Surface orbitronics: new twists from orbital Rashba physics

When the inversion symmetry is broken at a surface, spin-orbit interaction gives rise to spin-dependent energy shifts - a phenomenon which is known as the spin Rashba effect. Recently, it has been recognized that an orbital counterpart of the spin Rashba effect - the orbital Rashba effect - can be realized at surfaces even without spin- orbit coupling. Here, we propose a mechanism for the orbital Rashba effect based on sp orbital hybridization, which ultimately leads to the electric polarization of surface states. As a proof of principle, we show from first principles that this effect leads to chiral orbital textures in $\mathbf{k}$-space of the BiAg$_2$ monolayer. In predicting the magnitude of the orbital moment arising from the orbital Rashba effect, we demonstrate the crucial role that the Berry phase theory plays for the magnitude and variation of the orbital textures. As a result, we predict a pronounced manifestation of various orbital effects at surfaces, and proclaim the orbital Rashba effect to be a key platform for surface orbitronics

First-principles calculation of orbital Hall effect by Wannier interpolation: Role of orbital dependence of the anomalous position

The position operator in a Bloch representation acquires a gauge correction in the momentum space on top of the canonical position, which is called the anomalous position. We show that the anomalous position is generally orbital-dependent and thus plays a crucial role in the description of the intrinsic orbital Hall effect in terms of Wannier basis. We demonstrate this from the first-principles calculation of orbital Hall conductivities of transition metals by Wannier interpolation. Our results show that consistent treatment of the velocity operator by adding the additional term originating from the anomalous position predicts the orbital Hall conductivities different from those obtained by considering only the group velocity. We find the difference is crucial in several metals. For example, we predict the negative sign of the orbital Hall conductivities for elements in the groups X and XI such as Cu, Ag, Au, and Pd, for which the previous studies predicted the positive sign. Our work suggests the importance of consistently describing the spatial dependence of basis functions by first-principles methods as it is fundamentally missing in the tight-binding approximation

Orbital Pumping by Magnetization Dynamics in Ferromagnets

We show that dynamics of the magnetization in ferromagnets can pump the orbital angular momentum, which we denote by orbital pumping. This is the reciprocal phenomenon to the orbital torque that induces magnetization dynamics by the orbital angular momentum in non-equilibrium. The orbital pumping is analogous to the spin pumping established in spintronics but requires the spin-orbit coupling for the orbital angular momentum to interact with the magnetization. We develop a formalism that describes the generation of the orbital angular momentum by magnetization dynamics within the adiabatic perturbation theory. Based on this, we perform first-principles calculation of the orbital pumping in prototypical $3d$ ferromagnets, Fe, Co, and Ni. The results show that the ratio between the orbital pumping and the spin pumping ranges from 5 to 15 percents, being smallest in Fe and largest in Ni. This implies that ferromagnetic Ni is a good candidate for measuring the orbital pumping. Implications of our results on experiments are also discussed

Detection of long-range orbital-Hall torques

We report and quantify a large orbital-Hall torque generated by Nb and Ru, which we identify from a strong dependence of torques on the ferromagnets. This is manifested as a sign reversal and strong enhancement in the damping-like torques measured in Nb (or Ru)/Ni bilayers as compared to Nb (or Ru)/FeCoB bilayers. The long-range nature of orbital transport in the ferromagnet is revealed by the thickness dependences of Ni in Nb (or Ru)/Ni bilayers which are markedly different from the regular spin absorption in the ferromagnet that takes place within a few angstroms and thus it uniquely distinguishes the orbital Hall torque from the spin Hall torque

Theory of Current-Induced Angular Momentum Transfer Dynamics in Spin-Orbit Coupled Systems

Motivated by the importance of understanding competing mechanisms to current-induced spin-orbit torque in complex magnets, we develop a unified theory of current-induced spin-orbital coupled dynamics. The theory describes angular momentum transfer between different degrees of freedom in solids, e.g., the electron orbital and spin, the crystal lattice, and the magnetic order parameter. Based on the continuity equations for the spin and orbital angular momenta, we derive equations of motion that relate spin and orbital current fluxes and torques describing the transfer of angular momentum between different degrees of freedom. We then propose a classification scheme for the mechanisms of the current-induced torque in magnetic bilayers. Based on our first-principles implementation, we apply our formalism to two different magnetic bilayers, Fe/W(110) and Ni/W(110), which are chosen such that the orbital and spin Hall effects in W have opposite sign and the resulting spin- and orbital-mediated torques can compete with each other. We find that while the spin torque arising from the spin Hall effect of W is the dominant mechanism of the current-induced torque in Fe/W(110), the dominant mechanism in Ni/W(110) is the orbital torque originating in the orbital Hall effect of W. It leads to negative and positive effective spin Hall angles, respectively, which can be directly identified in experiments. This clearly demonstrates that our formalism is ideal for studying the angular momentum transfer dynamics in spin-orbit coupled systems as it goes beyond the "spin current picture" by naturally incorporating the spin and orbital degrees of freedom on an equal footing. Our calculations reveal that, in addition to the spin and orbital torque, other contributions such as the interfacial torque and self-induced anomalous torque within the ferromagnet are not negligible in both material systems.Comment: 26 pages, 13 figure

Inverse Orbital Torque via Spin-Orbital Entangled States

While current-induced torque by orbital current has been experimentally found in various structures, evidence for its reciprocity has been missing so far. Here, we report experimental evidences of strong inverse orbital torque in YIG/Pt/CuOx (YIG = Y3Fe5O12) mediated by spin-orbital entangled electronic states in Pt. By injecting spin current from YIG to Pt by the spin pumping via ferromagnetic resonance and by the spin Seebeck effect, we find a pronounced inverse spin Hall effect-like signal. While a part of the signal is explained as due to the inverse spin-orbital Hall effect in Pt, we also find substantial increase of the signal in YIG/Pt/CuOx structures compared to the signal in YIG/Pt. We attribute this to the inverse orbital Edelstein effect at Pt/CuOx interface mediated by the spin-orbital entangled states in Pt. Our work paves the way toward understanding of spin-orbital entangled physics in nonequilibrium and provides a way for electrical detection of the orbital current in orbitronic device applications.Comment: 8 pages, four figure

Time-domain observation of ballistic orbital-angular-momentum currents with giant relaxation length in tungsten

The emerging field of orbitronics exploits the electron orbital momentum $\textit{L}$. Compared to spin-polarized electrons, $\textit{L}$ may allow magnetic-information transfera with significantly higher density over longer distances in more materials. However, direct experimental observation of $\textit{L}$ currents, their extended propagation lengths and their conversion into charge currents has remained challenging. Here, we optically trigger ultrafast angular-momentum transport in Ni|W|SiO$_2$ thin-film stacks. The resulting terahertz charge-current bursts exhibit a marked delay and width that grow linearly with W thickness. We consistently ascribe these observations to a ballistic $\textit{L}$ current from Ni through W with giant decay length (~80 nm) and low velocity (~0.1 nm/fs). At the W/SiO$_2$ interface, the $\textit{L}$ flow is efficiently converted into a charge current by the inverse orbital Rashba-Edelstein effect, consistent with ab-initio calculations. Our findings establish orbitronic materials with long-distance ballistic $\textit{L}$ transport as possible candidates for future ultrafast devices and an approach to discriminate Hall- and Rashba-Edelstein-like conversion processes