First-Principles Quantum Transport Modeling of Spin-Transfer and Spin-Orbit Torques in Magnetic Multilayers

  • Branislav K. NikolićEmail author
  • Kapildeb Dolui
  • Marko D. Petrović
  • Petr Plecháč
  • Troels Markussen
  • Kurt Stokbro
Living reference work entry


A unified approach for computing (i) spin-transfer torque in magnetic trilayers like spin valve and magnetic tunnel junction, where injected charge current flows perpendicularly to interfaces, and (ii) spin-orbit torque in magnetic bilayers of the type ferromagnet/spin-orbit-coupled material, where injected charge current flows parallel to the interface, was reviewed. The experimentally explored and technologically relevant spin-orbit-coupled materials include 5d heavy metals, topological insulators, Weyl semimetals, and transition metal dichalcogenides. This approach requires to construct the torque operator for a given Hamiltonian of the device and the steady-state nonequilibrium density matrix, where the latter is expressed in terms of the nonequilibrium Green’s functions and split into three contributions. Tracing these contributions with the torque operator automatically yields field-like and damping-like components of spin-transfer torque or spin-orbit torque vector, which is particularly advantageous for spin-orbit torque where the direction of these components depends on the unknown-in-advance orientation of the current-driven nonequilibrium spin density in the presence of spin-orbit coupling. Illustrative examples are provided by computing spin-transfer torque in a one-dimensional toy model of a magnetic tunnel junction and realistic Co/Cu/Co spin valve, both of which are described by first-principles Hamiltonians obtained from noncollinear density functional theory calculations, as well as by computing spin-orbit torque in a ferromagnetic layer described by a tight-binding Hamiltonian which includes spin-orbit proximity effect within ferromagnetic monolayers assumed to be generated by the adjacent monolayer transition metal dichalcogenide. In addition, it is shown here how spin-orbit proximity effect, quantified by computing (via first-principles retarded Green’s function) spectral functions and spin textures on monolayers of realistic ferromagnetic material like Co in contact with heavy metal or monolayer transition metal dichalcogenide, can be tailored to enhance the magnitude of spin-orbit torque. Errors made in the calculation of spin-transfer torque are quantified when using Hamiltonian from collinear density functional theory, with rigidly rotated magnetic moments to create noncollinear magnetization configurations, instead of proper (but computationally more expensive) self-consistent Hamiltonian obtained from noncollinear density functional theory.



We are grateful to K. D. Belashchenko, K. Xia, and Z. Yuan for illuminating discussions and P.-H. Chang, F. Mahfouzi, and J.-M. Marmolejo-Tejada for the collaboration. B. K. N. and K. D. were supported by DOE Grant No. DE-SC0016380 and NSF Grant No. ECCS 1509094. M. P. and P. P. were supported by ARO MURI Award No. W911NF-14-0247. K. S. and T. M. acknowledge support from the European Commission Seventh Framework Programme Grant Agreement IIIV-MOS, Project No. 61932, and Horizon 2020 research and innovation program under grant agreement SPICE, Project No. 713481. The supercomputing time was provided by XSEDE, which is supported by NSF Grant No. ACI-1548562.


  1. Ado IA, Tretiakov OA, Titov M (2017) Microscopic theory of spin-orbit torques in two dimensions. Phys Rev B 95:094401CrossRefADSGoogle Scholar
  2. Areshkin DA, Nikolić BK (2010) Electron density and transport in top-gated graphene nanoribbon devices: first-principles Green function algorithms for systems containing a large number of atoms. Phys Rev B 81:155450CrossRefADSGoogle Scholar
  3. Aronov AG, Lyanda-Geller YB (1989) Nuclear electric resonance and orientation of carrier spins by an electric field. JETP Lett 50:431ADSGoogle Scholar
  4. Atomistix Toolkit (ATK) 2017.2.
  5. Bahramy M, King PDC, de la Torre A, Chang J, Shi M, Patthey L, Balakrishnan G, Hofmann P, Arita R, Nagaosa N, Baumberger F (2012) Emergent quantum confinement at topological insulator surfaces. Nat Commun 3:1159CrossRefGoogle Scholar
  6. Bansil A, Lin H, Das T (2016) Colloquium: topological band theory. Rev Mod Phys 88:021004CrossRefADSGoogle Scholar
  7. Bastin A, Lewiner C, Betbeder-Matibet O, Noziéres P (1971) Quantum oscillations of the Hall effect of a fermion gas with random impurity scattering. J Phys Chem Solids 32:1811CrossRefADSGoogle Scholar
  8. Baumgartner M, Garello K, Mendil J, Avci CO, Grimaldi E, Murer C, Feng J, Gabureac M, Stamm C, Acremann Y, Finizio S, Wintz S, Raabe J, Gambardella P (2017) Spatially and time-resolved magnetization dynamics driven by spin-orbit torques. Nat Nanotech 12:980CrossRefADSGoogle Scholar
  9. Belashchenko KD, Kovalev AA, van Schilfgaarde M (2016) Theory of spin loss at metallic interfaces. Phys Rev Lett 117:207204CrossRefADSGoogle Scholar
  10. Berger L (1996) Emission of spin waves by a magnetic multilayer traversed by a current. Phys Rev B 54:9353CrossRefADSGoogle Scholar
  11. Berkov DV, Miltat J (2008) Spin-torque driven magnetization dynamics: micromagnetic modeling. J Magn Magn Mater 320:1238CrossRefADSGoogle Scholar
  12. Blöchl PE (1994) Projector augmented-wave method. Phys Rev B 50:17953CrossRefADSGoogle Scholar
  13. Borders WA, Akima H, Fukami S, Moriya S, Kurihara S, Horio Y, Sato S, Ohno H (2017) Analogue spin-orbit torque device for artificial-neural-network-based associative memory operation. Appl Phys Expr 10:013007CrossRefADSGoogle Scholar
  14. Brandbyge M, Mozos JL, Ordejón P, Taylor J, Stokbro K (2002) Density-functional method for nonequilibrium electron transport. Phys Rev B 65:165401CrossRefADSGoogle Scholar
  15. Brataas A, Bauer GEW, Kelly PJ (2006) Non-collinear magnetoelectronics. Phys Rep 427:157CrossRefADSGoogle Scholar
  16. Bulik IW, Scalmani G, Frisch MJ, Scuseria GE (2013) Oncollinear density functional theory having proper invariance and local torque properties. Phys Rev B 87:035117CrossRefADSGoogle Scholar
  17. Capelle K, Vignale G, Györffy BL (2001) Spin currents and spin dynamics in time-dependent density-functional theory. Phys Rev Lett 87:206403CrossRefADSGoogle Scholar
  18. Carva K, Turek I (2009) Landauer theory of ballistic torkances in noncollinear spin-valves. Phys Rev B 80:104432CrossRefADSGoogle Scholar
  19. Ceperley DM, Alder BJ (1980) Ground state of the electron gas by a stochastic method. Phys Rev Lett 45:566CrossRefADSGoogle Scholar
  20. Chang P-H, Markussen T, Smidstrup S, Stokbro K, Nikolić BK (2015) Nonequilibrium spin texture within a thin layer below the surface of current-carrying topological insulator Bi2Se3: a first-principles quantum transport study. Phys Rev B 92:201406(R)Google Scholar
  21. Chantis AN, Belashchenko KD, Tsymbal EY, van Schilfgaarde M (2007) Tunneling anisotropic magnetoresistance driven by resonant surface states: first-principles calculations on an Fe(001) surface. Phys Rev Lett 98:046601CrossRefADSGoogle Scholar
  22. Christen T, Büttiker M (1996) Gauge-invariant nonlinear electric transport in mesoscopic conductors. Europhys Lett 35:523CrossRefADSGoogle Scholar
  23. Dolui K, Nikolić BK (2017) Spin-memory loss due to spin-orbit coupling at ferromagnet/heavy-metal interfaces: ab initio spin-density matrix approach. Phys Rev B 96:220403(R)Google Scholar
  24. Edelstein VM (1990) Spin polarization of conduction electrons induced by electric current in two-dimensional asymmetric electron systems. Solid State Commun 73:233CrossRefADSGoogle Scholar
  25. Eich FG, Gross EKU (2013) Transverse spin-gradient functional for noncollinear spin-density-functional theory. Phys Rev Lett 111:156401CrossRefADSGoogle Scholar
  26. Eich FG, Pittalis S, Vignale G (2013) Transverse and longitudinal gradients of the spin magnetization in spin-density-functional theory. Phys Rev B 88:245102CrossRefADSGoogle Scholar
  27. Ellis MOA, Stamenova M, Sanvito S (2017) Multiscale modeling of current-induced switching in magnetic tunnel junctions using ab initio spin-transfer torques. Phys Rev B 96:224410CrossRefADSGoogle Scholar
  28. Evans RFL, Fan WJ, Chureemart P, Ostler TA, Ellis MOA, Chantrell RW (2014) Atomistic spin model simulations of magnetic nanomaterials. J Phys Condens Matter 26:103202CrossRefADSGoogle Scholar
  29. Fan Y, Upadhyaya P, Kou X, Lang M, Takei S, Wang Z, Tang J, He L, Chang L-T, Montazeri M, Jiang GY, Nie T, Schwartz RN, Tserkovnyak Y, Wang KL (2014) Magnetization switching through giant spin-orbit torque in a magnetically doped topological insulator heterostructure. Nat Mater 13:699CrossRefADSGoogle Scholar
  30. Freimuth F, Blügel S, Mokrousov Y (2014) Spin-orbit torques in Co/Pt(111) and Mn/W(001) magnetic bilayers from first principles. Phys Rev B 90:174423CrossRefADSGoogle Scholar
  31. Garello K, Miron IM, Avci CO, Freimuth F, Mokrousov Y, Blügel S, Auffret S, Boulle O, Gaudin G, Gambardella P (2013) Symmetry and magnitude of spin-orbit torques in ferromagnetic heterostructures. Nat Nanotech 8:587CrossRefADSGoogle Scholar
  32. Ghosh S, Manchon A (2018) Spin-orbit torque in a three-dimensional topological insulator-ferromagnet heterostructure: crossover between bulk and surface transport. Phys Rev B 97:134402CrossRefADSGoogle Scholar
  33. Guimarães MHD, Stiehl GM, MacNeill D, Reynolds ND, Ralph DC (2018) Spin-orbit torques in NbSe2/Permalloy bilayers. Nano Lett 18:1311CrossRefADSGoogle Scholar
  34. Han J, Richardella A, Siddiqui SA, Finley J, Samarth N, Liu L (2017) Room-temperature spin-orbit torque switching induced by a topological insulator. Phys Rev Lett 119:077702CrossRefADSGoogle Scholar
  35. Haney PM, Stiles MD (2010) Current-induced torques in the presence of spin-orbit coupling. Phys Rev Lett 105:126602CrossRefADSGoogle Scholar
  36. Haney PM, Waldron D, Duine RA, Núñez AS, Guo H, MacDonald AH (2007) Current-induced order parameter dynamics: microscopic theory applied to Co/Cu/Co spin-valves. Phys Rev B 76:024404CrossRefADSGoogle Scholar
  37. Haney PM, Lee H-W, Lee K-J, Manchon A, Stiles MD (2013) Current induced torques and interfacial spin-orbit coupling: semiclassical modeling. Phys Rev B 87:174411CrossRefADSGoogle Scholar
  38. Heiliger C, Stiles MD (2008) Ab Initio studies of the spin-transfer torque in magnetic tunnel junctions. Phys Rev Lett 100:186805CrossRefADSGoogle Scholar
  39. Heiliger C, Czerner M, Yavorsky BY, Mertig I, Stiles MD (2008) Implementation of a nonequilibrium Green’s function method to calculate spin-transfer torque. J Appl Phys 103:07A709Google Scholar
  40. Hernández AR, Lewenkopf CH (2013) Nonlinear electronic transport in nanoscopic devices: nonequilibrium Green’s functions versus scattering approach. Eur Phys J B 86:131CrossRefADSMathSciNetGoogle Scholar
  41. Jia X, Xia K, Ke Y, Guo H (2011) Nonlinear bias dependence of spin-transfer torque from atomic first principles. Phys Rev B 84:014401CrossRefADSGoogle Scholar
  42. Johansson A, Henk J, Mertig I (2018) Edelstein effect in Weyl semimetals. Phys Rev B 97:085417CrossRefADSGoogle Scholar
  43. Junquera J, Paz O, Sánchez-Portal D, Artacho E (2001) Numerical atomic orbitals for linear-scaling calculations. Phys Rev B 64:235111CrossRefADSGoogle Scholar
  44. Kalitsov A, Nikolaev SA, Velev J, Chshiev M, Mryasov O (2017) Intrinsic spin-orbit torque in a single-domain nanomagnet. Phys Rev B 96:214430CrossRefADSGoogle Scholar
  45. Karrasch C, Meden V, Schönhammer K (2010) Finite-temperature linear conductance from the Matsubara Green’s function without analytic continuation to the real axis. Phys Rev B 82:125114CrossRefADSGoogle Scholar
  46. Katine JA, Albert FJ, Buhrman RA, Myers EB, Ralph DC (2000) Current-driven magnetization reversal and spin-wave excitations in Co∕Cu∕Co pillars. Phys Rev Lett 84:3149CrossRefADSGoogle Scholar
  47. Kent AD, Worledge DC (2015) A new spin on magnetic memories. Nat Nanotech 10:187CrossRefADSGoogle Scholar
  48. Kim J, Sinha J, Hayashi M, Yamanouchi M, Fukami S, Suzuki T, Mitani S, Ohno H (2013) Layer thickness dependence of the current-induced effective field vector in Ta|CoFeB|MgO. Nat Mater 12:240CrossRefADSGoogle Scholar
  49. Kim K-W, Lee K-J, Sinova J, Lee H-W, Stiles MD (2017) Spin-orbit torques from interfacial spin-orbit coupling for various interfaces. Phys Rev B 96:104438CrossRefADSGoogle Scholar
  50. Kresse G, Furthmüller J (1996) Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput Mater Sci 6:15CrossRefGoogle Scholar
  51. Kresse G, Hafner J (1993) Ab initio molecular dynamics for liquid metals. Phys Rev B 47:558CrossRefADSGoogle Scholar
  52. Kresse G, Joubert D (1999) From ultrasoft pseudopotentials to the projector augmented-wave method. Phys Rev B 59:1758CrossRefADSGoogle Scholar
  53. Kubota H, Fukushima A, Yakushiji K, Nagahama T, Yuasa S, Ando K, Maehara H, Nagamine Y, Tsunekawa K, Djayaprawira DD, Watanabe N, Suzuki Y (2008) Quantitative measurement of voltage dependence of spin-transfer torque in MgO-based magnetic tunnel junctions. Nat Phys 4:37CrossRefGoogle Scholar
  54. Kurebayashi H, Fang JSD, Irvine AC, Skinner TD, Wunderlich J, Novák V, Campion RP, Gallagher BL, Vehstedt EK, Zârbo LP, Výborný K, Ferguson AJ, Jungwirth T (2014) An antidamping spin-orbit torque originating from the Berry curvature. Nat Nanotech 9:211CrossRefADSGoogle Scholar
  55. Lee K-S, Go D, Manchon A, Haney PM, Stiles MD, Lee H-W, Lee K-J (2015) Angular dependence of spin-orbit spin-transfer torques. Phys Rev B 91:144401CrossRefADSGoogle Scholar
  56. Levy PM, Fert A (2006) Spin transfer in magnetic tunnel junctions with hot electrons. Phys Rev Lett 97:097205CrossRefADSGoogle Scholar
  57. Li H, Gao H, Zârbo LP, Výborný K, Wang X, Garate I, Doǧan F, Čejchan A, Sinova J, Jungwirth T, Manchon A (2015) Intraband and interband spin-orbit torques in noncentrosymmetric ferromagnets. Phys Rev B 91:134402Google Scholar
  58. Liu L, Pai C-F, Li Y, Tseng HW, Ralph DC, Buhrman RA (2012) Spin-torque switching with the giant spin Hall effect of tantalum. Science 336:555CrossRefADSGoogle Scholar
  59. Locatelli N, Cros V, Grollier J (2014) Spin-torque building blocks. Nat Mater 13:11CrossRefADSGoogle Scholar
  60. Lv W, Jia Z, Wang B, Lu Y, Luo X, Zhang B, Zeng Z, Liu Z (2018) Electric-field control of spin-orbit torques in WS2/permalloy bilayers. ACS Appl Mater Interfaces 10:2843CrossRefGoogle Scholar
  61. MacNeill D, Stiehl GM, Guimaraes MHD, Buhrman RA, Park J, Ralph DC (2017a) Control of spin-orbit torques through crystal symmetry in WTe2/ferromagnet bilayers. Nat Phys 13:300CrossRefGoogle Scholar
  62. MacNeill D, Stiehl GM, Guimarães MHD, Reynolds ND, Buhrman RA, Ralph DC (2017b) Thickness dependence of spin-orbit torques generated by WTe2. Phys Rev B 96:054450CrossRefADSGoogle Scholar
  63. Mahfouzi F, Kioussis N (2017) Current-induced damping of nanosized quantum moments in the presence of spin-orbit interaction. Phys Rev B 95:184417CrossRefADSGoogle Scholar
  64. Mahfouzi F, Kioussis N (2018) First-principles study of angular dependence of spin-orbit torque in Pt/Co and Pd/Co bilayers. Phys Rev B 97: 224426CrossRefADSGoogle Scholar
  65. Mahfouzi F, Nikolić BK (2013) How to construct the proper gauge-invariant density matrix in steady-state nonequilibrium: applications to spin-transfer and spin-orbit torques. SPIN 3:1330002CrossRefADSGoogle Scholar
  66. Mahfouzi F, Nikolić BK (2014) Signatures of electron-magnon interaction in charge and spin currents through magnetic tunnel junctions: a nonequilibrium many-body perturbation theory approach. Phys Rev B 90:045115CrossRefADSGoogle Scholar
  67. Mahfouzi F, Nikolić BK, Kioussis N (2016) Antidamping spin-orbit torque driven by spin-flip reflection mechanism on the surface of a topological insulator: a time-dependent nonequilibrium Green function approach. Phys Rev B 93:115419CrossRefADSGoogle Scholar
  68. Mahfouzi F, Kim J, Kioussis N (2017) Intrinsic damping phenomena from quantum to classical magnets: an ab initio study of Gilbert damping in a Pt/Co bilayer. Phys Rev B 96:214421CrossRefADSGoogle Scholar
  69. Manchon A, Zhang S (2008) Theory of nonequilibrium intrinsic spin torque in a single nanomagnet. Phys Rev B 78:212405CrossRefADSGoogle Scholar
  70. Manchon A, Miron IM, Jungwirth T, Sinova J, Zelezný J, Thiaville A, Garello K, Gambardella P (2018) Current-induced spin-orbit torques in ferromagnetic and antiferromagnetic systems.
  71. Manchon A, Ryzhanova N, Vedyayev A, Chschiev M, Dieny B (2008) Description of current-driven torques in magnetic tunnel junctions. J Phys Condens Matter 20:145208CrossRefADSGoogle Scholar
  72. Manchon A, Zhang S, Lee K-J (2010) Signatures of asymmetric and inelastic tunneling on the spin torque bias dependence. Phys Rev B 82:174420CrossRefADSGoogle Scholar
  73. Marmolejo-Tejada JM, Chang P-H, Lazić P, Smidstrup S, Stradi D, Stokbro K, Nikolić BK (2017) Proximity band structure and spin textures on both sides of topological-insulator/ferromagnetic-metal interface and their charge transport probes. Nano Lett 17:5626CrossRefADSGoogle Scholar
  74. Marzari N, Mostofi AA, Yates JR, Souza I, Vanderbilt D (2012) Maximally localized Wannier functions: theory and applications. Rev Mod Phys 84:1419CrossRefADSGoogle Scholar
  75. Mellnik AR, Lee JS, Richardella A, Grab JL, Mintun PJ, Fischer MH, Vaezi A, Manchon A, Kim E-A, Samarth N, Ralph DC (2014) Spin-transfer torque generated by a topological insulator. Nature 511:449CrossRefADSGoogle Scholar
  76. Mikuszeit N, Boulle O, Miron IM, Garello K, Gambardella P, Gaudin G, Buda-Prejbeanu LD (2015) Spin-orbit torque driven chiral magnetization reversal in ultrathin nanostructures. Phys Rev B 92:144424CrossRefADSGoogle Scholar
  77. Miron IM, Garello K, Gaudin G, Zermatten P-J, Costache MV, Auffret S, Bandiera S, Rodmacq B, Schuhl A, Gambardella P (2011) Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection. Nature 476:189CrossRefADSGoogle Scholar
  78. Myers EB, Ralph DC, Katine JA, Louie RN, Buhrman RA (1999) Current-induced switching of domains in magnetic multilayer devices. Science 285:867CrossRefGoogle Scholar
  79. Ndiaye PB, Akosa CA, Fischer MH, Vaezi A, Kim E-A, Manchon A (2017) Dirac spin-orbit torques and charge pumping at the surface of topological insulators. Phys Rev B 96:014408CrossRefADSGoogle Scholar
  80. Nikolić BK, Zârbo LP, Souma S (2006) Imaging mesoscopic spin Hall flow: spatial distribution of local spin currents and spin densities in and out of multiterminal spin-orbit coupled semiconductor nanostructures. Phys Rev B 73:075303CrossRefADSGoogle Scholar
  81. Nordström L, Singh DJ (1996) Noncollinear intra-atomic magnetism. Phys Rev Lett 76:4420CrossRefADSGoogle Scholar
  82. Oh S-C, Manchon S-YPA, Chshiev M, Han J-H, Lee H-W, Lee J-E, Nam K-T, Jo Y, Kong Y-C, Dieny B, Lee K-J (2009) Bias-voltage dependence of perpendicular spin-transfer torque in asymmetric MgO-based magnetic tunnel junctions. Nat Phys 5:898CrossRefGoogle Scholar
  83. Okuma N, Nomura K (2017) Microscopic derivation of magnon spin current in a topological insulator/ferromagnet heterostructure. Phys Rev B 95:115403CrossRefADSGoogle Scholar
  84. Ozaki T (2003) Variationally optimized atomic orbitals for large-scale electronic structures. Phys Rev B 67:155108CrossRefADSGoogle Scholar
  85. Ozaki T (2007) Continued fraction representation of the Fermi-Dirac function for large-scale electronic structure calculations. Phys Rev B 75:035123CrossRefADSGoogle Scholar
  86. Perdew JP, Burke K, Ernzerhof M (1996) Generalized gradient approximation made simple. Phys Rev Lett 77:3865CrossRefADSGoogle Scholar
  87. Perez N, Martinez E, Torres L, Woo S-H, Emori S, Beach GSD (2014) Chiral magnetization textures stabilized by the Dzyaloshinskii-Moriya interaction during spin-orbit torque switching. Appl Phys Lett 104:213909CrossRefGoogle Scholar
  88. Pesin DA, MacDonald AH (2012a) Quantum kinetic theory of current-induced torques in Rashba ferromagnets. Phys Rev B 86:014416CrossRefADSGoogle Scholar
  89. Pesin D, MacDonald AH (2012b) Spintronics and pseudospintronics in graphene and topological insulators. Nat Mater 11:409CrossRefADSGoogle Scholar
  90. Petrović M, Popescu BS, Plecháč P, Nikolić BK (2018) Spin and charge pumping by steady or pulse current driven magnetic domain wall: a self-consistent multiscale time-dependent-quantum/time-dependent-classical approach.
  91. Ralph D, Stiles M (2008) Spin transfer torques. J Magn Magn Mater 320:1190CrossRefADSGoogle Scholar
  92. Rungger I, Sanvito S (2008) Algorithm for the construction of self-energies for electronic transport calculations based on singularity elimination and singular value decomposition. Phys Rev B 78:035407CrossRefADSGoogle Scholar
  93. Sankey JC, Cui Y-T, Sun JZ, Slonczewski JC, Buhrman RA, Ralph DC (2008) Measurement of the spin-transfer-torque vector in magnetic tunnel junctions. Nat Phys 4:67CrossRefGoogle Scholar
  94. Sanvito S (2011) Electron transport theory for large systems. In: Bichoutskaia E (ed) Computational nanoscience. RSC Publishing, CambridgeGoogle Scholar
  95. Schlipf M, Gygi F (2015) Optimization algorithm for the generation of ONCV pseudopotentials. Comput Phys Commun 196:36CrossRefADSzbMATHGoogle Scholar
  96. Shao Q, Yu G, Lan Y-W, Shi Y, Li M-Y, Zheng C, Zhu X, Li L-J, Khalili Amiri P, Wang KL (2016) Strong Rashba-Edelstein effect-induced spin-orbit torques in monolayer transition metal dichalcogenide/ferromagnet bilayers. Nano Lett 16:7514CrossRefADSGoogle Scholar
  97. Shelley M, Poilvert N, Mostofi AA, Marzari N (2011) Automated quantum conductance calculations using maximally-localised Wannier functions. Comput Phys Commun 182:2174CrossRefADSGoogle Scholar
  98. Sinova J, Valenzuela SO, Wunderlich J, Back CH, Jungwirth T (2015) Spin Hall effects. Rev Mod Phys 87:1260CrossRefGoogle Scholar
  99. Sklenar J, Zhang W, Jungfleisch MB, Jiang W, Saglam H, Pearson JE, Ketterson JB, Hoffmann A (2016) Perspective: Interface generation of spin-orbit torques. J Appl Phys 120:180901CrossRefADSGoogle Scholar
  100. Slonczewski JC (1996) Current-driven excitation of magnetic multilayers. J Magn Magn Mater 159:L1CrossRefADSGoogle Scholar
  101. Soumyanarayanan A, Reyren N, Fert A, Panagopoulos C (2016) Emergent phenomena induced by spin-orbit coupling at surfaces and interfaces. Nature 539:509CrossRefGoogle Scholar
  102. Stamenova M, Mohebbi R, Seyed-Yazdi J, Rungger I, Sanvito S (2017) First-principles spin-transfer torque in CuMnAs|GaP|CuMnAs junctions. Phys Rev B 95:060403CrossRefADSGoogle Scholar
  103. Stefanucci G, van Leeuwen R (2013) Nonequilibrium many-body theory of quantum systems: a modern introduction. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  104. Stiles MD, Zangwill A (2002) Anatomy of spin-transfer torque. Phys Rev B 66:014407CrossRefADSGoogle Scholar
  105. Theodonis I, Kioussis N, Kalitsov A, Chshiev M, Butler WH (2006) Anomalous bias dependence of spin torque in magnetic tunnel junctions. Phys Rev Lett 97:237205CrossRefADSGoogle Scholar
  106. Theurich G, Hill NA (2001) Self-consistent treatment of spin-orbit coupling in solids using relativistic fully separable ab initio pseudopotentials. Phys Rev B 64:073106CrossRefADSGoogle Scholar
  107. Thygesen K, Jacobsen K (2005) Molecular transport calculations with Wannier functions. Chem Phys 319:111CrossRefGoogle Scholar
  108. Timopheev AA, Sousa R, Chshiev M, Buda-Prejbeanu LD, Dieny B (2015) Respective influence of in-plane and out-of-plane spin-transfer torques in magnetization switching of perpendicular magnetic tunnel junctions. Phys Rev B 92:104430CrossRefADSGoogle Scholar
  109. Tsoi M, Jansen AGM, Bass J, Chiang W-C, Seck M, Tsoi V, Wyder P (1998) Excitation of a magnetic multilayer by an electric current. Phys Rev Lett 80:4281CrossRefADSGoogle Scholar
  110. Velev J, Butler W (2004) On the equivalence of different techniques for evaluating the green function for a semi-infinite system using a localized basis. J Phys Condens Matter 16:R637CrossRefADSGoogle Scholar
  111. Vienna Ab initio Simulation Package (VASP) 5.4.
  112. Vignale G (2010) Ten years of spin Hall effect. J Supercond Nov Magn 23:3CrossRefGoogle Scholar
  113. Virtual Nanolab (VNL) 2017.2.
  114. Wang S, Xu Y, Xia K (2008) First-principles study of spin-transfer torques in layered systems with noncollinear magnetization. Phys Rev B 77:184430CrossRefADSGoogle Scholar
  115. Wang C, Cui Y-T, Katine JA, Buhrman RA, Ralph DC (2011) Time-resolved measurement of spin-transfer-driven ferromagnetic resonance and spin torque in magnetic tunnel junctions. Nat Phys 7:496CrossRefGoogle Scholar
  116. Wang L, Wesselink RJH, Liu Y, Yuan Z, Xia K, Kelly PJ (2016) Giant room temperature interface spin Hall and inverse spin Hall effects. Phys Rev Lett 116:196602CrossRefADSGoogle Scholar
  117. Wang Y, Zhu D, Wu Y, Yang Y, Yu J, Ramaswamy R, Mishra R, Shi S, Elyasi M, Teo K-L, Wu Y, Yang H (2017) Room temperature magnetization switching in topological insulator-ferromagnet heterostructures by spin-orbit torques. Nat Commun 8:1364CrossRefADSGoogle Scholar
  118. Winkler R (2003) Spin-Orbit coupling effects in two-dimensional electron and Hole systems. Springer, BerlinCrossRefGoogle Scholar
  119. Xiao J, Bauer GEW, Brataas A (2008) Spin-transfer torque in magnetic tunnel junctions: scattering theory. Phys Rev B 77:224419CrossRefADSGoogle Scholar
  120. Xie Y, Rungger I, Munira K, Stamenova M, Sanvito S, Ghosh AW (2016) Spin transfer torque: a multiscale picture. In: Atulasimha J, Bandyopadhyay S (eds) Nanomagnetic and spintronic devices for energy-efficient memory and computing. Wiley, HobokenGoogle Scholar
  121. Yang HX, Chshiev M, Kalitsov A, Schuhl A, Butler WH (2010) Effect of structural relaxation and oxidation conditions on interlayer exchange coupling in Fe/MgO/Fe tunnel junctions. Appl Phys Lett 96:262509CrossRefADSGoogle Scholar
  122. Yasuda K, Tsukazaki A, Yoshimi R, Kondou K, Takahashi KS, Otani Y, Kawasaki M, Tokura Y (2017) Current-nonlinear Hall effect and spin-orbit torque magnetization switching in a magnetic topological insulator. Phys Rev Lett 119:137204CrossRefADSGoogle Scholar
  123. Yoon J, Lee S-W, Kwon JH, Lee JM, Son J, Qiu X, Lee K-J, Yang H (2017) Anomalous spin-orbit torque switching due to field-like torque-assisted domain wall reflection. Sci Adv 3:e1603099CrossRefADSGoogle Scholar
  124. Zhang SS-L, Vignale G, Zhang S (2015) Anisotropic magnetoresistance driven by surface spin-orbit scattering. Phys Rev B 92:024412CrossRefADSGoogle Scholar
  125. Zholud A, Freeman R, Cao R, Srivastava A, Urazhdin S (2017) Spin transfer due to quantum magnetization fluctuations. Phys Rev Lett 119:257201CrossRefADSGoogle Scholar
  126. Zhu ZY, Cheng YC, Schwingenschlögl U (2011) Giant spin-orbit-induced spin splitting in two-dimensional transition-metal dichalcogenide semiconductors. Phys Rev B 84:153402CrossRefADSGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Branislav K. Nikolić
    • 1
    Email author
  • Kapildeb Dolui
    • 1
  • Marko D. Petrović
    • 2
  • Petr Plecháč
    • 2
  • Troels Markussen
    • 3
  • Kurt Stokbro
    • 3
  1. 1.Department of Physics and AstronomyUniversity of DelawareNewarkUSA
  2. 2.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  3. 3.Synopsys QuantumWiseCopenhagenDenmark

Section editors and affiliations

  • Stefano Sanvito
    • 1
  1. 1.Department of PhysicsTrinity CollegeDublinIreland

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