Spin Hall Conductivity Research Articles

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.

Giant Spin Hall Effect and Spin-Orbit Torques in 5d Transition Metal-Aluminum Alloys from Extrinsic Scattering.

The generation of spin currents from charge currents via the spin Hall effect (SHE) is of fundamental and technological interest. Here, some of the largest SHEs yet observed via extrinsic scattering are found in a large class of binary compounds formed from a 5d element and aluminum, with a giant spin Hall angle (SHA) of ≈1 in the compound Os22 Al78 . A critical composition of the 5d element is found at which there is a structural phase boundary between poorly and highly textured crystalline material, where the SHA exhibits its largest value. Furthermore, a systematic increase is found in the spin Hall conductivity (SHC) and SHA at this critical composition as the atomic number of the 5d element is systematically increased. This clearly shows that the SHE and SHC are derived from extrinsic scattering mechanisms related to the potential mismatch between the 5d element and Al. These studies show the importance of extrinsic mechanisms derived from potential mismatch as a route to obtaining large spin Hall angles with high technological impact. Indeed, it is demonstrated that a state-of-the-art racetrack device has a several-fold increased current-induced domain wall efficiency using these materials as compared to prior-art materials.

Spin Hall conductivity and anomalous Hall conductivity in full Heusler compounds

The spin Hall conductivity (SHC) and anomalous Hall conductivity (AHC) in about 120 full Heusler compounds are calculated using the density functional theory in a high-throughput way. The electronic structures are mapped to the Wannier basis and the linear response theory is used to get the conductivity. Our results show that the mechanism under the SHC or AHC cannot be simply related to the valence electron numbers or atomic weights. It is related to the very details of the electronic structures, which can only be obtained by calculations. A high-throughput calculation is efficient to screen out the desired materials. According to our present results, Rh2MnAl and Cu2CoSn, as well as Co2MnAl and Co2MnGa are candidates in spintronic materials regarding their high SHC and AHC values, which can benefit the spin-torque-driven nanodevices.

Momentum-Space Spin Antivortex and Spin Transport in Monolayer Pb.

Nontrivial momentum-space spin texture of electrons can be induced by spin-orbit coupling and underpins various spin transport phenomena, such as current-induced spin polarization and the spin Hall effect. In this work, we find a nontrivial spin texture, spin antivortex, can appear at certain momenta on the Γ-K line in a 2D monolayer Pb on top of SiC. Different from spin vortex due to the band degeneracy in the Rashba model, the existence of this spin antivortex is guaranteed by the Poincaré-Hopf theorem and thus topologically stable. Accompanied with this spin antivortex, a Lifshitz transition of Fermi surfaces occurs at certain momenta on the K-M line, and both phenomena are originated from the anticrossing between the j=1/2 and j=3/2 bands. A rapid variation of the response coefficients for both the current-induced spin polarization and spin Hall conductivity is found when the Fermi energy is tuned around the spin antivortex. Our work demonstrates the monolayer Pb as a potentially appealing platform for spintronic applications.

Regulating Interfacial Spin Hall Conductivity with Ferroelectricity.

We report a new physical phenomenon of active and enhanced control of the spin Hall conductivity (SHC) in a family of metal-ferroelectric multilayers. Rather than the direction of the built-in electric field, such ferroelectric regulation of SHC originates from the drastic change of the interfacial electronic state along with its intrinsic Berry phase near the Fermi level due to the distinct hybridization between the metal film and substrate when the ferroelectric polarization reverses. Using Pt/PbZrTiO3 multilayers as a representative model, we demonstrate the controllability of a large magnitude and even the sign of SHC in the Pt film at the two interfaces with an antitype conducting carrier in a ferroelectric substrate. The interfacial Rashba effect plays a role in contributing to the change of SHC through spin-projected band analysis. The present work makes a fundamental theoretical discovery and opens up a new direction to manipulate spin-charge conversion of thin-film layered structures by ferroelectricity, which is crucial for designing future electric and spintronic devices.

Tunable spin–charge conversion in class-I topological Dirac semimetals

We theoretically demonstrate that class-I topological Dirac semimetals (TDSMs) can provide a platform for realizing both electrically and magnetically tunable spin–charge conversion. With time-reversal symmetry, the spin component along the uniaxial rotation axis (z axis) is approximately conserved, which leads to an anisotropic spin Hall effect; the resulting spin Hall current relies on the relative orientation between the external electric field and the z axis. The application of a magnetic field, on the other hand, breaks time-reversal symmetry, driving the TDSM into a Weyl semimetal phase and, consequently, partially converting the spin current to a charge Hall current. Using the Kubo formulas, we numerically evaluate the spin and charge Hall conductivities based on a low-energy TDSM Hamiltonian together with the Zeeman coupling. Besides the conventional tensor element of the spin Hall conductivity σxyz, we find that unconventional components, such as σxyx and σxyy, also exist and vary as the magnetic field is rotated. Likewise, the charge Hall conductivity also exhibits appreciable tunability upon variation of the magnetic field. We show that such tunability—as well as large spin–charge conversion efficiency—arises from the interplay of symmetry and band topology of the TDSMs.

Nonlinear optical Hall effect of few-layered NbSe2

${\mathrm{NbSe}}_{2}$ is one of metallic two-dimensional (2D) transition-metal dichalcogenide (TMDC) materials. Because of broken crystal inversion symmetry, large spin splitting is induced by Ising-type spin-orbit coupling in odd-number-layered ${\mathrm{NbSe}}_{2}$, but absent for even-number-layered ${\mathrm{NbSe}}_{2}$ with the inversion symmetry. In this paper we numerically calculate nonlinear optical charge and spin Hall conductivities of few-layered ${\mathrm{NbSe}}_{2}$ based on an effective tight-binding model which includes ${d}_{{z}^{2}}$, ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$, and ${d}_{xy}$ orbitals of Nb atoms. We show that the nonlinear optical Hall conductivity for the second harmonic generation (SHG) process has a nonvanishing value in odd-number-layered ${\mathrm{NbSe}}_{2}$. Also, we provide a nonlinear optical selection rule in few-layered ${\mathrm{NbSe}}_{2}$ and their polarization dependencies. Furthermore, for the even-number-layered case, the nonlinear optical Hall currents can be generated by applying electric fields which break inversion symmetry. We also discuss that the nonlinear optical Hall effect is expected to occur in TMDC materials in general. Thus, our results will serve to design potential opt-spintronics devices based on 2D materials to generate the spin Hall current by SHG.

Global and local topological quantized responses from geometry, light, and time

To describe a spin-$\frac{1}{2}$ particle on the Bloch sphere with a radial magnetic field and topological states of matter from the reciprocal space, we introduce $C$ square (${C}^{2}$) as a local formulation of the global topological invariant. For the Haldane model on the honeycomb lattice, this ${C}^{2}$ can be measured from the Dirac points through circularly polarized light related to the high-symmetry $M$ point(s). For the quantum spin Hall effect and the Kane-Mele model, the ${\mathbb{Z}}_{2}$ topological number robust to interactions can be measured locally from a correspondence between the Pfaffian and light. We address a relation with a spin pump and the quantum spin Hall conductance. The analogy between light and magnetic nuclear resonance may be applied for imaging, among other applications.

Edge spin transport in the disordered two-dimensional topological insulator WTe2

The spin conductance of two-dimensional topological insulators (2D TIs) is not expected to be quantized in the presence of perturbations that break the spin-rotational symmetry. However, the deviation from the pristine-limit quantization has yet to be studied in detail. In this paper, we define the spin current operator for the helical edge modes of a 2D TI and introduce a four-terminal setup to measure spin conductances. Using the developed formalism, we consider the effects of disorder terms that break spin-rotational symmetry or give rise to edge-to-edge coupling. We identify a key role played by spin torque in an out-of-equilibrium edge. We then utilize a tight-binding model of topological monolayer ${\mathrm{WTe}}_{2}$ and scattering matrix formalism to numerically study spin transport in a four-terminal 2D TI device. In particular, we calculate the spin conductances and characteristic spin decay length in the presence of magnetic disorder. In addition, we study the effects of interedge scattering in a quantum point contact geometry. We find that the spin Hall conductance is surprisingly robust to spin symmetry-breaking perturbations, as long as time-reversal symmetry is preserved and interedge scattering is weak.

Topological properties and optical conductivities tuned by spin-orbit coupling and strain in kagome lattices

The spin–orbit coupling (SOC) effect is the dominant origin of the topological properties in a topological insulator which can be induced and enhanced through various methods. Here we theoretically study the effects of varying SOC on the topological properties of two-dimensional kagome lattice based on the tight-binding approximation. We find that the system can undergo a transition between two non-trivial topological states by tuning the SOC strength. In addition, the topological phase transition can be achieved with a weaker SOC under a uniform tensile strain, which suggests that strain engineering can be used to regulate topological states. Besides, the characteristics of optical spin Hall conductivity are also studied, which supplies an experimental determination to reveal the different topological states. Furthermore, we propose an external mechanical strain modulated spin-switch device, which can generate opposite spin currents under stretch and compressed strain. Our results indicate that with the combination of SOC and strain, kagome lattice materials have potential applications on tunable spintronics and optoelectronic devices.

Anomalous Spin Transport Properties of Gapped Dirac Electrons with Tilting

The anomalous spin transport coefficients of gapped Dirac electrons are studied with application to a quasi-two-dimensional organic conductor $\alpha$-(BETS)$_2$I$_3$ in mind. In the presence of a gap induced by spin-orbit interaction, we show that the effective Hamiltonian is similar to the model considered by Kane and Mele with additional tilting. With this effective Hamiltonian, conductivity tensors up to the linear order of the applied magnetic field are obtained analytically using the microscopic linear response theory or Kubo formula. It is shown that spin Hall conductivity and anomalous diagonal spin conductivity proportional to the magnetic field become nonzero in this system, which are written in terms of the Berry curvature and orbital magnetic moment.The estimated values of spin conductivities using typical parameters turn out to be comparable to the spin Hall conductivity in Pt.