEM: Modular Finite Element Methods [Software]. Available online: https:\/\/mfem.org."},{"key":"ref_19","unstructured":"Ender, J., Fiorentini, S., de Orio, R.L., Had\u00e1mek, T., Bendra, M., J\u00f8rstad, N.P., and Loch, W.J. (2023, April 19). ViennaSpinMag. Available online: https:\/\/www.iue.tuwien.ac.at\/viennaspinmag."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Hu, G., Lauer, G., Sun, J.Z., Hashemi, P., Safranski, C., Brown, S.L., Buzi, L., Edwards, E.R.J., D\u2019Emic, C.P., and Galligan, E. (2021, January 11\u201316). 2X Reduction of STT-MRAM Switching Current Using Double Spin-Torque Magnetic Tunnel Junction. Proceedings of the IEDM Conference, San Francisco, CA, USA.","DOI":"10.1109\/IEDM19574.2021.9720691"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Jinnai, B., Igarashi, J., Watanabe, K., Funatsu, T., Sato, H., Fukami, S., and Ohno, H. (2020, January 12\u201318). High-Performance Shape-Anisotropy Magnetic Tunnel Junctions down to 2.3 nm. Proceedings of the IEDM Conference, San Francisco, CA, USA.","DOI":"10.1109\/IEDM13553.2020.9371972"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"582","DOI":"10.1038\/s41928-018-0160-7","article-title":"Field-Free Switching of a Perpendicular Magnetic Tunnel Junction Through the Interplay of Spin\u2013Orbit and Spin-Transfer Torques","volume":"1","author":"Wang","year":"2018","journal-title":"Nat. Electron."},{"key":"ref_23","first-page":"153","article-title":"On the Theory of the Dispersion of Magnetic Permeability in Ferromagnetic Bodies","volume":"8","author":"Landau","year":"1935","journal-title":"Phys. Z. Sowjetunion"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3443","DOI":"10.1109\/TMAG.2004.836740","article-title":"A Phenomenological Theory of Damping in Ferromagnetic Materials","volume":"40","author":"Gilbert","year":"2004","journal-title":"IEEE Trans. Magn."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/j.jmmm.2007.03.178","article-title":"Micromagnetic Simulations with Thermal Noise: Physical and Numerical Aspects","volume":"316","author":"Martinez","year":"2007","journal-title":"J. Magn. Magn. Mater."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1016\/j.jmmm.2004.09.126","article-title":"Micromagnetic Computations of Spin Polarized Current-Driven Magnetization Processes","volume":"286","author":"Torres","year":"2005","journal-title":"J. Magn. Magn. Mater."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"093907","DOI":"10.1063\/1.2800999","article-title":"Micromagnetic Simulations of Current-Induced Magnetization Switching in Co\/Cu\/Co Nanopillars","volume":"102","author":"Xiao","year":"2007","journal-title":"J. Appl. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"063914","DOI":"10.1063\/1.2496202","article-title":"Micromagnetic Modeling of Magnetization Switching Driven by Spin-Polarized Current in Magnetic Tunnel Junctions","volume":"101","author":"Finocchio","year":"2007","journal-title":"J. Appl. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"07B117","DOI":"10.1063\/1.2838285","article-title":"Modeling of Fast Switching Processes in Nanoscale Spin Valves","volume":"103","author":"Carpentieri","year":"2008","journal-title":"J. Appl. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"117204","DOI":"10.1103\/PhysRevLett.109.117204","article-title":"Unified Drift-Diffusion Theory for Transverse Spin Currents in Spin Valves, Domain Walls, and Other Textured Magnets","volume":"109","author":"Petitjean","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"15692","DOI":"10.1038\/s41598-021-95267-1","article-title":"Nonresonant Amplification of Spin Waves Through Interface Magnetoelectric Effect and Spin-Transfer Torque","volume":"11","author":"Graczyk","year":"2021","journal-title":"Sci. Rep."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"174411","DOI":"10.1103\/PhysRevB.87.174411","article-title":"Current Induced Torques and Interfacial Spin-Orbit Coupling: Semiclassical Modeling","volume":"87","author":"Haney","year":"2013","journal-title":"Phys. Rev. B"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1140\/epjb\/e2019-90599-6","article-title":"Micromagnetics and Spintronics: Models and Numerical Methods","volume":"92","author":"Abert","year":"2019","journal-title":"Eur. Phys. J. B"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"236601","DOI":"10.1103\/PhysRevLett.88.236601","article-title":"Mechanisms of Spin-Polarized Current-Driven Magnetization Switching","volume":"88","author":"Zhang","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","unstructured":"Luc, D. (2016). Th\u00e9orie Unifi\u00e9e du Transport de Spin, Charge et Chaleur. [Ph.D. Thesis, Universit\u00e9 Grenoble Alpes]."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/j.physb.2015.09.003","article-title":"Coupling of Dynamical Micromagnetism and a Stationary Spin Drift-Diffusion Equation: A Step Towards a Fully Self-Consistent Spintronics Framework","volume":"486","author":"Ruggeri","year":"2016","journal-title":"Physica B"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Braess, D. (2007). Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press. [3rd ed.].","DOI":"10.1017\/CBO9780511618635"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Larson, M.G., and Bengzon, F. (2013). The Finite Element Method: Theory, Implementation, and Applications, Springer.","DOI":"10.1007\/978-3-642-33287-6"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1142\/S0218202506001169","article-title":"Convergence of a Finite Element Discretization for the Landau-Lifshitz Equations in Micromagnetism","volume":"16","author":"Alouges","year":"2006","journal-title":"Math. Models Methods Appl. Sci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1090\/S0025-5718-07-02079-0","article-title":"Numerical Analysis of an Explicit Approximation Scheme for the Landau-Lifshitz-Gilbert Equation","volume":"77","author":"Bartels","year":"2008","journal-title":"Math. Comput."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"187","DOI":"10.3934\/dcdss.2008.1.187","article-title":"A New Finite Element Scheme for Landau-Lifchitz Equations","volume":"1","author":"Alouges","year":"2008","journal-title":"Discrete Contin. Dyn. Syst. S"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1345","DOI":"10.1016\/j.physb.2011.11.031","article-title":"A Convergent Finite Element Approximation for Landau\u2013Lifschitz\u2013Gilbert Equation","volume":"407","author":"Alouges","year":"2012","journal-title":"Physica B"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2627","DOI":"10.1142\/S0218202514500328","article-title":"Multiscale Modeling in Micromagnetics: Existence of Solutions and Numerical Integration","volume":"24","author":"Bruckner","year":"2014","journal-title":"Math. Models Methods Appl. Sci."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1016\/j.camwa.2014.07.010","article-title":"Spin-Polarized Transport in Ferromagnetic Multilayers: An Unconditionally Convergent FEM Integrator","volume":"68","author":"Abert","year":"2014","journal-title":"Comp. Math. Appl."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"108103","DOI":"10.1016\/j.sse.2021.108103","article-title":"Coupled Spin and Charge Drift-Diffusion Approach Applied to Magnetic Tunnel Junctions","volume":"186","author":"Fiorentini","year":"2021","journal-title":"Solid-State Electron."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"20958","DOI":"10.1038\/s41598-022-25586-4","article-title":"Spin and Charge Drift-Diffusion in Ultra-Scaled MRAM Cells","volume":"12","author":"Fiorentini","year":"2022","journal-title":"Sci. Rep."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"104422","DOI":"10.1103\/PhysRevB.92.104422","article-title":"Analytical Description of Ballistic Spin Currents and Torques in Magnetic Tunnel Junctions","volume":"92","author":"Chshiev","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/0375-9601(75)90174-7","article-title":"Tunneling Between Ferromagnetic Films","volume":"54","author":"Julliere","year":"1975","journal-title":"Phys. Lett. A"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"409","DOI":"10.6028\/jres.049.044","article-title":"Methods of Conjugate Gradients for Solving Linear Systems","volume":"49","author":"Hestenes","year":"1952","journal-title":"J. Res. Natl. Bur. Stand."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Camsari, K.Y., Ganguly, S., Datta, D., and Datta, S. (2014, January 15\u201317). Physics-Based Factorization of Magnetic Tunnel Junctions for Modeling and Circuit Simulation. Proceedings of the IEDM Conference, San Francisco, CA, USA.","DOI":"10.1109\/IEDM.2014.7047177"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"856","DOI":"10.1137\/0907058","article-title":"GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems","volume":"7","author":"Saad","year":"1986","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1329","DOI":"10.1007\/s10444-019-09667-z","article-title":"Convergent Tangent Plane Integrators for the Simulation of Chiral Magnetic Skyrmion Dynamics","volume":"45","author":"Hrkac","year":"2019","journal-title":"Adv. Comput. Math."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/j.jmmm.2013.05.051","article-title":"magnum.fe: A Micromagnetic Finite-Element Simulation Code Based on FEniCS","volume":"345","author":"Abert","year":"2013","journal-title":"J. Magn. Magn. Mater."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1659","DOI":"10.1109\/20.104482","article-title":"An Original Solution for Unbounded Electromagnetic 2D- and 3D-Problems Throughout the Finite Element Method","volume":"26","author":"Imhoff","year":"1990","journal-title":"IEEE Trans. Magn."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1663","DOI":"10.1109\/20.124021","article-title":"Finite Element Modeling of Unbounded Problems Using Transformations: A Rigorous, Powerful and Easy Solution","volume":"28","author":"Brunotte","year":"1992","journal-title":"IEEE Trans. Magn."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"1434","DOI":"10.1109\/20.767235","article-title":"Finite Element Modelling with Transformation Techniques","volume":"35","author":"Henrotte","year":"1999","journal-title":"IEEE Trans. Magn."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"180901","DOI":"10.1063\/1.5093730","article-title":"Tomorrow\u2019s Micromagnetic Simulations","volume":"125","author":"Leliaert","year":"2019","journal-title":"J. Appl. Phys."},{"key":"ref_58","unstructured":"Ender, J., Mohamedou, M., Fiorentini, S., de Orio, R.L., Selberherr, S., Goes, W., and Sverdlov, V. (October, January 23). Efficient Demagnetizing Field Calculation for Disconnected Complex Geometries in STT-MRAM Cells. Proceedings of the SISPAD Conference, Kobe, Japan."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1109\/20.106342","article-title":"Hybrid Method for Computing Demagnetizing Fields","volume":"26","author":"Fredkin","year":"1990","journal-title":"IEEE Trans. Magn."},{"key":"ref_60","unstructured":"Lirkov, I., Margenov, S., and Wa\u015bniewski, J. (2005, January 6\u201310). \u210b-Matrix Techniques for Stray-Field Computations in Computational Micromagnetics. Proceedings of the Large-Scale Scientific Computing Conference, Sozopol, Bulgaria."},{"key":"ref_61","unstructured":"(2023, April 19). H2Lib [Software]. Available online: http:\/\/www.h2lib.org."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"108335","DOI":"10.1016\/j.sse.2022.108335","article-title":"Double Reference Layer STT-MRAM Structures with Improved Performance","volume":"194","author":"Loch","year":"2022","journal-title":"Solid-State Electron."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"995","DOI":"10.1109\/TED.2020.2966731","article-title":"Novel Quad-Interface MTJ Technology and its First Demonstration With High Thermal Stability Factor and Switching Efficiency for STT-MRAM Beyond 2X nm","volume":"67","author":"Nishioka","year":"2020","journal-title":"IEEE Trans. Electron Devices"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1038\/s41565-019-0607-7","article-title":"Single-Shot Dynamics of Spin-Orbit Torque and Spin Transfer Torque Switching in Three-Terminal Magnetic Tunnel Junctions","volume":"15","author":"Grimaldi","year":"2020","journal-title":"Nat. Nanotechnol."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"496","DOI":"10.1038\/nphys3304","article-title":"Role of Transparency of Platinum\u2013Ferromagnet Interfaces in Determining the Intrinsic Magnitude of the Spin Hall Effect","volume":"11","author":"Zhang","year":"2015","journal-title":"Nat. Phys."}],"container-title":["Micromachines"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-666X\/14\/5\/898\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:21:08Z","timestamp":1760124068000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-666X\/14\/5\/898"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,22]]},"references-count":65,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["mi14050898"],"URL":"https:\/\/doi.org\/10.3390\/mi14050898","relation":{},"ISSN":["2072-666X"],"issn-type":[{"value":"2072-666X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,22]]}}}