Modeling Spin-Orbit Torques
in Advanced Magnetoresistive Devices

B.5 Analytical Solution for Spin Accumulation

In this section, the analytical solutions for the spin accumulation components are summarized.

\begin{multline} V^N_{dz} = -\alpha _\mathrm {SH}\lambda ^N_{sf}E_x\frac {\cosh \left (z/\lambda ^N_{sf} + \xi ^N\right )}{\sinh (\xi ^N)}\frac {(1-e^{-\xi ^N})^2}{1+e^{-2\xi ^N}} \times \frac {\vert \tilde {G}^N_{\uparrow \downarrow }\vert ^2 +\operatorname {Re}\{\tilde {G}^N_{\uparrow \downarrow }\}\tanh ^2\left (\xi ^N \right )}{\left (\tanh ^2(\xi ^N )+\operatorname {Re}\{\tilde {G}^N_{\uparrow \downarrow }\}\right )^2 +\operatorname {Im}\{\tilde {G}^N_{\uparrow \downarrow }\}^2} \\ - \alpha _\mathrm {SH}E_x\lambda ^N_{sf}\left [-e^{z/\lambda ^N_{sf} + \xi ^N}- \left (1 -e^{\xi ^N}\right )\frac {\cosh \left (z/\lambda ^N_{sf} + \xi ^N\right )}{\sinh (\xi ^N)}\right ] \end{multline}

\begin{equation} V^N_{fz} = -\alpha _\mathrm {SH}\lambda ^N_{sf} E_x \frac {\cosh \left (z/\lambda ^N_{sf}+\xi ^N\right )}{\sinh \left (\xi ^N\right )}\frac {(1-e^{-\xi ^N })^2}{1+e^{-2\xi ^N }} \times \frac {\operatorname {Im}\{\tilde {G}^N_{\uparrow \downarrow }\}\tanh ^2(\xi ^N )}{\left (\tanh ^2(\xi ^N )+\operatorname {Re}\{\tilde {G}^N_{\uparrow \downarrow }\}\right )^2 + \operatorname {Im}\{\tilde {G}^N_{\uparrow \downarrow }\}^2}. \end{equation}

\begin{equation} V^N_{lz} = -\alpha _\mathrm {SH}\lambda ^N_{sf} E_x \frac {\sinh \left (z/\lambda ^N_{sf}+\xi ^N\right )}{\sinh \left (\xi ^N\right )} \frac {(1-e^{-\xi ^N})^2}{1-e^{-2\xi ^N}}\frac {\tilde {G_{\|}}^N\tanh (\xi ^N)}{2 - \tilde {G_{\|}}^F - \tilde {G_{\|}}^N} + \alpha _\mathrm {SH}\lambda ^N_{sf} E_x\left [-e^{z/\lambda ^N_{sf}+\xi ^N} - \left (1-e^{\xi ^N}\right )\frac {\sinh \left (z/\lambda ^N_{sf}+\xi ^N\right )}{\sinh \left (\xi ^N\right )} \right ] \end{equation}

\begin{equation} V^F_{lz} = -\alpha _\mathrm {SH}\frac {\sigma ^N }{\sigma ^F}\lambda ^F_{sdl}E_x\frac {\cosh \left (z/\lambda _{sdl}-\xi ^F\right )}{\sinh \left (-\xi ^F\right )}\frac {(1-e^{-\xi ^N})^2}{1-e^{-2\xi ^N}} \frac {\tilde {G_{\|}}^N\tanh (\xi ^N)}{2 - \tilde {G_{\|}}^F - \tilde {G_{\|}}^N} \end{equation}

Bibliography

[1]  R. Pollie, “Nanosheet chips poised to rescue Moore’s law,” Engineering, vol. 7, no. 12, pp. 1655–1656, 2021. doi: 10.1016/j.eng.2021.11.008.

[2]  J. Shalf, “The future of computing beyond Moore’s law,” Philosophical Transactions of the Royal Society A, vol. 378, no. 2166, p. 20190061, 2020. doi: 10.1098/rsta.2019.0061.

[3]  H. Wong, W. Li, J. Zhang, W. Bao, L. Wu, and J. Liu, “Driving for more Moore on computing devices with advanced non-volatile memory technology,” Electronics, vol. 14, no. 17, 2025. doi: 10.3390/electronics14173456.

[4]  K. Rupp and S. Selberherr, “The economic limit to Moore’s law,” IEEE Transactions on Semiconductor Manufacturing, vol. 24, no. 1, pp. 1–4, 2010. doi: 10.1109/TSM.2010.2089811.

[5]  N. Kim et al., “Leakage current: Moore’s law meets static power,” Computer, vol. 36, no. 12, pp. 68–75, 2003. doi: 10.1109/MC.2003.1250885.

[6]  G. Kamiya and P. Bertoldi, Energy Consumption in Data Centres and Broadband Communication Networks in the EU. Publications Office of the European Union Luxembourg, 2024. doi: 10.2760/706491. . ISBN 978-92-68-12554-0

[7]  T. Hanyu et al., “Standby-power-free integrated circuits using MTJ-based VLSI computing,” Proceedings of the IEEE, vol. 104, no. 10, pp. 1844–1863, 2016. doi: 10.1109/JPROC.2016.2574939.

[8]  P. Mannocci et al., “In-memory computing with emerging memory devices: Status and outlook,” APL Machine Learning, vol. 1, no. 1, 2023. doi: 10.1063/5.0136403.

[9]  S. Bhatti, R. Sbiaa, A. Hirohata, H. Ohno, S. Fukami, and S. Piramanayagam, “Spintronics based random access memory: A review,” Materials Today, vol. 20, no. 9, pp. 530–548, 2017. doi: 10.1016/j.mattod.2017.07.007.

[10]  “Everspin spin-transfer torque DDR products,” https://www.everspin.com/spin-transfer-torque-ddr-products, accessed: 01-10-2025.

[11]  V. Nguyen, S. Rao, K. Wostyn, and S. Couet, “Recent progress in spin-orbit torque magnetic random-access memory,” npj Spintronics, vol. 2, no. 1, p. 48, 2024. doi: 10.1038/s44306-024-00044-1.

[12]  M. Stettler et al., “State-of-the-art TCAD: 25 years ago and today,” in 2019 IEEE International Electron Devices Meeting (IEDM), 2019, pp. 39.1.1–39.1.4. doi: 10.1109/IEDM19573.2019.8993451.

[13]  A. Hirohata et al., “Review on spintronics: Principles and device applications,” Journal of Magnetism and Magnetic Materials, vol. 509, p. 166711, 2020. doi: 10.1016/j.jmmm.2020.166711.

[14]  W. v. Pauli Jr., “Zur quantenmechanik des magnetischen elektrons,” Zeitschrift für Physik, vol. 43, no. 9, pp. 601–623, 1927. doi: 10.1007/BF01397326.

[15]  J. M. Coey, Magnetism and Magnetic Materials. Cambridge University Press, 2010. doi: 10.1017/CBO9780511845000. . ISBN 9780511845000

[16]  A. Manchon et al., “Current-induced spin-orbit torques in ferromagnetic and antiferromagnetic systems,” Reviews of Modern Physics, vol. 91, p. 035004, 2019. doi: 10.1103/RevModPhys.91.035004.

[17]  M. N. Baibich et al., “Giant magnetoresistance of (001)Fe/(001)Cr magnetic superlattices,” Physical Review Letters, vol. 61, pp. 2472–2475, 1988. doi: 10.1103/PhysRevLett.61.2472.

[18]  G. Binasch, P. Grünberg, F. Saurenbach, and W. Zinn, “Enhanced magnetoresistance in layered magnetic structures with antiferromagnetic interlayer exchange,” Physical Review B, vol. 39, pp. 4828–4830, 1989. doi: 10.1103/PhysRevB.39.4828.

[19]  S. M. Thompson, “The discovery, development and future of gmr: The Nobel prize 2007,” Journal of Physics D: Applied Physics, vol. 41, no. 9, p. 093001, 2008. doi: 10.1088/0022-3727/41/9/093001.

[20]  M. Julliere, “Tunneling between ferromagnetic films,” Physics Letters A, vol. 54, no. 3, pp. 225–226, 1975. doi: 10.1016/0375-9601(75)90174-7.

[21]  T. Miyazaki and N. Tezuka, “Giant magnetic tunneling effect in Fe/Al2O3/Fe junction,” Journal of Magnetism and Magnetic Materials, vol. 139, no. 3, pp. L231–L234, 1995. doi: 10.1016/0304-8853(95)90001-2.

[22]  J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meservey, “Large magnetoresistance at room temperature in ferromagnetic thin film tunnel junctions,” Physical Review Letters, vol. 74, pp. 3273–3276, 1995. doi: 10.1103/PhysRevLett.74.3273.

[23]  W. H. Butler, X.-G. Zhang, T. C. Schulthess, and J. M. MacLaren, “Spin-dependent tunneling conductance of \(\mathrm {Fe}|\mathrm {MgO}|\mathrm {Fe}\) sandwiches,” Physical Review B, vol. 63, p. 054416, 2001. doi: 10.1103/PhysRevB.63.054416.

[24]  J. Mathon and A. Umerski, “Theory of tunneling magnetoresistance of an epitaxial Fe/MgO/Fe(001) junction,” Physical Review B, vol. 63, p. 220403, 2001. doi: 10.1103/PhysRevB.63.220403.

[25]  S. S. Parkin et al., “Giant tunnelling magnetoresistance at room temperature with MgO (100) tunnel barriers,” Nature Materials, vol. 3, no. 12, pp. 862–867, 2004. doi: 10.1038/nmat1256.

[26]  D. D. Djayaprawira et al., “230% room-temperature magnetoresistance in CoFeB/MgO/CoFeB magnetic tunnel junctions,” Applied Physics Letters, vol. 86, no. 9, p. 092502, 2005. doi: 10.1063/1.1871344.

[27]  J. Hayakawa, S. Ikeda, F. Matsukura, H. Takahashi, and H. Ohno, “Dependence of giant tunnel magnetoresistance of sputtered CoFeB/MgO/CoFeB magnetic tunnel junctions on MgO barrier thickness and annealing temperature,” Japanese Journal of Applied Physics, vol. 44, no. 4L, p. L587, 2005. doi: 10.1143/JJAP.44.L587.

[28]  S. Ikeda et al., “Magnetic tunnel junctions for spintronic memories and beyond,” IEEE Transactions on Electron Devices, vol. 54, no. 5, pp. 991–1002, 2007. doi: 10.1109/TED.2007.894617.

[29]  J. Slonczewski, “Current-driven excitation of magnetic multilayers,” Journal of Magnetism and Magnetic Materials, vol. 159, no. 1, pp. L1–L7, 1996. doi: 10.1016/0304-8853(96)00062-5.

[30]  L. Berger, “Emission of spin waves by a magnetic multilayer traversed by a current,” Physical Review B, vol. 54, pp. 9353–9358, 1996. doi: 10.1103/PhysRevB.54.9353.

[31]  J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, “Current-driven magnetization reversal and spin-wave excitations in Co \(/\)Cu \(/\)Co pillars,” Physical Review Letters, vol. 84, pp. 3149–3152, 2000. doi: 10.1103/PhysRevLett.84.3149.

[32]  Y. Huai, F. Albert, P. Nguyen, M. Pakala, and T. Valet, “Observation of spin-transfer switching in deep submicron-sized and low-resistance magnetic tunnel junctions,” Applied Physics Letters, vol. 84, no. 16, pp. 3118–3120, 2004. doi: 10.1063/1.1707228.

[33]  M. I. Dyakonov and V. I. Perel, “Possibility of orienting electron spins with current,” Soviet Journal of Experimental and Theoretical Physics Letters, vol. 13, p. 467, 1971.

[34]  M. I. Dyakonov and V. I. Perel, “Current-induced spin orientation of electrons in semiconductors,” Physics Letters A, vol. 35, no. 6, pp. 459–460, 1971. doi: 10.1016/0375-9601(71)90196-4.

[35]  J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, “Universal intrinsic spin Hall effect,” Physical Review Letters, vol. 92, p. 126603, 2004. doi: 10.1103/PhysRevLett.92.126603.

[36]  A. Bakun, B. Zakharchenya, A. Rogachev, M. Tkachuk, and V. Fleisher, “Observation of a surface photocurrent caused by optical orientation of electrons in a semiconductor,” JETP Letters, vol. 40, no. 11, 1984.

[37]  Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, “Observation of the spin Hall effect in semiconductors,” Science, vol. 306, no. 5703, pp. 1910–1913, 2004. doi: 10.1126/science.1105514.

[38]  K. Ando et al., “Electric manipulation of spin relaxation using the spin Hall effect,” Physical Review Letters, vol. 101, p. 036601, 2008. doi: 10.1103/PhysRevLett.101.036601.

[39]  I. M. Miron et al., “Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection,” Nature, vol. 476, no. 7359, pp. 189–193, 2011. doi: 10.1038/nature10309.

[40]  L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, “Spin-torque switching with the giant spin Hall effect of tantalum,” Science, vol. 336, no. 6081, pp. 555–558, 2012. doi: 10.1126/science.1218197.

[41]  G. Dresselhaus, “Spin-orbit coupling effects in zinc blende structures,” Physical Review, vol. 100, no. 2, pp. 580–586, 1955. doi: 10.1103/PhysRev.100.580.

[42]  É. I. Rashba and V. I. Sheka, “Symmetry of energy bands in crystals of wurtzite type: II. symmetry of bands including spin-orbit interaction,” Fizika Tverdogo Tela, vol. 1, p. 162, 1959.

[43]  Y. A. Bychkov and É. I. Rashba, “Properties of a 2D electron gas with lifted spectral degeneracy,” JETP lett, vol. 39, no. 2, pp. 78–81, 1984.

[44]  V. Edelstein, “Spin polarization of conduction electrons induced by electric current in two-dimensional asymmetric electron systems,” Solid State Communications, vol. 73, no. 3, pp. 233–235, 1990. doi: 10.1016/0038-1098(90)90963-C.

[45]  A. Manchon and S. Zhang, “Theory of nonequilibrium intrinsic spin torque in a single nanomagnet,” Physical Review B, vol. 78, p. 212405, 2008. doi: 10.1103/PhysRevB.78.212405.

[46]  J. C. Rojas Sánchez et al., “Spin-to-charge conversion using Rashba coupling at the interface between non-magnetic materials,” Nature Communications, vol. 4, p. 2944, 2013. doi: 10.1038/ncomms3944.

[47]  A. Manchon, H. Koo, J. Nitta, S. Frolov, and R. Duine, “New perspectives for Rashba spin-orbit coupling,” Nature Materials, vol. 14, pp. 871–882, 2015. doi: 10.1038/nmat4360.

[48]  “II. the doctor’s dissertation (text and translation),” in EARLY WORK (1905–1911), ser. Niels Bohr Collected Works, L. Rosenfeld and J. R. Nielsen, Eds. Elsevier, 1972, vol. 1, pp. 163–393. doi: 10.1016/S1876-0503(08)70015-X.

[49]  H.-J. Van Leeuwen, “Problemes de la théorie électronique du magnétisme,” Journal de Physique et le Radium, vol. 2, no. 12, pp. 361–377, 1921.

[50]  W. Heisenberg, “Zur Theorie des Ferromagnetismus,” Zeitschrift für Physik, vol. 49, no. 9, pp. 619–636, 1928. doi: 10.1007/BF01328601.

[51]  L. Landau and E. Lifshitz, “On the theory of the dispersion of magnetic permeability in ferromagnetic bodies,” Physikalische Zeitschrift der Sowjetunion, vol. 8, pp. 153–169, 1935.

[52]  T. L. Gilbert, “A phenomenological theory of damping in ferromagnetic materials,” IEEE Transactions on Magnetics, vol. 40, no. 6, 2004. doi: 10.1109/TMAG.2004.836740.

[53]  W. Brown, Micromagnetics. Interscience Publishers, New York, 1963. ISBN 9780470110379

[54]  C. Abert, “Micromagnetics and spintronics: Models and numerical methods,” The European Physical Journal B, vol. 92, no. 6, p. 120, 2019. doi: 10.1140/epjb/e2019-90599-6.

[55]  J. Ender, “Machine learning-enhanced numerical modeling of modern magnetoresistive memories,” Ph.D. dissertation, 2024, doctoral thesis supervised by Sverdlov, Viktor, Technische Universität Wien (TU Wien) 2024. [Online]. Available: https://doi.org/10.34726/hss.2024.118820

[56]  G. Hrkac et al., “Influence of eddy currents on the effective damping parameter,” Journal of Applied Physics, vol. 99, no. 8, p. 08B902, 2006. doi: 10.1063/1.2168436.

[57]  S. Zhang, P. M. Levy, and A. Fert, “Mechanisms of spin-polarized current-driven magnetization switching,” Physical Review Letters, vol. 88, p. 236601, 2002. doi: 10.1103/PhysRevLett.88.236601.

[58]  J. Xiao, A. Zangwill, and M. D. Stiles, “Macrospin models of spin transfer dynamics,” Physical Review B, vol. 72, p. 014446, 2005. doi: 10.1103/PhysRevB.72.014446.

[59]  T. Valet and A. Fert, “Theory of the perpendicular magnetoresistance in magnetic multilayers,” Physical Review B, vol. 48, pp. 7099–7113, 1993. doi: 10.1103/PhysRevB.48.7099.

[60]  S. Datta, Electronic Transport in Mesoscopic Systems, ser. Cambridge Studies in Semiconductor Physics and Microelectronic Engineering. Cambridge University Press, 1995. ISBN 9780521416047

[61]  M. Ruggeri, C. Abert, G. Hrkac, D. Suess, and D. Praetorius, “Coupling of dynamical micromagnetism and a stationary spin drift-diffusion equation: A step towards a fully self-consistent spintronics framework,” Physica B: Condensed Matter, vol. 486, pp. 88–91, 2016. doi: 10.1016/j.physb.2015.09.003.

[62]  D. Luc, “Théorie unifiée du transport de spin, charge et chaleur,” Ph.D. dissertation, 2016, thèse de doctorat dirigée par Waintal, Xavier et Chshiev, Mairbek, Nanophysique Université Grenoble Alpes (ComUE) 2016. [Online]. Available: http://www.theses.fr/2016GREAY007

[63]  C. Petitjean, D. Luc, and X. Waintal, “Unified drift-diffusion theory for transverse spin currents in spin valves, domain walls, and other textured magnets,” Physical Review Letters, vol. 109, p. 117204, 2012. doi: 10.1103/PhysRevLett.109.117204.

[64]  M. Chshiev et al., “Analytical description of ballistic spin currents and torques in magnetic tunnel junctions,” Physical Review B, vol. 92, p. 104422, 2015. doi: 10.1103/PhysRevB.92.104422.

[65]  A. Brataas, G. E. Bauer, and P. J. Kelly, “Non-collinear magnetoelectronics,” Physics Reports, vol. 427, no. 4, pp. 157–255, 2006. doi: 10.1016/j.physrep.2006.01.001.

[66]  M. I. Dyakonov, “Magnetoresistance due to edge spin accumulation,” Physical Review Letters, vol. 99, p. 126601, 2007. doi: 10.1103/PhysRevLett.99.126601.

[67]  J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, “Spin Hall effects,” Reviews of Modern Physics, vol. 87, pp. 1213–1260, 2015. doi: 10.1103/RevModPhys.87.1213.

[68]  A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, “Finite-element theory of transport in ferromagnet–normal metal systems,” Physical Review Letters, vol. 84, pp. 2481–2484, 2000. doi: 10.1103/PhysRevLett.84.2481.

[69]  A. Brataas, Y. Nazarov, and G. Bauer, “Spin-transport in multi-terminal normal metal-ferromagnet systems with non-collinear magnetizations,” The European Physical Journal B, vol. 22, no. 1, pp. 99–110, 2001. doi: 10.1007/pl00011139.

[70]  P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and M. D. Stiles, “Current induced torques and interfacial spin-orbit coupling: Semiclassical modeling,” Physical Review B, vol. 87, p. 174411, 2013. doi: 10.1103/PhysRevB.87.174411.

[71]  A. Brataas, G. Zaránd, Y. Tserkovnyak, and G. E. W. Bauer, “Magnetoelectronic spin echo,” Physical Review Letters, vol. 91, p. 166601, 2003. doi: 10.1103/PhysRevLett.91.166601.

[72]  K. Y. Camsari, S. Ganguly, D. Datta, and S. Datta, “Physics-based factorization of magnetic tunnel junctions for modeling and circuit simulation,” in 2014 IEEE International Electron Devices Meeting, 2014, pp. 35.6.1–35.6.4. doi: 10.1109/IEDM.2014.7047177.

[73]  S. Fiorentini et al., “Spin and charge drift-diffusion in ultra-scaled MRAM cells,” Scientific Reports, vol. 12, no. 1, p. 20958, 2022. doi: 10.1038/s41598-022-25586-4.

[74]  S. Fiorentini, “Computation of torques in magnetic tunnel junctions,” Ph.D. dissertation, 2023, doctoral thesis supervised by Sverdlov, Viktor and Selberherr, Siegfried, Technische Universität Wien (TU Wien) 2023. [Online]. Available: http://www.doi.org/10.34726/hss.2023.109800

[75]  V. P. Amin and M. D. Stiles, “Spin transport at interfaces with spin-orbit coupling: Formalism,” Physical Review B, vol. 94, p. 104419, 2016. doi: 10.1103/PhysRevB.94.104419.

[76]  V. P. Amin and M. D. Stiles, “Spin transport at interfaces with spin-orbit coupling: Phenomenology,” Physical Review B, vol. 94, p. 104420, 2016. doi: 10.1103/PhysRevB.94.104420.

[77]  K.-W. Kim, K.-J. Lee, J. Sinova, H.-W. Lee, and M. D. Stiles, “Spin-orbit torques from interfacial spin-orbit coupling for various interfaces,” Physical Review B, vol. 96, no. 10, p. 104438, 2017. doi: 10.1103/PhysRevB.96.104438.

[78]  T. Valet and A. Fert, “Theory of the perpendicular magnetoresistance in magnetic multilayers,” Physical Review B, vol. 48, pp. 7099–7113, 1993. doi: 10.1103/PhysRevB.48.7099.

[79]  N. P. Jørstad, B. Pruckner, W. Goes, V. Sverdlov, and S. Selberherr, “Spin currents and torques in ferromagnetic systems with strong interfacial spin-orbit coupling,” Scientific Reports, vol. 15, no. 1, p. 38749, 2025. doi: 10.1038/S41598-025-22567-1.

[80]  V. P. Amin, P. M. Haney, and M. D. Stiles, “Interfacial spin-orbit torques,” Journal of Applied Physics, vol. 128, no. 15, p. 151101, 2020. doi: 10.1063/5.0024019.

[81]  V. I. Lebedev and D. N. Laikov, “A quadrature formula for the sphere of the 131st algebraic order of accuracy,” in Doklady Mathematics, vol. 59, no. 3, 1999, pp. 477–481.

[82]  P. N. Swarztrauber, “On computing the points and weights for Gauss–Legendre quadrature,” SIAM Journal on Scientific Computing, vol. 24, no. 3, pp. 945–954, 2003. doi: 10.1137/S1064827500379690.

[83]  T. Sauer, Numerical Analysis, 3rd ed. Pearson, 2017. ISBN 9780134696454; 013469645X

[84]  C. R. Harris et al., “Array programming with NumPy,” Nature, vol. 585, no. 7825, pp. 357–362, 2020. doi: 10.1038/s41586-020-2649-2.

[85]  D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd ed. Cambridge University Press, 2007.

[86]  Vienna University of Technology, “ViennaSpinMag.” [Online]. Available: https://www.iue.tuwien.ac.at/viennaspinmag

[87]  R. Anderson et al., “MFEM: A modular finite element methods library,” Computers & Mathematics with Applications, vol. 81, pp. 42–74, 2021. doi: 10.1016/j.camwa.2020.06.009.

[88]  J. Schöberl, “NETGEN an advancing front 2D/3D-mesh generator based on abstract rules,” Computing and visualization in science, vol. 1, no. 1, pp. 41–52, 1997. doi: 10.1007/s007910050004.

[89]  G. Hrkac, C.-M. Pfeiler, D. Praetorius, M. Ruggeri, A. Segatti, and B. Stiftner, “Convergent tangent plane integrators for the simulation of chiral magnetic skyrmion dynamics,” Advances in Computational Mathematics, vol. 45, no. 3, pp. 1329–1368, 2019. doi: 10.1007/s10444-019-09667-z.

[90]  J. Ender et al., “Efficient demagnetizing field calculation for disconnected complex geometries in STT-MRAM cells,” in 2020 International Conference on Simulation of Semiconductor Processes and Devices (SISPAD), 2020, pp. 213–216. doi: 10.23919/SISPAD49475.2020.9241662.

[91]  M. Bendra et al., “Finite element method approach to MRAM modeling,” in 2021 44th International Convention on Information, Communication and Electronic Technology (MIPRO), 2021, pp. 70–73. doi: 10.23919/MIPRO52101.2021.9597194.

[92]  D. Cortés-Ortuño et al., “Proposal for a micromagnetic standard problem for materials with Dzyaloshinskii-Moriya interaction,” New Journal of Physics, vol. 20, no. 11, p. 113015, 2018. doi: 10.1088/1367-2630/aaea1c.

[93]  S. Rohart and A. Thiaville, “Skyrmion confinement in ultrathin film nanostructures in the presence of Dzyaloshinskii-Moriya interaction,” Physical Review B, vol. 88, p. 184422, 2013. doi: 10.1103/PhysRevB.88.184422.

[94]  E. Davoli, G. Di Fratta, D. Praetorius, and M. Ruggeri, “Micromagnetics of thin films in the presence of Dzyaloshinskii-Moriya interaction,” Mathematical Models and Methods in Applied Sciences, vol. 32, no. 05, pp. 911–939, 2022. doi: 10.1142/S0218202522500208.

[95]  A. Ghosh, K. Garello, C. O. Avci, M. Gabureac, and P. Gambardella, “Interface-enhanced spin-orbit torques and current-induced magnetization switching of \(\mathrm {Pd}/\mathrm {Co}/\mathrm {AlO}_{x}\) layers,” Physical Review Applied, vol. 7, p. 014004, 2017. doi: 10.1103/PhysRevApplied.7.014004.

[96]  A. J. Berger, E. R. J. Edwards, H. T. Nembach, O. Karis, M. Weiler, and T. J. Silva, “Determination of the spin hall effect and the spin diffusion length of Pt from self-consistent fitting of damping enhancement and inverse spin-orbit torque measurements,” Physical Review B, vol. 98, p. 024402, 2018. doi: 10.1103/PhysRevB.98.024402.

[97]  L. Zhu, D. C. Ralph, and R. A. Buhrman, “Maximizing spin-orbit torque generated by the spin Hall effect of Pt,” Applied Physics Reviews, vol. 8, no. 3, p. 031308, 2021. doi: 10.1063/5.0059171.

[98]  S. Dutta, A. Bose, A. A. Tulapurkar, R. A. Buhrman, and D. C. Ralph, “Interfacial and bulk spin Hall contributions to fieldlike spin-orbit torque generated by iridium,” Physical Review B, vol. 103, p. 184416, 2021. doi: 10.1103/PhysRevB.103.184416.

[99]  T. Fache, J. C. Rojas-Sanchez, L. Badie, S. Mangin, and S. Petit-Watelot, “Determination of spin Hall angle, spin mixing conductance, and spin diffusion length in CoFeB/Ir for spin-orbitronic devices,” Physical Review B, vol. 102, p. 064425, 2020. doi: 10.1103/PhysRevB.102.064425.

[100]  B. Wang et al., “The accurate measurement of spin orbit torque by utilizing the harmonic longitudinal voltage with Wheatstone bridge structure,” Applied Physics Letters, vol. 116, no. 22, p. 222402, 2020. doi: 10.1063/1.5145221.

[101]  K. K. Vudya Sethu et al., “Optimization of tungsten \(\ensuremath {\beta }\)-phase window for spin-orbit-torque magnetic random-access memory,” Physical Review Applied, vol. 16, p. 064009, 2021. doi: 10.1103/PhysRevApplied.16.064009.

[102]  W. Zhang, W. Han, X. Jiang, S.-H. Yang, and S. S. P. Parkin, “Role of transparency of platinum-ferromagnet interfaces in determining the intrinsic magnitude of the spin Hall effect,” Nature Physics, vol. 11, no. 6, pp. 496–502, 2015. doi: 10.1038/nphys3304.

[103]  H. Oshima, K. Nagasaka, Y. Seyama, Y. Shimizu, S. Eguchi, and A. Tanaka, “Perpendicular giant magnetoresistance of CoFeB/Cu single and dual spin-valve films,” Journal of Applied Physics, vol. 91, no. 10, pp. 8105–8107, 2002. doi: 10.1063/1.1448310.

[104]  L. Piraux, S. Dubois, A. Fert, and L. Belliard, “The temperature dependence of the perpendicular giant magnetoresistance in Co/Cu multilayered nanowires,” The European Physical Journal B: Condensed Matter and Complex Systems, vol. 4, pp. 413–420, 1998. doi: 10.1007/s100510050398.

[105]  M. Belmeguenai et al., “Ferromagnetic-resonance-induced spin pumping in Co20Fe60B20/Pt systems: Damping investigation,” Journal of Physics D: Applied Physics, vol. 51, no. 4, p. 045002, 2018. doi: 10.1088/1361-6463/aa9f55.

[106]  S. Cho, S.-h. Baek, Y. Jo, and B.-G. Park, “Large spin Hall magnetoresistance and its correlation to the spin-orbit torque in W/CoFeB/MgO structures,” Scientific Reports, vol. 5, 2015. doi: 10.1038/srep14668.

[107]  F. Xue et al., “Enhanced spin-torque efficiency by metal insertion in the Pt/Co/MgO system,” Physical Review B, vol. 110, p. 174437, 2024. doi: 10.1103/PhysRevB.110.174437.

[108]  S. Jiang et al., “Spin-torque nano-oscillators and their applications,” Applied Physics Reviews, vol. 11, no. 4, p. 041309, 2024. doi: 10.1063/5.0221877.

[109]  A. Kumar, A. Litvinenko, N. Behera, A. A. Awad, R. Khymyn, and J. Åkerman, Mutual Synchronization in Spin-Torque and Spin Hall Nano-oscillators. Springer Nature Switzerland, 2024, pp. 143–182. ISBN 978-3-031-73191-4

[110]  R. Arun, R. Gopal, V. K. Chandrasekar, and M. Lakshmanan, “Large amplitude spin-Hall oscillations due to field-like torque,” Journal of Physics: Condensed Matter, vol. 33, no. 16, p. 165402, 2021. doi: 10.1088/1361-648X/abf0c1.

[111]  W.-J. Wang, R.-X. Wang, Q. Wan, M.-Q. Cai, and P.-B. He, “Field-free self-oscillation of magnetization enabled by the fieldlike spin-orbit torque,” Physica Scripta, vol. 98, no. 11, p. 115968, 2023. doi: 10.1088/1402-4896/ad02ca.

[112]  R. Arun, R. Gopal, V. K. Chandrasekar, and M. Lakshmanan, “Exploration of field-like torque and field-angle tunability in coupled spin-torque nano oscillators for synchronization,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 34, no. 1, p. 013114, 2024. doi: 10.1063/5.0173943.

[113]  K. Garello et al., “Symmetry and magnitude of spin–orbit torques in ferromagnetic heterostructures,” Nature Nanotechnology, vol. 8, no. 8, pp. 587–593, 2013. doi: 10.1038/nnano.2013.145.

[114]  E.-S. Park et al., “Strong higher-order angular dependence of spin-orbit torque in W/CoFeB bilayer,” Physical Review B, vol. 107, p. 064411, 2023. doi: 10.1103/PhysRevB.107.064411.

[115]  K. D. Belashchenko, A. A. Kovalev, and M. van Schilfgaarde, “First-principles calculation of spin-orbit torque in a Co/Pt bilayer,” Physical Review Materials, vol. 3, p. 011401, 2019. doi: 10.1103/PhysRevMaterials.3.011401.

[116]  C. Ortiz Pauyac, X. Wang, M. Chshiev, and A. Manchon, “Angular dependence and symmetry of Rashba spin torque in ferromagnetic heterostructures,” Applied Physics Letters, vol. 102, no. 25, p. 252403, 2013. doi: 10.1063/1.4812663.

[117]  K.-S. Lee et al., “Angular dependence of spin-orbit spin-transfer torques,” Physical Review B, vol. 91, p. 144401, 2015. doi: 10.1103/PhysRevB.91.144401.

[118]  S. Woo et al., “Observation of room-temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic ferromagnets,” Nature Materials, vol. 15, pp. 501–506, 2016. doi: 10.1038/NMAT4593.

[119]  E. Montoya et al., “Spin transport in tantalum studied using magnetic single and double layers,” Physical Review B, vol. 94, p. 054416, 2016. doi: 10.1103/PhysRevB.94.054416.

[120]  A. Ghosh, S. Auffret, U. Ebels, and W. E. Bailey, “Penetration depth of transverse spin current in ultrathin ferromagnets,” Physical Review Letters, vol. 109, p. 127202, 2012. doi: 10.1103/PhysRevLett.109.127202.

[121]  K. Litzius et al., “The role of temperature and drive current in skyrmion dynamics,” Nature Electronics, vol. 3, pp. 30–36, 2020. doi: 10.1038/s41928-019-0359-2.

[122]  G. Choi et al., “Thickness dependence of spin-orbit torques in Pt/Co structures on epitaxial substrates,” APL Materials, vol. 10, no. 1, p. 011105, 2022. doi: 10.1063/5.0077074.

[123]  M. S. Gabor, M. Belmeguenai, and I. M. Miron, “Bulk and interface spin-orbit torques in Pt/Co/MgO thin film structures,” Physical Review B, vol. 109, p. 104407, 2024. doi: 10.1103/PhysRevB.109.104407.

[124]  L. Zhou, V. L. Grigoryan, S. Maekawa, X. Wang, and J. Xiao, “Spin Hall effect by surface roughness,” Physical Review B, vol. 91, p. 045407, 2015. doi: 10.1103/PhysRevB.91.045407.

[125]  G. G. B. Flores and K. D. Belashchenko, “Effect of interfacial intermixing on spin-orbit torque in Co/Pt bilayers,” Physical Review B, vol. 105, p. 054405, 2022. doi: 10.1103/PhysRevB.105.054405.

[126]  K. Gupta, R. J. H. Wesselink, R. Liu, Z. Yuan, and P. J. Kelly, “Disorder dependence of interface spin memory loss,” Physical Review Letters, vol. 124, p. 087702, 2020. doi: 10.1103/PhysRevLett.124.087702.

[127]  M. Cecot et al., “Influence of intermixing at the Ta/CoFeB interface on spin Hall angle in Ta/CoFeB/MgO heterostructures,” Scientific Reports, vol. 7, no. 1, p. 968, 2017. doi: 10.1038/s41598-017-00994-z.

[128]  G. C. La Rocca, N. Kim, and S. Rodriguez, “Effect of uniaxial stress on the electron spin resonance in zinc-blende semiconductors,” Physical Review B, vol. 38, pp. 7595–7601, 1988. doi: 10.1103/PhysRevB.38.7595.

[129]  C. O. Avci et al., “Fieldlike and antidamping spin-orbit torques in as-grown and annealed Ta/CoFeB/MgO layers,” Physical Review B, vol. 89, p. 214419, 2014. doi: 10.1103/PhysRevB.89.214419.

[130]  A. Lamperti, S.-M. Ahn, B. Ocker, R. Mantovan, and D. Ravelosona, “Interface width evaluation in thin layered CoFeB/MgO multilayers including Ru or Ta buffer layer by X-ray reflectivity,” Thin Solid Films, vol. 533, pp. 79–82, 2013. doi: 10.1016/j.tsf.2012.11.130.

[131]  D. K. Ojha, L.-J. Chang, Y.-H. Wu, W.-Y. Jang, and Y.-C. Tseng, “Enhancing spin-orbit torque in tri-layer devices with an ultra-thin light metal spacer,” Applied Physics Letters, vol. 126, no. 13, p. 132405, 04 2025. doi: 10.1063/5.0253761.

[132]  S.-h. Baek et al., “Spin currents and spin-orbit torques in ferromagnetic trilayers,” Nature Materials, vol. 17, 2018. doi: 10.1038/s41563-018-0041-5.

[133]  J. Ryu et al., “Efficient spin–orbit torque in magnetic trilayers using all three polarizations of a spin current,” Nature Electronics, vol. 5, no. 4, pp. 217–223, 2022. doi: 10.1038/s41928-022-00735-9.

[134]  D. Kubler, D. A. Smith, T. Nguyen, F. Ramos-Diaz, S. Emori, and V. P. Amin, “Large-amplitude easy-plane spin-orbit torque oscillators driven by out-of-plane spin current: A micromagnetic study,” Physical Review B, vol. 111, p. 054425, 2025. doi: 10.1103/PhysRevB.111.054425.

[135]  Y. Liu et al., “Field-free switching of perpendicular magnetization at room temperature using out-of-plane spins from TaIrTe4,” Nature Electronics, vol. 6, no. 10, pp. 732–738, 2023. doi: 10.1038/s41928-023-01039-2.

[136]  P. Gupta et al., “Symmetry enhanced unconventional spin current anisotropy in a collinear antiferromagnet,” Advanced Functional Materials, p. e06816, 2025. doi: 10.1002/adfm.202506816.

[137]  S. Hu et al., “Efficient perpendicular magnetization switching by a magnetic spin Hall effect in a noncollinear antiferromagnet,” Nature Communications, vol. 13, no. 1, p. 4447, 2022. doi: 10.1038/s41467-022-32179-2.

[138]  P. Gupta et al., “Generation of out-of-plane polarized spin current in (permalloy, Cu)/EuS interfaces,” Physical Review B, vol. 109, p. L060405, 2024. doi: 10.1103/PhysRevB.109.L060405.

[139]  A. Kumar et al., “Interfacial origin of unconventional spin-orbit torque in Py/\(\gamma \)-IrMn3,” Advanced Quantum Technologies, vol. 6, no. 7, p. 2300092, 2023. doi: 10.1002/qute.202300092.

[140]  B. Pruckner, N. P. Jørstad, W. Goes, S. Selberherr, and V. Sverdlov, “Field-free magnetization switching in SOT-MRAM devices with noncollinear antiferromagnets,” Microelectronic Engineering, vol. 300, p. 112372, 2025. doi: 10.1016/j.mee.2025.112372.

[141]  D. M. Lattery, D. Zhang, J. Zhu, X. Hang, J.-P. Wang, and X. Wang, “Low gilbert damping constant in perpendicularly magnetized W/CoFeB/MgO films with high thermal stability,” Scientific reports, vol. 8, no. 1, p. 13395, 2018. doi: 10.1038/s41598-018-31642-9.

[142]  T. Devolder, J.-V. Kim, L. Nistor, R. Sousa, B. Rodmacq, and B. Diény, “Exchange stiffness in ultrathin perpendicularly magnetized CoFeB layers determined using the spectroscopy of electrically excited spin waves,” Journal of Applied Physics, vol. 120, no. 18, p. 183902, 2016. doi: 10.1063/1.4967826.

[143]  J.-S. Kim et al., “Enhancement in interfacial Dzyaloshinskii–Moriya interaction in Pt/CoFe (B)/MgO structures by suppression of FePt interface phases with the addition of boron,” ACS Applied Electronic Materials, vol. 5, no. 10, pp. 5453–5462, 2023. doi: 10.1021/acsaelm.3c00667.

[144]  J. Torrejon et al., “Interface control of the magnetic chirality in CoFeB/MgO heterostructures with heavy-metal underlayers,” Nature Communications, vol. 5, no. 1, p. 4655, 2014. doi: 10.1038/ncomms5655.

[145]  M. Baumgartner et al., “Spatially and time-resolved magnetization dynamics driven by spin–orbit torques,” Nature Nanotechnology, vol. 12, no. 10, pp. 980–986, 2017. doi: 10.1038/nnano.2017.151.

[146]  A. van den Brink et al., “Spin-Hall-assisted magnetic random access memory,” Applied Physics Letters, vol. 104, no. 1, p. 012403, 2014. doi: 10.1063/1.4858465.

[147]  Z. Wang, W. Zhao, E. Deng, J.-O. Klein, and C. Chappert, “Perpendicular-anisotropy magnetic tunnel junction switched by spin-Hall-assisted spin-transfer torque,” Journal of Physics D: Applied Physics, vol. 48, no. 6, p. 065001, 2015. doi: 10.1088/0022-3727/48/6/065001.

[148]  M. Wang et al., “Field-free switching of a perpendicular magnetic tunnel junction through the interplay of spin–orbit and spin-transfer torques,” Nature Electronics, vol. 1, no. 11, pp. 582–588, 2018. doi: 10.1038/s41928-018-0160-7.

[149]  K. Garello et al., “Manufacturable 300mm platform solution for field-free switching SOT-MRAM,” in 2019 Symposium on VLSI Technology, 2019, pp. T194–T195. doi: 10.23919/VLSIT.2019.8776537.

[150]  Q. Yang et al., “Field-free spin–orbit torque switching in ferromagnetic trilayers at sub-ns timescales,” Nature Communications, vol. 15, no. 1, p. 1814, 2024. doi: 10.1038/s41467-024-46113-1.

[151]  K. Cai et al., “First demonstration of field-free perpendicular SOT-MRAM for ultrafast and high-density embedded memories,” in 2022 International Electron Devices Meeting (IEDM). San Francisco, CA, USA: IEEE, 2022, pp. 36.2.1–36.2.4. doi: 10.1109/IEDM45625.2022.10019360. . ISBN 978-1-6654-8959-1

[152]  M. Stiles, “Interlayer exchange coupling,” Journal of Magnetism and Magnetic Materials, vol. 200, no. 1, pp. 322–337, 1999. doi: 10.1016/S0304-8853(99)00334-0.

[153]  X. Zhao and J. Zhao, “Interlayer Dzyaloshinskii-Moriya interaction in spintronics,” npj Spintronics, vol. 3, p. 25, 06 2025. doi: 10.1038/s44306-025-00091-2.

[154]  W. R. Inc., “Mathematica, Version 14.3,” Champaign, IL, 2025. [Online]. Available: https://www.wolfram.com/mathematica

List of Publications

Journal Articles

• [1] N. P. Jørstad, B. Pruckner, W. Goes, S. Selberherr, V. Sverdlov, "Spin currents and torques in ferromagnetic systems with strong interfacial spin-orbit coupling", Scientific Reports, vol. 15, no. 1, p. 38749, 2025, doi: 10.1038/s41598-025-22567-1.

• [2] B. Pruckner, N. P. Jørstad, W. Goes, S. Selberherr, V. Sverdlov, "Field-free magnetization switching in SOT-MRAM devices with noncollinear antiferromagnets",Microelectronic Engineering, vol. 300, p. 112372, 2025, doi: 10.1016/j.mee.2025.112372.

• [3] N. P. Jørstad, W. Goes, S. Selberherr, V. Sverdlov, "Optimizing unconventional trilayer SOTs for field-free switching", Solid-State Electronics, vol. 228, p. 109135, 2025, doi: 10.1016/j.sse.2025.109135.

• [4] N. P. Jørstad, W. Goes, S. Selberherr, V. Sverdlov, "Modeling torques in systems with spin-orbit coupling", IEEE Transactions on Magnetics, vol. 61, no. 6, pp. 1300404-1–1300404-4, 2025, doi: 10.1109/TMAG.2025.3525810.

• [5] N. P. Jørstad, S. Fiorentini, J. Ender, G. Wolfgang, S. Selberherr, V. Sverdlov, "Micromagnetic modeling of SOT-MRAM dynamics", Physica B: Condensed Matter, vol. 676, pp. 415612-1–415612-10, 2024, doi: 10.1016/j.physb.2023.415612.

• [6] T. Hadámek, N. P. Jørstad, W. Goes, S. Selberherr, V. Sverdlov, "Numerical study of two-terminal SOT-MRAM", Physica B: Condensed Matter, vol. 673, pp. 415362-1–415362-6, 2023, doi: 10.1016/j.physb.2023.415362.

• [7] T. Hadámek, N. P. Jørstad, R. Lacerda de Orio, W. Goes, S. Selberherr, V. Sverdlov, "A comprehensive study of temperature and its effects in SOT-MRAM devices", Micromachines, vol. 14, no. 8, p. 1581, 2023,
  doi: 10.3390/mi14081581.

• [8] S. Fiorentini, N. P. Jørstad, J. Ender, R. Lacerda de Orio, S. Selberherr, M. Bendra, W. Goes, V. Sverdlov, "Finite element approach for the simulation of modern MRAM devices", Micromachines, vol. 14, no. 5, p. 898, 2023,
  doi: 10.3390/mi14050898.

• [9] W. J. Loch, S. Fiorentini, N. P. Jørstad, W. Goes, S. Selberherr, V. Sverdlov, "Double reference layer STT-MRAM structures with improved performance", Solid-State Electronics, vol. 194, pp. 108335-1–108335-4, 2022,
  doi: 10.1016/j.sse.2022.108335.

• [10] S. Fiorentini, R. Lacerda de Orio, J. Ender, S. Selberherr, M. Bendra, N. P. Jørstad, W. Goes, V. Sverdlov, "Finite element method for MRAM switching simulations", WSEAS Transactions on Systems and Controls, vol. 17, pp. 585–588, 2022, doi: 10.37394/23203.2022.17.64.

• [11] N. P. Jørstad, S. Fiorentini , W. J. Loch, W. Goes, S. Selberherr, V. Sverdlov, "Finite element modeling of spin-orbit torques", Solid-State Electronics, vol. 194, pp. 108323-1–108323-4, 2022, doi: 10.1016/j.sse.2022.108323.

Conference Contributions (with Proceedings-Entry)

• [12] V. Sverdlov, N. P. Jørstad, W. Goes, B. Pruckner, "Deterministic switching in SOT-MRAM with an additional magnetic layer", in Book of Abstracts APS Global Physics Summit Conf., 2025, p. 1

• [13] N. P. Jørstad, B. Pruckner, W. Goes, S. Selberherr, V. Sverdlov, “Magnetic field free SOT-MRAM switching”, in Book of Abstracts WINDS Conf., 2024, pp. 44–45.

• [14] B. Pruckner, N. P. Jørstad, M. Bendra, T. Hadámek, W. Goes, S. Selberherr, V. Sverdlov, “Simulation of advanced MRAM devices for sub-ns switching”, in Proc. SISPAD, 2024, pp. 1–4.

• [15] B. Pruckner, N. P. Jørstad, W. Goes, S. Selberherr, V. Sverdlov, “Field-free magnetization switching in SOT-MRAM devices with noncollinear antiferromagnets”, in Proc. Austrochip Conf., 2024, pp. 1–4.

• [16] B. Pruckner, N. P. Jørstad, W. Goes, S. Selberherr, V. Sverdlov, “Field-free perpendicular magnetization switching of SOT-MRAM devices with non-collinear antiferromagnets”, in Book of Abstracts ICM Conf., 2024, p. 1842.

• [17] N. P. Jørstad, W. Goes, S. Selberherr, V. Sverdlov, “Optimization of spin-orbit torque for field-free switching of ferromagnetic trilayers”, in Book of Abstracts ICM Conf., 2024, p. 1286.

• [18] N. P. Jørstad, B. Pruckner, W. Goes, S. Selberherr, V. Sverdlov, “Out-of-plane polarized spin current generation for field-free switching of perpendicular SOT-MRAM”, in Book of Abstracts DRC Conf., 2024, p. 143, (Best Student Poster Award).

• [19] N. P. Jørstad, B. Pruckner, M. Bendra, W. Goes, V. Sverdlov, “Modeling advanced perpendicular MRAM cells: Generating spin currents for fast field-free cell switching”, in Book of Abstracts AMSE Conf., 2024, p. 46.

• [20] V. Sverdlov, N. P. Jørstad, M. Bendra, B. Pruckner, T. Hadámek, W. Goes, S. Selberherr, “Modeling spin and charge transport in magnetic multilayers: From emerging memories to terahertz emitters”, in Book of Abstracts TeraTech Conf., 2024, pp. 13–14.

• [21] B. Pruckner, N. P. Jørstad, T. Hadámek, W. Goes, S. Selberherr, V. Sverdlov, “Field-free perpendicular magnetization switching of SOT-MRAM devices by magnetic spin Hall effect”, in Proc. MIPRO Conf., 2024, pp. 1584–1589.

• [22] B. Pruckner, N. P. Jørstad, W. Goes, S. Selberherr, V. Sverdlov, “Magnetic spin Hall induced field-free magnetization switching in SOT-MRAM devices”, in Book of Abstracts EuroSOI-ULIS Conf., 2024, pp. 154–155.

• [23] N. P. Jørstad, W. Goes, S. Selberherr, V. Sverdlov, “Optimizing unconventional trilayer SOTs for field-free switching”, in Book of Abstracts EuroSOI-ULIS Conf., 2024, pp. 84–85.

• [24] M. Bendra, S. Fiorentini, J. Ender, R. Lacerda de Orio, T. Hadámek, N. P. Jørstad, B. Pruckner, S. Selberherr, W. Goes, V. Sverdlov, “Back-hopping in ultra-scaled MRAM cells”, in Proc. MIPRO Conf., 2023, pp. 159–162.

• [25] N. P. Jørstad, S. Fiorentini, W. Goes, S. Selberherr, V. Sverdlov, “Micromagnetic modeling of SOT-MRAM dynamics”, in Book of Abstracts HMM Conf., 2023, p. 1.

• [26] V. Sverdlov, N. P. Jørstad, M. Bendra, T. Hadámek, W. Goes, “Modeling emerging spintronic devices and spintronic THz emitters”, in Book of Abstracts TeraTech Conf., 2023, pp. 50–51.

• [27] V. Sverdlov, M. Bendra, N. P. Jørstad, B. Pruckner, T. Hadámek, W. Goes, S. Selberherr, “Multi-bit operation in an MRAM cell with a composite free layer”, in Book of Abstracts WINDS Conf., 2023, pp. 143–144.

• [28] N. P. Jørstad, T. Hadámek, M. Bendra, J. Ender, B. Pruckner, W. Goes, V. Sverdlov, “Numerical simulations of spintronic magnetoresistive memories”, in Proc. SURGE Virtual Event, 2023, p. 1.

• [29] T. Hadámek, N. P. Jørstad, W. Goes, S. Selberherr, V. Sverdlov, “Numerical study of two-terminal SOT-MRAM”, in Book of Abstracts HMM Conf., 2023, p. 1.

• [30] N. P. Jørstad, W. Goes, S. Selberherr, V. Sverdlov, “Spin drift-diffusion boundary conditions for FEM modeling of multilayer SOT devices”, in Proc. SISPAD Conf., 2023, pp. 357–360.

• [31] V. Sverdlov, M. Bendra, B. Pruckner, N. P. Jørstad, T. Hadámek, J. Ender, R. Lacerda de Orio, W. Goes, “Spin and charge transport in ultra-scaled MRAM cells”, in Proc. ICMNE Conf., 2023, p. 55.

• [32] T. Hadámek, N. P. Jørstad, W. Goes, S. Selberherr, V. Sverdlov, “Study of self-heating and its effects in SOT-STT-MRAM”, in Proc. SISPAD Conf., 2023, pp. 337–340.

• [33] M. Bendra, S. Fiorentini, T. Hadámek, N. P. Jørstad, J. Ender, R. Lacerda de Orio, S. Selberherr, W. Goes, V. Sverdlov, “Switching composite free layers in ultra-scaled MRAM cells”, in Abstract Book International Winterschool on New Developments in Solid State Physics, 2023, pp. 184–185.

• [34] B. Pruckner, S. Fiorentini, N. P. Jørstad, T. Hadámek, S. Selberherr, W. Goes, V. Sverdlov, “Switching performance of Mo-based pMTJ and dsMTJ structures”, in Book of Abstracts IWCN Conf., 2023, pp. 144–145.

• [35] M. Bendra, N. P. Jørstad, R. Lacerda de Orio, S. Selberherr, W. Goes, V. Sverdlov, “Unified modeling of ultra-scaled STT-MRAM cells: Harnessing parasitic effects for enhanced data storage dynamics”, in Proc. IEDM Conf. (Special MRAM Poster Session), 2023.

• [36] V. Sverdlov, M. Bendra, S. Fiorentini, J. Ender, R. Lacerda de Orio, T. Hadámek, W. Loch, N. P. Jørstad, W. Goes, S. Selberherr, “Advanced modeling of emerging devices for digital spintronics”, in Proc. Int. Conf. on Nanosc. and Nanotec., 2022, p. 40.

• [37] V. Sverdlov, M. Bendra, S. Fiorentini, J. Ender, R. Lacerda de Orio, T. Hadámek, W. Loch, N. P. Jørstad, W. Goes, S. Selberherr, “Advanced modeling of emerging magneto-resistive memory”, in Proc. NANOMEET Conf., 2022, pp. 78–79.

• [38] S. Fiorentini, W. J. Loch, M. Bendra, N. P. Jørstad, J. Ender, R. Lacerda de Orio, T. Hadámek, W. Goes, V. Sverdlov, and S. Selberherr, “Design analysis of ultra-scaled MRAM cells”, in Proc. ICSICT Conf., 2022.

• [39] V. Sverdlov, M. Bendra, S. Fiorentini, J. Ender, R. Lacerda de Orio, T. Hadámek, W.J. Loch, N. P. Jørstad, W. Goes, and S. Selberherr, “Modeling advanced spintronic based magnetoresistive memory”, in Book of Abstracts IRPhE Conf., 2022.

• [40] M. Bendra, W. Loch, N. P. Jørstad, S. Fiorentini, S. Selberherr, W. Goes, V. Sverdlov, “Modeling ultra-scaled multi-layer STT-MRAM cells: A unified spin and charge drift-diffusion approach”, in Proc. IEDM Conf. (Special MRAM Poster Session), 2022.

• [41] V. Sverdlov, W. J. Loch, M. Bendra, S. Fiorentini, J. Ender, R. Lacerda de Orio, T. Hadámek, N. P. Jørstad, W. Goes, and S. Selberherr, “Modeling approach to ultra-scaled MRAM cells”, in Book of Abstracts ASETMEET Conf., 2022, pp. 7–8.

• [42] T. Hadámek, S. Fiorentini, M. Bendra, R. Lacerda de Orio, W. Loch, N. P. Jørstad, S. Selberherr, W. Goes, V. Sverdlov, “Temperature modeling in STT-MRAM: A fully three-dimensional finite element approach”, in Book of Abstracts NANO Conf., 2022.

• [43] N. P. Jørstad, S. Fiorentini, S. Selberherr, W. Goes, and V. Sverdlov, “Modeling interfacial and bulk spin-orbit torques”, in Book of Abstracts NANO Conf., 2022.

• [44] V. Sverdlov, M. Bendra, S. Fiorentini, J. Ender, R. Lacerda de Orio, T. Hadámek, W. J. Loch, N. P. Jørstad, W. Goes, and S. Selberherr, “Emerging devices for digital spintronics”, in Proc. GECNN Conf., 2022, pp. 32–33.

• [45] M. Bendra, S. Fiorentini, J. Ender, R. Orio, T. Hadámek, W. J. Loch, N. P. Jørstad, S. Selberherr, W. Goes, and V. Sverdlov, “Spin transfer torques in ultra-scaled MRAM cells”, in Proc. MIPRO Conf., 2022, pp. 129–132.

• [46] M. Bendra, S. Fiorentini, J. Ender, R. Lacerda de Orio, T. Hadámek, W. J. Loch, N. P. Jørstad, W. Goes, and S. Selberherr, “Interface effects in ultra-scaled MRAM cells”, in Book of Abstracts EUROSOI-ULIS Conf., 2022.

• [47] N. P. Jørstad, S. Fiorentini, J. Ender, R. Lacerda de Orio, T. Hadámek, W. Loch, M. Bendra, W. Goes, S. Selberherr, V. Sverdlov, “Finite element modeling of spin-orbit torques”, in Book of Abstracts EUROSOI-ULIS Conf., 2022.

• [48] V. Sverdlov, M. Bendra, S. Fiorentini, J. Ender, R. Lacerda de Orio, T. Hadámek, W. J. Loch, N. P. Jørstad, and S. Selberherr, “Modeling advanced magnetoresistive memory: A journey from finite element methods to machine learning approaches”, in Proc. Global Webinar on Nanosc. and Nanotec., 2022.

• [49] N. P. Jørstad, S. Fiorentini, W. Goes, V. Sverdlov, “Efficient finite element method approach to model spin-orbit torque MRAM,” in Book of Abstracts MOS-AK Workshop, 2021, p. 1.

Conference Contributions (no Proceedings-Entry)

• [50] M. Bendra, S. Fiorentini, J. Ender, R. Lacerda de Orio, T. Hadamek, N. P. Jørstad, B. Pruckner, S. Selberherr, W. Goes, V. Sverdlov, “Back-hopping in ultra-scaled MRAM cells”, in Proc. MIPRO Conf., 2023.

• [51] S. Fiorentini, R. Lacerda de Orio, J. Ender, S. Selberherr, M. Bendra, N. P. Jørstad, W. Goes, V. Sverdlov, “Finite element method for MRAM switching simulations”, in Proc. MMCTSE Conf., 2023.

Curriculum Vitae

Nils Petter Jørstad, M.Sc.

01/2015 – 01/2016

Military service
The Norwegian Armed Forces
Troms, Norway

08/2016 – 08/2019

Bachelor’s degree, Physics
The Norwegian University of Science and Technology
Trondheim, Norway

08/2019 – 08/2021

Master’s degree, Physics
Thesis titled: "Reflection of diffuse light from two-dimensional rough surfaces & The light scattering properties of Gaussian-cosine correlated surfaces"
The Norwegian University of Science and Technology
Trondheim, Norway

09/2021 – present

Doctoral Candidate and Research Assistant
Christian Doppler Laboratory for Nonvolatile Magnetoresistive Memory and Logic
Institute for Microelectronics, TU Wien
Vienna, Austria