How to construct the proper gauge-invariant density matrix in steady-state nonequilibrium: Applications to spin-transfer and spin-orbit torques

Abstract

Experiments observing spin density and spin currents (responsible for, e.g., spin-transfer torque) in spintronic devices measure only the nonequilibrium contributions to these quantities, typically driven by injecting unpolarized charge current or by applying external time-dependent fields. On the other hand, theoretical approaches to calculate them operate with both the nonequilibrium (carried by electrons around the Fermi surface) and the equilibrium (carried by the Fermi sea electrons) contributions. Thus, an unambiguous procedure should remove the equilibrium contributions, thereby rendering the nonequilibrium ones which are measurable and satisfy the gauge-invariant condition according to which expectation values of physical quantities should not change when electric potential everywhere is shifted by a constant amount. Using the framework of nonequilibrium Green functions, we delineate such procedure which yields the proper gauge-invariant nonequilibrium density matrix in the linear-response and elastic transport regime for current-carrying steady state of an open quantum system connected to two macroscopic reservoirs. Its usage is illustrated by computing: (i) conventional spin-transfer torque (STT) in asymmetric F/I/F magnetic tunnel junctions (MTJs); (ii) unconventional STT in asymmetric N/I/F semi-MTJs with the strong Rashba spin-orbit coupling (SOC) at the I/F interface and injected current perpendicular to that plane; and (iii) current-driven spin density within a clean ferromagnetic Rashba spin-split two-dimensional electron gas (2DEG) which generates SO torque in laterally patterned N/F/I heterostructures when such 2DEG is located at the N/F interface and injected charge current flows parallel to the plane.

18 pages, 5 color EPS figures; mini-review prepared for SPIN (World Scientific); typos corrected and references added in v2

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Cited by in corpus (13)

• First-principles calculation of spin-orbit torque in a Co/Pt bilayer
• Spin and charge pumping by steady or pulse current-driven magnetic domain wall: A self-consistent multiscale time-dependent-quantum/time-dependent-classical approach
• Twist- and gate-tunable proximity spin-orbit coupling, spin relaxation anisotropy, and charge-to-spin conversion in heterostructures of graphene and transition-metal dichalcogenides
• Nonequilibrium spin texture within a thin layer below the surface of current-carrying topological insulator Bi${}_2$Se${}_3$: A first-principles quantum transport study
• Spintronics meets nonadiabatic molecular dynamics: Geometric spin torque and damping on noncollinear classical magnetism due to electronic open quantum system
• First-principles quantum transport modeling of spin-transfer and spin-orbit torques in magnetic multilayers
• Spin-orbit-proximitized ferromagnetic metal by monolayer transition metal dichalcogenide: Atlas of spectral functions, spin textures and spin-orbit torques in Co/MoSe${}_2$, Co/WSe${}_2$ and Co/TaSe${}_2$ heterostructures
• Negative differential resistance in graphene-nanoribbon/carbon-nanotube crossbars: A first-principles multiterminal quantum transport study
• Steering Magnetic Skyrmions with Nonequilibrium Green's Functions
• Robustness of quantized transport through edge states of finite length: Imaging current density in Floquet topological vs. quantum spin and anomalous Hall insulators
• Nonequilibrium dynamics in a spin valve with noncollinear magnetization
• Quantum transport by spin-polarized edge states in graphene nanoribbons in the quantum spin Hall and quantum anomalous Hall regimes
• Reconciling Kubo and Keldysh Approaches to Fermi-Sea-Dependent Nonequilibrium Observables: Application to Spin Hall Current and Spin-Orbit Torque in Spintronics