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Quantum transport studies of spin-dependent phenomena in solids commonly employ the Kubo or Keldysh formulas for the nonequilibrium density operator in the steady-state linear-response regime. Its trace with operators of interest, such as the spin density, spin current density, etc., gives expectation values of experimentally accessible observables. For local quantities, these formulas require summing over the manifolds of {\em both} Fermi-surface and Fermi-sea states. However, debates have been raging in the literature about the vastly different physics the two formulations can apparently produce, even when applied to the same system. Here, we revisit this problem using infinite-size graphene with proximity-induced spin-orbit and magnetic exchange effects as a testbed. By splitting this system into semi-infinite leads and central active region, in the spirit of Landauer formulation of quantum transport, we prove the {\em numerically exact equivalence} of the Kubo and Keldysh approaches via the computation of spin Hall current density and spin-orbit torque in both clean and disordered limits. The key to reconciling the two approaches are the numerical frameworks we develop for: ({\em i}) evaluation of Kubo(-Bastin) formula for a system attached to semi-infinite leads, which ensures continuous energy spectrum and evades the need for commonly used phenomenological broadening introducing ambiguity; and ({\em ii}) proper evaluation of Fermi-sea term in the Keldysh approach, which {\em must} include the voltage drop across the central active region even if it is disorder free., Comment: 9 pages, 3 figures, 115 references; Supplemental Material is available from https://wiki.physics.udel.edu/qttg/Publications

The Landau-Zener problem, where a minimum energy separation is passed with constant rate in a two-state quantum-mechanical system, is an excellent model quantum system for a computational project. It requires a low-level computational effort, but has a number of complex numerical and algorithmic issues that can be resolved through dedicated work. It can be used to teach computational concepts such as accuracy, discretization, and extrapolation, and it reinforces quantum concepts of time-evolution via a time-ordered product and of extrapolation to infinite time via time-dependent perturbation theory. In addition, we discuss the concept of compression algorithms, which are employed in many advanced quantum computing strategies, and easy to illustrate with the Landau-Zener problem., Comment: (9 pages, accepted for publication in the American Journal of Physics)

We describe coupled electron-phonon systems semiclassically - Ehrenfest dynamics for the phonons and quantum mechanics for the electrons - using a classical Monte Carlo approach that determines the nonequilibrium response to a large pump field. The semiclassical approach is quite accurate, because the phonons are excited to average energies much higher than the phonon frequency, eliminating the need for a quantum description. The numerical efficiency of this method allows us to perform a self-consistent time evolution out to very long times (tens of picoseconds) enabling us to model pump-probe experiments of a charge density wave (CDW) material. Our system is a half-filled, one-dimensional (1D) Holstein chain that exhibits CDW ordering due to a Peierls transition. The chain is subjected to a time-dependent electromagnetic pump field that excites it out of equilibrium, and then a second probe pulse is applied after a time delay. By evolving the system to long times, we capture the complete process of lattice excitation and subsequent relaxation to a new equilibrium, due to an exchange of energy between the electrons and the lattice, leading to lattice relaxation at finite temperatures. We employ an indirect (impulsive) driving mechanism of the lattice by the pump pulse due to the driving of the electrons by the pump field. We identify two driving regimes, where the pump can either cause small perturbations or completely invert the initial CDW order. Our work successfully describes the ringing of the amplitude mode in CDW systems that has long been seen in experiment, but never successfully explained by microscopic theory.

The standard model of spin-transfer torque (STT) in antiferromagnetic spintronics considers exchange of angular momentum between quantum spins of flowing electrons and noncollinear-to-them localized spins treated as classical vectors. These vectors are assumed to realize N\'{e}el order in equilibrium, $\uparrow \downarrow \ldots \uparrow \downarrow$, and their STT-driven dynamics is described by the Landau-Lifshitz-Gilbert (LLG) equation. However, many experimentally employed materials (such as archetypal NiO) are strongly electron-correlated antiferromagnetic Mott insulators (AFMI) where localized spins form a ground state quite different from the unentangled N\'{e}el state $|\!\! \uparrow \downarrow \ldots \uparrow \downarrow \rangle$. The true ground state is entangled by quantum spin fluctuations, leading to expectation value of all localized spins being zero, so that LLG dynamics of classical vectors of fixed length rotating due to STT cannot even be initiated. Instead, a fully quantum treatment of both conduction electrons and localized spins is necessary to capture exchange of spin angular momentum between them, denoted as quantum STT. We use a recently developed time-dependent density matrix renormalization group approach to quantum STT to predict how injection of a spin-polarized current pulse into a normal metal layer coupled to AFMI overlayer via exchange interaction and possibly small interlayer hopping -- which mimics, e.g., topological-insulator/NiO bilayer employed experimentally -- will induce nonzero expectation value of AFMI localized spins. This new nonequilibrium phase is a spatially inhomogeneous ferromagnet with zigzag profile of localized spins. The total spin absorbed by AFMI increases with electron-electron repulsion in AFMI, as well as when the two layers do not exchange any charge., Comment: 7 pages, 5 figures; Supplemental Material, including a movie, is available from this URL https://wiki.physics.udel.edu/qttg/Publications

The recent experiment [Y. Wang et al., Science 366, 1125 (2019)] on magnon-mediated spin-transfer torque (MSTT) was interpreted in terms of a picture where magnons are excited within an antiferromagnetic insulator (AFI), by applying nonequilibrium electronic spin density at one of its surfaces, so that their propagation across AFI deprived of conduction electrons eventually leads to reversal of magnetization of a ferromagnetic metal (FM) attached to the opposite surface of AFI. We employ a recently developed time-dependent nonequilibrium Green functions combined with the Landau-Lifshitz-Gilbert equation (TDNEGF+LLG) formalism to evolve conduction electrons quantum-mechanically while they interact via self-consistent back-action with localized magnetic moments described classically by atomistic spin dynamics solving a system of LLG equations. Upon injection of square current pulse as the initial condition, TDNEGF+LLG simulations of FM-polarizer/AFI/FM-analyzer junctions show that reversal of localized magnetic moments within FM-analyzer is less efficient, in the sense of requiring larger pulse height and its longer duration, than conventional electron-mediated STT (ESTT) driving magnetization switching in standard FM-polarizer/normal-metal/FM-analyzer spin valve. Since both electronic, generated by spin pumping from AFI, and magnonic, generated by direct transmission from AFI, spin currents are injected into the FM-analyzer, its localized magnetic moments will experience combined MSTT and ESTT. Nevertheless, by artificially turning off ESTT we demonstrate that MSTT plays a dominant role whose understanding, therefore, paves the way for all-magnon-driven magnetization switching devices with no electronic parts., Comment: 12 pages, 7 figures, 3 embedded videos

We introduce time-dependent density matrix renormalization group (tDMRG) as a solution to long standing problem in spintronics -- how to describe spin-transfer torque (STT) between flowing spins of conduction electrons and localized spins within a magnetic material by treating the dynamics of both spin species fully quantum-mechanically. In contrast to conventional Slonczewski-Berger STT, where the localized spins are viewed as classical vectors obeying the Landau-Lifshitz-Gilbert equation and where their STT-driven dynamics is initiated only when the spin-polarization of flowing electrons and localized spins are noncollinear, quantum STT can occur when these vectors are collinear but antiparallel. Using tDMRG, we simulate the time evolution of a many-body quantum state of electrons and localized spins, where the former are injected as a spin-polarized current pulse while the latter comprise a quantum Heisenberg ferromagnetic metallic (FM) spin-$\frac{1}{2}$ XXZ chain initially in the ground state with spin-polarization antiparallel to that of injected electrons. The quantum STT reverses the direction of localized spins, but without rotation from the initial orientation, when the number of injected electrons exceeds the number of localized spins. Such nonclassical reversal, which is absent from LLG dynamics, is strikingly inhomogeneous across the FM chain and it can be accompanied by reduction of the magnetization associated with localized spins, even to zero at specific locations. This is because quantum STT generates a highly entangled nonequilibrium many-body state of all flowing and localized spins, despite starting from the initially unentangled ground state of a mundane FM. Furthermore, the mutual information between localized spins at the FM edges remains nonzero even at infinite separation as the signature of dynamical buildup of long-range entanglement., Comment: 11 pages, 8 figures; four movie available from https://wiki.physics.udel.edu/qttg/Publications

Graphene sandwiched between semiconducting monolayers of ferromagnet Cr$_2$Ge$_2$Te$_6$ and transition-metal dichalcogenide WS$_2$ acquires both spin-orbit (SO), of valley-Zeeman and Rashba types, and exchange couplings. Using first-principles combined with quantum transport calculations, we predict that such doubly proximitized graphene within van der Waals heterostructure will exhibit SO torque driven by unpolarized charge current. This system lacking spin Hall current, putatively considered to be necessary for efficient damping-like (DL) SO torque that plays a key role in magnetization switching, demonstrates how DL torque component can be generated solely by skew-scattering off spin-independent potential barrier or impurities in purely two-dimensional electronic transport due to the presence of proximity SO coupling and its spin texture tilted out-of-plane. This leads to current-driven nonequilibrium spin density emerging in all spatial directions, whose cross product with proximity magnetization yields DL SO torque, unlike the ballistic regime with no scatterers in which only field-like (FL) SO torque appears. In contrast to SO torque on conventional metallic ferromagnets in contact with three dimensional SO-coupled materials, the ratio of FL and DL torque can be tuned by more than an order of magnitude via combined top and back gates., Comment: 12 pages, 9 figures

The recently discovered two-dimensional (2D) magnetic insulator CrI$_3$ is an intriguing case for basic research and spintronic applications since it is a ferromagnet in the bulk, but an antiferromagnet in bilayer form, with its magnetic ordering amenable to external manipulations. Using first-principles quantum transport approach, we predict that injecting unpolarized charge current parallel to the interface of bilayer-CrI$_3$/monolayer-TaSe$_2$ van der Waals heterostructure will induce spin-orbit torque (SOT) and thereby driven dynamics of magnetization on the first monolayer of CrI$_3$ in direct contact with TaSe$_2$. By combining calculated complex angular dependence of SOT with the Landau-Lifshitz-Gilbert equation for classical dynamics of magnetization, we demonstrate that current pulses can switch the direction of magnetization on the first monolayer to become parallel to that of the second monolayer, thereby converting CrI$_3$ from antiferromagnet to ferromagnet while not requiring any external magnetic field. We explain the mechanism of this reversible current-driven nonequilibrium phase transition by showing that first monolayer of CrI$_3$ carries current due to evanescent wavefunctions injected by metallic transition metal dichalcogenide TaSe$_2$, while concurrently acquiring strong spin-orbit coupling (SOC) via such proximity effect, whereas the second monolayer of CrI$_3$ remains insulating. The transition can be detected by passing vertical read current through the vdW heterostructure, encapsulated by bilayer of hexagonal boron nitride and sandwiched between graphite electrodes, where we find tunneling magnetoresistance of $\simeq 240$%., Comment: 8 pages, 6 figures, one movie (available from https://wiki.physics.udel.edu/qttg/Publications), PDFLaTeX

We not only reproduce burst of short-wavelength spin waves (SWs) observed in recent experiment [S. Woo et al., Nat. Phys. 13, 448 (2017)] on magnetic-field-driven annihilation of two magnetic domain walls (DWs) but, furthermore, we predict that this setup additionally generates highly unusual} pumping of electronic spin currents in the absence of any bias voltage. Prior to the instant of annihilation, their power spectrum is ultrabroadband, so they can be converted into rapidly changing in time charge currents, via the inverse spin Hall effect, as a source of THz radiation of bandwidth $\simeq 27$ THz where the lowest frequency is controlled by the applied magnetic field. The spin pumping stems from time-dependent fields introduced into the quantum Hamiltonian of electrons by the classical dynamics of localized magnetic moments (LMMs) comprising the domains. The pumped currents carry spin-polarized electrons which, in turn, exert backaction on LMMs in the form of nonlocal damping which is more than twice as large as conventional local Gilbert damping. The nonlocal damping can substantially modify the spectrum of emitted SWs when compared to widely-used micromagnetic simulations where conduction electrons are completely absent. Since we use fully microscopic (i.e., Hamiltonian-based) framework, self-consistently combining time-dependent electronic nonequilibrium Green functions with the Landau-Lifshitz-Gilbert equation, we also demonstrate that previously derived phenomenological formulas miss ultrabroadband spin pumping while underestimating the magnitude of nonlocal damping due to nonequilibrium electrons., Comment: 7 pages, 4 figures, Supplemental Material (one movie + one note available from https://wiki.physics.udel.edu/qttg/Publications)

The standard spin-transfer torque (STT)---where spin-polarized current drives dynamics of magnetization viewed as a classical vector---requires noncollinearity between electron spins carried by the current and magnetization of a ferromagnetic layer. However, recent experiments [A. Zholud et al., Phys. Rev. Lett. 119, 257201 (2017)] observing magnetization dynamics in spin valves at cryogenic temperatures, even when electron spin is collinear to magnetization, point at overlooked quantum effects in STT which can lead to highly nonclassical magnetization states. Using fully quantum many-body treatment, where an electron injected as spin-polarized wave packet interacts with local spins comprising the anisotropic quantum Heisenberg ferromagnetic chain, we define quantum STT as any time evolution of local spins due to initial many-body state not being an eigenstate of electron+local-spins system. For time evolution caused by injected spin-down electron scattering off local up-spins, entanglement between electron subsystem and local spins subsystem takes place leading to decoherence and, therefore, shrinking of the total magnetization but without rotation from its initial orientation which explains the experiments. Furthermore, the same processes---entanglement and thereby induced decoherence---are present also in standard noncollinear geometry, together with the usual magnetization rotation. This is because STT in quantum many-body picture is caused only by electron spin-down factor state, and the only difference between collinear and noncollinear geometries is in relative size of the contribution of initial many-body state containing such factor state to superpositions of separable many-body quantum states generated during time evolution., Comment: 6 pages, 4 figures; PDFLaTeX

We introduce a multiscale framework which combines time-dependent nonequilibrium Green function (TD-NEGF) algorithms, scaling linearly in the number of time steps and describing quantum-mechanically conduction electrons in the presence of time-dependent fields of arbitrary strength or frequency, with classical time evolution of localized magnetic moments described by the Landau-Lifshitz-Gilbert (LLG) equation. The TD-NEGF+LLG framework can be applied to a variety of problems where current-driven spin torque induces dynamics of magnetic moments as the key resource for next generation spintronics. Using magnetic domain wall (DW) as an example, we predict that its motion will pump time-dependent spin and charge currents (on the top of unpolarized DC charge current injected through normal metal leads to drive the DW motion). The conversion of AC components of spin current, whose amplitude increases (decreases) as the DW approaches (distances from) the normal metal lead, into AC voltage via the inverse spin Hall effect offers a tool to precisely track the DW position along magnetic nanowire. We also quantify the DW transient inertial displacement due to its acceleration and deceleration by pulse current and the entailed spin and charge pumping. Finally, TD-NEGF+LLG as a nonperturbative (i.e., numerically exact) framework allows us to establish the limits of validity of the so-called spin-motive force (SMF) theory for pumped charge current by time-dependent magnetic textures---the perturbative analytical formula of SMF theory becomes inapplicable for large frequencies (but unrealistic in magnetic system) and, more importantly, for increasing noncollinearity when the angles between neighboring magnetic moments exceed $\simeq 10^\circ$., Comment: 11 pages, 8 figures, PDFLaTeX; three additional movies of domain wall motion driven by injecting DC or pulse charge current are available from https://wiki.physics.udel.edu/qttg/Publications

We review a unified approach for computing: (i) spin-transfer torque in magnetic trilayers like spin-valves 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. Our 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. We provide illustrative examples 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 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., Comment: 22 pages, 9 figures, PDFLaTeX; prepared for Springer Handbook of Materials Modeling, Volume 2 Applications: Current and Emerging Materials

We have investigated spin relaxation times in CdTe quantum dots doped with a single Mn atom, a prototype of a system where the interaction between a single charge carrier and a single spin takes place. A theoretical model that was used includes the electron-Mn spin exchange interaction responsible for mixing of the states of different spin in the basic Hamiltonian and electron-phonon interaction as a perturbation responsible for transitions between the states. It was found that the dominant electron-phonon interaction mechanism responsible for spin relaxation is the interaction with acoustic phonons through deformation potential. Electron and Mn spin relaxation times at room temperature take values in the range from microseconds at a magnetic field of 0.5 T down to nanoseconds at a magnetic field of 10 T and become three orders of magnitude larger at cryogenic temperatures. It was found that electron spin-orbit interaction has a negligible effect on spin relaxation times, while the changes in the position of the Mn atom within the dot and in the dot dimensions can change the spin relaxation times by up to one order of magnitude., Comment: 9 pages, 5 figures