Spin Hall Conductivity Research Articles

Converging tetrahedron method calculations for the nondissipative parts of spectral functions

Many physical quantities in solid-state physics are calculated from $k$-space summation. For spectral functions, the frequency-dependent factor can be decomposed into the energy-conserving $\ensuremath{\delta}$-function part and the nondissipative principal value part. A very useful scheme for this $k$-space summation is the tetrahedron method. Tetrahedron methods have been widely used to calculate the summation of the energy-conserving $\ensuremath{\delta}$-function part, such as the imaginary part of the dielectric function. On the other hand, the corresponding tetrahedron method for the nondissipative part, such as the real part of the dielectric function has not been used much. In this paper, we address the technical difficulties in the tetrahedron method for the nondissipative part and present an easy-to-implement stable method to overcome those difficulties. We demonstrate our method by calculating the static and dynamical spin Hall conductivity of platinum. Our method can be widely applied to calculate linear static or dynamical conductivity, self-energy of an electron, and electric polarizability, to name a few.

Energy-Efficient W100−xTax/ Co-Fe-B/MgO Spin Hall Nano-Oscillators

We investigate a W-Ta alloying route to reduce the auto-oscillation current densities and the power consumption of nano-constriction based spin Hall nano oscillators. Using spin-torque ferromagnetic resonance (ST-FMR) measurements on microbars of W$_{\text{100-x}}$Ta$_{\text{x}}$(5 nm)/CoFeB(t)/MgO stacks with t = 1.4, 1.8, and 2.0 nm, we measure a substantial improvement in both the spin-orbit torque efficiency and the spin Hall conductivity. We demonstrate a 34\% reduction in threshold auto-oscillation current density, which translates into a 64\% reduction in power consumption as compared to pure W-based SHNOs. Our work demonstrates the promising aspects of W-Ta alloying for the energy-efficient operation of emerging spintronic devices.

Giant intrinsic spin Hall effect in layered material Pt2B

Spin Hall effect (SHE) is highly promising for future spintronic applications. Using ab initio calculations , we predict that the hexagonal layered material Pt 2 B exhibits a giant intrinsic SHE with an intrinsic spin Hall conductivity (SHC) of 2159 ( ħ / e ) S/cm, which can reach its maximum value of 2274 ( ħ / e ) S/cm at 0.04 eV above the intrinsic Fermi level. The large SHC mainly comes from the contributions of the two occupied bands below the Fermi level, and the energy difference between the occupied and unoccupied states determines the energy position of the maximal SHC. Pt 2 B’s spin Hall angle (SHA) is of the same order as pure metal Pt. Our finds suggest that Pt 2 B is a fascinating material for producing pure spin current. • Pt 2 B could be a ideal system for achieving large SHC. It has an intrinsic SHC σ x y z as large as 2159 ( ħ / e ) S/cm. • The large SHC mainly come from the contribution of the two occupied bands below the Fermi level, and the energy difference between occupied and unoccupied states determines the energy position of maximal SHC. • The bands around the Fermi level do not have the SOC-induced anti-cross. Therefore, the mechanism of the large SHC of Pt 2 B is different from many previous reported materials.

Spin Hall angle of rhodium and its effects on magnetic damping of Ni80Fe20 in Rh/Ni80Fe20 bilayer

We report the investigation of the spin Hall angle (SHA) of Rhodium (Rh) and the effective magnetic damping constant of Ni80Fe20 in Rh (10 nm)/Ni80Fe20 (10 nm) bilayer system using Spin Torque Ferromagnetic Resonance (STFMR) spectroscopy and density functional theory based analysis. The filtered DC output voltages from Rh (10 nm)/Ni80Fe20 (10 nm) bilayer system at different input signal frequencies and powers for a complete 360° angle were measured. The average magnetic damping constant of Ni80Fe20 is found to be 0.017 and the angle-dependent study shows a 2-fold symmetry with the in-plane anisotropy field of 120 Oe. The SHA of Rh is found to be 0.2 %. The results agrees well with the calculations based on density functional theory where we found spin Hall conductivity (σxyspinz) at the charge-neutral Fermi level (EF) is positive and it’s magnitude is 423ℏe(S/cm).

Geometric origin of intrinsic spin hall effect in an inhomogeneous electric field

In recent years, the spin Hall effect has received great attention because of its potential application in spintronics and quantum information processing and storage. However, this effect is usually studied under the external homogeneous electric field. Understanding how the inhomogeneous electric field affects the spin Hall effect is still lacking. Here, we investigate a two-dimensional two-band time-reversal symmetric system and give an expression for the intrinsic spin Hall conductivity in the presence of the inhomogeneous electric field, which is shown to be expressed through the geometric quantities: quantum metric and interband Berry connection. We show that for Rashba and Dresselhaus systems, the inhomogeneous intrinsic spin Hall conductivity can be tuned with the Fermi energy. On the other hand, when people get physical intuition on transport phenomena from the wave packet, one issue appears. It is shown that the conductivity obtained from the conventional wave packet approach cannot be fully consistent with the one predicted by the Kubo-Greenwood formula. Here, we attempt to solve this problem.

Magnus spin Hall and spin Nernst effects in gapped two-dimensional Rashba systems

We study the Magnus transport in a gapped two-dimensional electron gas with Rashba spin-orbit coupling using semiclassical Boltzmann transport formalism. Apart from its signature in the charge transport coefficients, the inclusion of Magnus velocity in the spin current operator enables us to study Magnus spin transport in the system. In particular, we study the roles of mass gap and Fermi surface topology on the behavior of Magnus Hall and Nernst conductivities and their spin counterparts. We find that the Magnus spin Hall conductivity vanishes in the limit of zero gap, unlike the universal spin Hall conductivity ${\ensuremath{\sigma}}_{s}=e/(8\ensuremath{\pi})$. The Magnus spin currents with spin polarization perpendicular to the applied bias (electrical/thermal) are finite while with polarization along the bias vanishes. Each Magnus conductivity displays a plateau as Fermi energy sweeps through the gap and has peaks (whose magnitudes decrease with the gap) when the Fermi energy is at the gap edges.

Nonlinear spin Hall effect in PT -symmetric collinear magnets

We theoretically investigate a nonlinear spin Hall effect in $\mathcal{PT}$-symmetric antiferromagnetic metals, which serve as an efficient spin current generator. We elucidate that an emergent spin-dependent Berry curvature dipole is a microscopic origin of the nonlinear spin Hall effect, which becomes nonzero with neither relativistic spin-orbit coupling, uniform magnetization, nor spin-split band structure. By analyzing a microscopic antiferromagnetic model without spin-orbit coupling for an intuitive understanding of the phenomena, we elucidate essential hopping processes and a condition to enhance the nonlinear spin Hall conductivity. We also provide a complete table to include useful correspondence among the N\'eel vector, odd-parity multipoles, nonlinear spin conductivity tensor, and candidate materials in all the $\mathcal{PT}$-symmetric black-and-white magnetic point groups.

Spin Swapping Effect of Band Structure Origin in Centrosymmetric Ferromagnets.

We theoretically demonstrate the spin swapping effect of band structure origin in centrosymmetric ferromagnets. It is mediated by an orbital degree of freedom but does not require inversion asymmetry or impurity spin-orbit scattering. Analytic and tight-binding models reveal that it originates mainly from k points where bands with different spins and different orbitals are nearly degenerate, and thus it has no counterpart in normal metals. First-principle calculations for centrosymmetric 3d transition-metal ferromagnets show that the spin swapping conductivity of band structure origin can be comparable in magnitude to the intrinsic spin Hall conductivity of Pt. Our theory generalizes transverse spin currents generated by ferromagnets and emphasizes the important role of the orbital degree of freedom in describing spin-orbit-coupled transport in centrosymmetric materials.

Large interfacial contribution to ultrafast THz emission by inverse spin Hall effect in CoFeB/Ta heterostructure

SummaryUltrafast THz radiation generation from ferromagnetic/nonmagnetic (FM/NM) spintronic heterostructures generally exploits the spin-charge conversion within the nonmagnetic layer and its interface with the ferromagnetic layer. Various possible sub-contributions to the underlying mechanism need to be exploited not only for investigating the intricacies at the fundamental level in the material properties themselves but also for improving their performance for broadband and high-power THz emission. Here, we report ultrafast THz emission from (CoFeB,Fe)/(Ta,Pt) bilayers at varying sample temperatures to unravel the role of intrinsic and extrinsic spin-charge conversion processes through the extracted values of spin-Hall conductivities. An enhanced THz emission along with temperature-dependent THz signal polarity reversal is observed in the case of annealed CoFeB/Ta. These results demonstrate a large interfacial contribution to the overall spin-Hall angle arising from the modified interface in the annealed CoFeB/Ta.

Full calculation of inter-conversion between charge, spin, and heat current using a common partial differential equation platform

We present a simple implementation of calculation of spin current profiles using a partial differential equation platform. By solving multiple scalar potentials, spin injection, spin/charge inter-conversion, and thermal spin injection phenomena can be well reproduced numerically. As a demonstration, we show spin current generation and detection in a composite of ferromagnetic, spin conducting, and spin-Hall-metallic materials. Furthermore, we present a model extended to three-dimensionally polarized spin current and describe the matrix for spin/charge current inter-conversion in a conductive ferromagnet, which allows for numerical reproduction of anomalous and planar Hall effects. It is found that the planar Hall voltage is in part generated by spin Hall conductivities, though its magnitude is orders smaller than that induced by the anisotropic magnetoresistance. Our method will contribute to further development of effective and feasible simulations of spin-current-mediated systems.

High Spin Hall Conductivity Induced by Ferromagnet and Interface

AbstractSpin‐orbit‐torque (SOT) offers a highly attractive perspective for manipulating magnetization dynamics in magnetic nanostructures. SOT is been observed and studied in various systems. However, limited by the efficiency, SOT‐induced switching of the ultrahard ferromagnet is still extremely difficult and a further improvement in efficiency is requested. Here, the SOT is reported in chemically disordered soft Fe0.5Pt0.5 and Pt/Fe0.5Pt0.5 bilayers. Due to the magnetization‐strengthened spin Hall effect, the damping‐like torque efficiency and spin Hall conductivity (SHC) in Fe0.5Pt0.5 reach 1.11 and 1.08 × 106 ℏ/2e Ω−1 m−1 respectively, much higher than those in conventional materials. Furthermore, the Pt/Fe0.5Pt0.5 interface enhances SHC to a total of 2.93 × 106 ℏ/2e Ω−1 m−1 due to the interfacial symmetry breaking. This system can be used to partially switch the magnetization of ultra‐hard exchange‐spring system with a switching field of 1T by applying a relatively low current. This finding will push forward the development of SOT devices with ultrahigh‐density and low‐power consumption.

Large intrinsic spin Hall conductivity and spin Hall angle in the nodal-line semimetals ThAl2 and ThGa2

The intrinsic spin Hall effect and topological properties of ${\mathrm{ThAl}}_{2}$ and ${\mathrm{ThGa}}_{2}$ have been investigated based on first-principles electronic structure calculations. The band structures calculated without the spin-orbital coupling (SOC) show that both ${\mathrm{ThAl}}_{2}$ and ${\mathrm{ThGa}}_{2}$ host multiple nodal lines near the Fermi level. Once the SOC is taken into account, these nodal lines are gapped and huge spin Berry curvatures can be generated. As a result, giant spin Hall conductances of ${\mathrm{ThAl}}_{2}$ and ${\mathrm{ThGa}}_{2}$ are induced around the Fermi level. Moreover, large spin Hall angles can be achieved due to the giant spin Hall conductance and the small electric conductivity. Our work indicates that ${\mathrm{ThAl}}_{2}$ and ${\mathrm{ThGa}}_{2}$ not only provide an ideal platform for exploring the interplay between spin Hall effect and topological property, but also have promising applications in spin-orbitronic devices.

Spin Hall effect driven by the spin magnetic moment current in Dirac materials

The spin Hall effect of a Dirac Hamiltonian system is studied using semiclassical analyses and the Kubo formula. In this system, the spin Hall conductivity is dependent on the definition of spin current. All components of the spin Hall conductivity vanish when spin current is defined as the flow of spin angular momentum. In contrast, the off-diagonal components of the spin Hall conductivity are non-zero and scale with the carrier velocity (and the effective $g$-factor) when spin current consists of the flow of spin magnetic moment. We derive analytical formula of the conductivity, carrier mobility and the spin Hall conductivity to compare with experiments. In experiments, we use Bi as a model system that can be characterized by the Dirac Hamiltonian. Te and Sn are doped into Bi to vary the electron and hole concentration, respectively. We find the spin Hall conductivity ($\sigma_\mathrm{SH}$) takes a maximum near the Dirac point and decreases with increasing carrier density ($n$). The sign of $\sigma_\mathrm{SH}$ is the same regardless of the majority carrier type. The spin Hall mobility, proportional to $\sigma_\mathrm{SH}/n$, increases with increasing carrier mobility with a scaling coefficient of $\sim$1.4. These features can be accounted for quantitatively using the derived analytical formula. The results demonstrate that the giant spin magnetic moment, with an effective $g$-factor that approaches 100, is responsible for the spin Hall effect in Bi.

Spin Hall effect in two-dimensional InSe: Interplay between Rashba and Dresselhaus spin-orbit couplings

Materials with inversion asymmetries can exhibit strong spin Hall effect (SHE) in the presence of Dresselhaus and Rashba spin-orbit coupling (D/R SOC) interactions. Ideally, in a two-dimensional crystal, inversion asymmetry could be modulated by stacking order and external perturbations. Here, using first-principles calculations, we systematically investigate the interplay between DSOC and RSOC and their influences on SHE in mono- and bilayer InSe. We show that in the presence of DSOC, the introduction of Rashba interaction through gate voltages in monolayer InSe increases Zeeman-like spin splitting around the Brillouin-zone center and contributes to the enhancement of spin Hall conductivity (SHC), which reaches a saturation point due to the RSOC-enforced spiral-spin texture. The SHC in the unperturbed centrosymmetric $\mathit{AB}$ stacked bilayer shows a peak associated with the Mexican-hat-like valence-band edge; however, in a wide energy range the SHC stays insignificant. In the $\mathit{AB}$\ensuremath{'} stacked bilayer with the intrinsic RSOC present, the value of SHC can be comparable to that of $\mathit{AB}$ stacked bilayer with an external electric field. Moreover, we show that the spin-momentum locking in the $\mathit{AB}$\ensuremath{'} stacked bilayer is switchable by a gate voltage. These findings provide a promising route for spintronic and magneto-optical applications by exploiting the rich physics of spin-orbit effects.

Vanishing of the quantum spin Hall phase in a semi-Dirac Kane-Mele model

We study the vanishing of the topological properties of a quantum spin Hall insulator induced by a deformation of the band structure that interpolates between the Dirac and the semi-Dirac limits of a tight-binding model on a honeycomb lattice. The above scenario is mimicked in a simple model, where there exists a differential hopping along one of the three neighbors (say, ${t}_{1}$) compared with the other two (say, $t$). For ${t}_{1}=t$, the properties of the quantum spin Hall phase is described by the familiar Kane-Mele model, while $t<{t}_{1}<2t$ denotes a situation in which the spin-resolved bands are continuously deformed. ${t}_{1}=2t$ represents a special case which is called as the semi-Dirac limit. Here, the spectral gaps between the conduction and the valence bands vanish. A closer inspection of the properties of such a deformed system yields insights on a topological phase transition occurring at the semi-Dirac limit, which continues to behave as a band insulator for ${t}_{1}>2t$. We demonstrate the evolution of the topological phase in presence of the Rashba and intrinsic spin-orbit couplings via computing the electronic band structure, edge modes in a nanoribbon and the ${\mathbb{Z}}_{2}$ invariant. The latter aids in arriving at the phase diagram which conclusively shows vanishing of the topological phase in the semi-Dirac limit. Furthermore, we demonstrate in gradual narrowing down of the plateau in the spin Hall conductivity, which along with a phase diagram provide robust support on the vanishing of the ${\mathbb{Z}}_{2}$ invariant and hence the quantum spin Hall phase.

Twisted nodal wires and three-dimensional quantum spin Hall effect in distorted square-net compounds

Recently, square-net materials have attracted lots of attention for the Dirac semimetal phase with negligible spin-orbit coupling (SOC) gap, e.g., ZrSiS/LaSbTe and ${\mathrm{CaMnSb}}_{2}$. In this paper, we demonstrate that the Jahn-Teller effect enlarges the nontrivial SOC gap in the distorted structure, e.g., LaAsS and ${\mathrm{SrZnSb}}_{2}$. Its distorted $X$ square-net layer ($X=$ P, As, Sb, Bi) resembles a quantum spin Hall (QSH) insulator. Since these QSH layers are simply stacked in the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}$ direction and weakly coupled, three-dimensional QSH effect can be expected in these distorted materials, such as insulating compounds ${\mathrm{CeAs}}_{1+x}{\mathrm{Se}}_{1\ensuremath{-}y}$ and ${\mathrm{EuCdSb}}_{2}$. Our detailed calculations show that it hosts two twisted nodal wires without SOC [each consists of two noncontractible time-reversal symmetry- and inversion symmetry-protected nodal lines touching at a fourfold degenerate point], while with SOC it becomes a topological crystalline insulator with symmetry indicators $(000;2)$ and mirror Chern numbers $(0,0)$. The nontrivial band topology is characterized by a generalized spin Chern number ${C}_{s+}=2$ when there is a gap between two sets of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{s}}_{x}$ eigenvalues. The nontrivial topology of these materials can be well reproduced by our tight-binding model and the calculated spin Hall conductivity is quantized to ${\ensuremath{\sigma}}_{yz}^{x}=(\frac{\ensuremath{\hbar}}{e})\frac{{G}_{x}{e}^{2}}{\ensuremath{\pi}h}$ with ${G}_{x}$ a reciprocal lattice vector.

Spin Hall conductivity of germanene supported by monolayer of different monochalcogenides and emergence of topologically insulating states

Similar to graphene, two-dimensional germanene possesses a honeycomb structure and Dirac cone in its band structure. Being inspired by these similarities, we have studied the electronic structure as well as spin Hall conductivity of germanene supported by monolayers of four monochalcogenides, namely, GaS, GaSe, GaTe, and InSe. By using the first principle calculations based on density functional theory we have investigated the structural stability, buckling of germanene, and electronic band structures of these systems with and without spin–orbit interactions. All the four systems possess Dirac cone in their band structures, but germanene on GaTe and InSe substrates become nonmetallic with very small band gaps of ∼ 28–70 meV. He calculated spin Hall conductivity and the nature of the edge states drive us to infer that germanene, supported by monolayered GaTe or InSe, is a spin Hall insulator with a nontrivial bandgap.

Superconducting properties of the spin Hall candidate Ta3Sb with eightfold degeneracy

We report the synthesis and characterization of phase pure Ta3Sb, a material predicted to be topological with eightfold degenerate fermionic states [Science 353, aaf5037 (2016)] and to exhibit a large spin Hall effect [Sci. Adv. 5, eaav8575 (2019]. We observe superconductivity in Ta3Sb with Tc~ 0.67 K in both electrical resistivity \r{ho}(T) and specific heat C(T) measurements. Field dependent measurements yield the superconducting phase diagram with an upper critical field of Hc2(0) ~ 0.95 T, corresponding to a superconducting coherence length of {\xi} ~18.6 nm. The gap ratio deduced from specific heat anomaly, 2{\Delta}0/kBTc is 3.46, a value close to the Bardeen-Cooper-Schrieffer (BCS) value of 3.53. From a detailed analysis of both the transport and thermodynamic data within the Ginsburg-Landau (GL) framework, a GL parameter of \k{appa} ~90 is obtained identifying Ta3Sb as an extreme type-II superconductor. The observation of superconductivity in an eightfold degenerate fermionic compound with topological surface states and predicted large spin Hall conductance positions Ta3Sb as an appealing platform to further explore exotic quantum states in multifold degenerate systems.

Spin accumulation without spin current

The spin Hall (SH) effect is a phenomenon in which the spin current flows perpendicular to an applied electric field and causes the spin accumulation at the boundaries. However, in the presence of spin-orbit couplings, the spin current is not well defined. Here, we calculate the spin response to an electric field gradient, which naturally appears at the boundaries. We derive a generic formula using the Bloch wave functions and the phenomenological relaxation time. We also calculate the response for the Rashba model with $\delta$-function nonmagnetic disorder within the first-order Born approximation and corresponding vertex corrections. We find the nonzero spin accumulation, although the SH conductivity exactly vanishes.

Transport and spin Hall conductivity in two-dimensional XY model on honeycomb lattice

Spin transport is studied in the frustrated two-dimensional XY model on honeycomb lattice. The calculations are performed taking into account next-nearest neighboring interactions J2 and single-ion anisotropy, using the SU(N) Schwinger boson approach. We consider the effect of next-nearest neighboring exchange interactions J2c and interlayer coupling J⊥ on continuum conductivity, where at critical line in the phase diagram, the system suffers a quantum phase transition between the Néel phase to paramagnetic phase. The spin Hall conductivity as a function of temperature, in the model with Dzyaloshinskii–Moriya interaction (DMI) is also analyzed.