The electric-field-induced resonant spin polarization of a two-dimensional hole gas described by Luttinger Hamiltonian with structural inversion asymmetry and Zeeman splitting in a perpendicular magnetic field was studied. The spin polarization arising from splitting between the light and the heavy hole bands shows a resonant peak at a certain magnetic field. Especially, the competition between the Luttinger term and the structural inversion asymmetry leads to a rich resonant peaks structure, and the required magnetic field for the resonance may be effectively reduced by enlarging the effective width of the quantum well. Furthermore, the Zeeman splitting tends to move the resonant spin polarization to a relative high magnetic field and destroy these rich resonant spin phenomena. Finally, both the height and the weight of the resonant peak increase as the temperature decreases. It is believed that such resonant spin phenomena can be verified in the sample of a two-dimensional hole gas, and it may provide an efficient way to control spin polarization by an external electric field.
Tunnelling density of states (DoS) in Luttinger liquid has a dip at zero energy, commonly known as the zero-bias anomaly (ZBA). In the presence of a magnetic field, in addition to the zero-bias anomaly, the DoS develops two peaks separated from the origin by the Zeeman energy. We find the shape of these peaks at arbitrary strength of the electron-electron interaction. The developed theory is applicable to various kinds of quantum wires, including carbon nanotubes.
We study a one-dimensional interacting electron gas with the strong Rashba spin-orbit coupling and Zeeman splitting in a quantum well. A bosonization theory is developed for this system. The tunneling current may deviate from a simple power law which is that in an ordinary Luttinger liquid. The microscopic interacting coupling and the spin-orbital parameter may be measured by varying the external magnetic field in the tunneling experiment.
We study the transport properties for a Luttinger-liquid (LL) quantum wire in the presence of both Rashba spin-orbit coupling (SOC) and a weak external in-plane magnetic field. The bosonized Hamiltonian of the system with an externally applied longitudinal electric field is established. And then the equations of motion for the bosonic phase fields are solved in the Fourier space, with which the both charge and spin conductivities for the system are calculated analytically based on the linear response theory. Generally, the ac conductivity is an oscillation function of the strengths of electron…
The Luttinger model of a quadratic-node semimetal for electrons with the $j=\frac32$ angular momentum under cubic symmetry is the parent, highest-symmetry low-energy model for a variety of topological and strongly correlated materials, such as HgTe, $α$-Sn, and iridate compounds. Previously, we have theoretically demonstrated that the Luttinger semimetal exhibits surface states. In the present work, we theoretically study the evolution of these surface states under symmetry-lowering perturbations: compressive strain and bulk-inversion asymmetry (BIA). This system is quite special in that each …
Zeeman splitting of quantum-confined states of excitons in InGaAs quantum wells (QWs) is experimentally found to depend strongly on quantization energy. Moreover, it changes sign when the quantization energy increases with a decrease in the QW width. In the 87-nm QW, the sign change is observed for the excited quantum-confined states, which are above the ground state only by a few meV. A two-step approach for the numerical solution of the two-particle Schroedinger equation, taking into account the Coulomb interaction and valence-band coupling, is used for a theoretical justification of the obs…
Luttinger holes with strong Zeemann and spin-orbit interactions in a wire proximity-coupled to a superconductor is a promising system for observation of Majorana fermions. Earlier treatments of confined Luttinger holes in wires ignored a mutual transformation of heavy and light holes upon reflection from the heteroboundaries. This effect is crucial for Zeemann and spin-orbit coupling in the ground subband of holes with several spin-orbit terms linear in momentum. We discuss the criterion for realizing Majorana modes in charge carrier hole systems. GaAs or InSb hole wires shall exhibit stronger…
We study theoretically the interplay effect of Zeeman field and modulated spin-orbit coupling on topological properties of a one-dimensional dimerized lattice, known as Su-Schrieffer-Heeger model. We find that in the weak (strong) modulated spin-orbit coupling regime, trivial regions or nontrivial ones with two pairs of zero-energy states can be turned into nontrivial regions by applying a uniform (staggered) perpendicular Zeeman field through a topological phase transition. Furthermore, the resulting nontrivial phase hosting a pair of zero-energy boundary states can survive within a certain r…
For a two-dimensional (2D) hole system (confined within Ge layers of a multilayered p-Ge/Ge_{1-x}Si_x heterostructure) described by Luttinger Hamiltonian with the g-factor highly anisotropic for orientations of magnetic field perpendicular and parallel to the 2D plane (g_perp >> g_par), reported is an observation of low-temperature transition from metallic (dR/dT > 0) to insulator (dR/dT < 0) behavior of resistivity R(T) induced by a perpendicular magnetic field B. The revealed positive magnetoresistance scales as a function of B/T. We attribute this finding to a suppression of the triplet cha…
Following a previous work [Shen, Ma, Xie and Zhang, Phys. Rev. Lett. 92, 256603 (2004)] on the resonant spin Hall effect, we present detailed calculations of the spin Hall conductance in two-dimensional quantum wells in a strong perpendicular magnetic field. The Rashba coupling, generated by spin-orbit interaction in wells lacking bulk inversion symmetry, introduces a degeneracy of Zeeman-split Landau levels at certain magnetic fields. This degeneracy, if occuring at the Fermi energy, will induce a resonance in the spin Hall conductance below a characteristic temperature of order of the Zeeman…
In few-layer (FL) transition metal dichalcogenides (TMDC), the conduction bands along the Gamma-K directions shift downward energetically in the presence of interlayer interactions, forming six Q valleys related by three-fold rotational symmetry and time reversal symmetry. In even-layers the extra inversion symmetry requires all states to be Kramers degenerate, whereas in odd-layers the intrinsic inversion asymmetry dictates the Q valleys to be spin-valley coupled. In this Letter, we report the transport characterization of prominent Shubnikov-de Hass (SdH) oscillations for the Q valley electr…
The theory for light-hole Zeeman splitting developed in [Physica E 44, 797 (2012)] is compared with experimental data found in literature for GaAs/AlGaAs, InGaAs/InP and CdTe/CdMgTe quantum wells. It is shown that the description of experiments is possible with account for excitonic effects and peculiarities of the hole energy spectrum in a quantum well including complex structure of the valence band and the interface mixing of light and heavy holes. It is demonstrated that the absolute values and the sign of the light-hole $g$-factor are extremely sensitive to the parametrization of the Lutti…
We investigate a Zeeman-split quantum dot (QD) containing a single spin 1/2 weakly coupled to a helical Luttinger liquid (HLL) within a generalized master equation approach. The HLL induces a tunable magnetization direction on the QD controlled by an applied bias voltage when the quantization axes of the QD and the HLL are noncollinear. The backscattering conductance (BSC) in the HLL is finite and shows a resonance feature when the bias voltage equals the Zeeman energy in magnitude. The observed BSC asymmetry in bias voltage directly reflects the quantization axis of the HLL spin.
We investigate spin-charge mixing effect on resonant tunneling in spin-polarized Tomonaga-Luttinger liquid with double impurities. The mixing arises from Fermi velocity difference between two spin species due to Zeeman effect. Zero bias conductance is calculated as a function of gate voltage $V_{\rm g}$, gate magnetic field $B_{\rm g}$, temperature and magnetic field applied to the system. Mixing effect is shown to cause rotation of the lattice pattern of the conductance peaks in $(V_{\rm g},B_{\rm g})$ plane, which can be observed in experiments. At low temperatures, the contour shapes are cl…
We have developed a theory of the longitudinal $g$ factor of light holes in semiconductor quantum wells. It is shown that the absolute value of the light-hole $g$-factor can strongly exceed its value in the bulk and, moreover, the dependence of the Zeeman splitting on magnetic field becomes non-linear in relatively low fields. These effects are determined by the proximity of the ground light-hole subband, $lh1$, to the first excited heavy-hole subband, $hh2$, in GaAs/AlGaAs-type structures. The particular calculations are performed in the framework of Luttinger Hamiltonian taking into account …
The Zeeman splitting of the conduction band in the HgTe quantum wells both with normal and inverted spectrum has been studied experimentally in a wide electron density range. The simultaneous analysis of the SdH oscillations in low magnetic fields at different tilt angles and of the shape of the oscillations in moderate magnetic fields gives a possibility to find the ratio of the Zeeman splitting to the orbital one and anisotropy of g-factor. It is shown that the ratios of the Zeeman splitting to the orbital one are close to each other for both types of structures, with a normal and inverted s…
We present a theory of the electron structure and the Zeeman effect for the helical edge states emerging in two-dimensional topological insulators based on HgTe/HgCdTe quantum wells with strong natural interface inversion asymmetry. The interface inversion asymmetry, reflecting the real atomistic structure of the quantum well, drastically modifies both bulk and edge states. For the in-plane magnetic field, this asymmetry leads to a strong anisotropy of the edge-state effective $g$-factor which becomes dependent on the edge orientation. The interface inversion asymmetry also couples the counter…
The phenomenon originating from spin-orbit coupling (SOC) provides energy-efficient strategies for spin manipulation and device applications. The broken inversion symmetry interface and resulting electric field induce a Rashba-type spin-orbit field (SOF), which has been demonstrated to generate spin-orbit torque for data storage applications. In this study, we found that spin flipping can be achieved by the valley-Zeeman SOF in monolayer WSe2 at room temperature, which manifests as a negative magnetoresistance in the vertical spin valve. Quantum transmission calculations based on an effective …
The inversion of a spin valve device is proposed. Opposite to a conventional spin valve of a non-magnetic spacer sandwiched between two ferromagnetic metals, an inverse spin valve is a ferromagnet sandwiched between two non-magnetic metals. It is predicted that, under a bias, the chemical potentials of spin-up and spin-down electrons in the metals split at metal-ferromagnet interfaces, a dynamical Zeeman effect. This split is of the order of an applied bias. Thus, there should be no problem of generating an $eV$ split that is not possible to be realized on the earth by the usual Zeeman effect.
Atomically thin transition metal dichalcogenides (TMDs) possess coupling of spin and valley degrees of freedom, making them promising for spin-valleytronics. ln monolayer TMDs, the emission helicity is locked to the valleys as a consequence of spin-orbit coupling and inversion symmetry breaking, leading to a valley analog of Zeeman effect in presence of out-of-plane magnetic field. As inversion symmetry is recovered in bilayers, the emission helicity should no longer be locked to the valleys. Here we show that Zeeman splitting, however, still persists in bilayers, as a result of an additional …
A monolayer of a transition metal dichalcogenide (TMD) such as WSe$_2$ is a two-dimensional (2D) direct band-gap valley-semiconductor having an effective Honeycomb lattice structure with broken inversion symmetry. The inequivalent valleys in the Brillouin zone could be selectively addressed using circularly-polarized light fields, suggesting the possibility for magneto-optical measurement and manipulation of the valley pseudospin degree of freedom. Here we report such experiments that demonstrate the valley Zeeman effect -- strongly anisotropic lifting of the degeneracy of the valley pseudospi…
We analyze theoretically two different setups for s-wave superconductivity (SC) proximitized $j=3/2$ particles in Luttinger materials that are able to host Majorana bound states (MBSs). First, we consider a one-dimensional SC wire with intrinsic bulk inversion asymmetry (BIA). In contrast to wires, modeled by a quadratic dispersion with Rashba spin-orbit coupling, there are two topological phase transitions in our systems at finite magnetic fields. Second, we analyze a two-dimensional Josephson junction on the Luttinger model finding a topological region even in the absence of BIA and Rashba s…
We consider the charge and spin effects in low dimensional superconducting weak links. The first part of the review deals with the effects of electron-electron interaction in Superconductor/Luttinger liquid/Superconductor junctions. The experimental realization of this mesoscopic hybrid system can be the individual single wall carbon nanotube that bridges the gap between two bulk superconductors. The dc Josephson current through a Luttinger liquid in the limits of perfectly and poorly transmitting junctions is evaluated. The relationship between the Josephson effect in a long SNS junction and …
We propose a straightforward and effective approach for quantifying the band inversion induced by spin-orbit coupling in band insulators. In this approach we define a quantity as a function of wavevector in the Brillouin zone measuring the spillage between the occupied states of a system with and without SOC. Plots of the spillage throughout the BZ provide a ready indication of the number and location of band inversions driven by SOC. To illustrate the method, we apply this approach to the 2D Kane-Mele model, a 2D Bi bilayer with applied Zeeman field, and to first-principles calculations of so…
The valley of transition metal dichalcogenides provides an additional platform to manipulate spin due to its unique selection rule. Normally, intralayer optical transitions in magnetic field show a Zeeman splitting with g factor of about $-4$. Here we report remarkable valley Zeeman effect exhibited by splitting of excitonic emission in a bilayer WS$_{2}$, with a value of g factor as large as $-16.5$. The observed large g factor results from the interlayer recombination, as the conduction band and valence band are modified in opposite directions by magnetic field in different layers. The inter…
This paper presents a theoretical description of both the valley Zeeman effect (g-factors) and Landau levels in two-dimensional H-phase transition metal dichalcogenides (TMDs) using the Luttinger-Kohn approximation with spin-orbit coupling. At the valley extrema in TMDs, energy bands split into Landau levels with a Zeeman shift in the presence of a uniform out-of-plane external magnetic field. The Landau level indices are symmetric in the $K$ and $K'$ valleys. We develop a numerical approach to compute the single band g-factors from first principles without the need for a sum over unoccupied b…
The electronic excitations at the edges of a Hall bar not much wider than a few magnetic lengths are studied theoretically at filling $ν= 2$. Both mean-field theory and Luttinger liquid theory techniques are employed for the case of a null Zeeman energy splitting. The first calculation yields a stable spin-density wave state along the bar, while the second one predicts dominant Wigner-crystal correlations along the edges of the bar. We propose an antiferromagnetic Wigner-crystal groundstate for the edge electrons that reconciles the two results. A net Zeeman splitting is found to produce canti…
We analyze the effect of magnetic fluctuations in superconductors with strong spin-orbit coupling and show that they drive a phase transition between two superconducting states: a conventional phase with zero center-of-mass momentum of Cooper pairs, and an exotic phase with non-zero pair momentum. The latter is found to exhibit persistent currents without magnetic field in doubly connected geometries such as rings. Surprisingly, the transition temperature into the superconducting state can be increased by applying a Zeeman magnetic field.
In honeycomb Dirac systems with broken inversion symmetry, orbital magnetic moments coupled to the valley degree of freedom arise due to the topology of the band structure, leading to valley-selective optical dichroism. On the other hand, in Dirac systems with prominent spin-orbit coupling, similar orbital magnetic moments emerge as well. These moments are coupled to spin, but otherwise have the same functional form as the moments stemming from spatial inversion breaking. After reviewing the basic properties of these moments, which are relevant for a whole set of newly discovered materials, su…
We have studied theoretically the effect of a tuneable lateral confinement on two-dimensional hole systems realised in III-V semiconductor heterostructures. Based on the 4x4 Luttinger description of the valence band, we have calculated quasi-onedimensional (quasi-1D) hole subband energies and anisotropic Lande g-factors. Confinement-induced band mixing results in the possibility to manipulate electronic and spin properties of quasi-1D hole states over a much wider range than is typically possible for confined conduction-band electrons. Our results are relevant for recent experiments where sour…
We study intrinsic spin Hall effect in p-type GaAs quantum well structure described by Luttinger Hamiltonian and a Rashba spin-orbit coupling arising from the structural inversion symmetry breaking. The Rashba term induces an energy level crossing in the lowest heavy hole subband, which gives rise to a resonant spin Hall conductance. The resonance may be used to identify the intrinsic spin Hall effect in experiments.
We report experimental evidence of excitonic spin-splitting, in addition to the conventional Zeeman effect, produced by a combination of the Rashba spin-orbit interaction, Stark shift and charge screening. The electric-field-induced modulation of the spin-splitting are studied during the charging and discharging processes of p-type GaAs/AlAs double barrier resonant tunneling diodes (RTD) under applied bias and magnetic field. The abrupt changes in the photoluminescence, with the applied bias, provide information of the charge accumulation effects on the device.
Many modern spin-based devices rely on the spin-orbit interaction, which is highly sensitive to the host semiconductor heterostructure and varies substantially depending on crystal direction, crystal asymmetry (Dresselhaus), and quantum confinement asymmetry (Rashba). One-dimensional quantum point contacts are a powerful tool to probe both energy and directional dependence of spin-orbit interaction through the effect on the hole $g$-factor. In this work we investigate the role of cubic crystal asymmetry in driving an oscillation in the in-plane hole $g$-factor anisotropy when the quantum point…
The charge and spin patterns of a quantum dot embedded into a spin-orbit coupled quantum wire subject to a magnetic field are investigated. A Luttinger liquid theory is developed, taking into account open boundaries and finite magnetic field. In the quasi-helical regime, when spin-orbit effects dominate over the Zeeman interaction, peculiar states develop at the Fermi surface of the dot. Anomalous Friedel oscillations with twice the expected wavelength develop in the wavefunction of collective excitations of such states, accompanied by peculiar spin patterns in their magnetization. Both effect…
The ratio of the Zeeman splitting to the cyclotron energy ($M=ΔE_Z / \hbar ω_c$), which characterizes the relative strength of the spin-orbit interaction in crystals, is examined for the narrow gap IV-VI semiconductors PbTe, SnTe, and their alloy Pb$_{1-x}$Sn$_x$Te on the basis of the multiband $k\cdot p$ theory. The inverse mass $α$, the g-factor $g$, and $M$ are calculated numerically by employing the relativistic empirical tight-binding band calculation. On the other hand, a simple but exact formula of $M$ is obtained for the six-band model based on the group theoretical analysis. It is sho…
We theoretically propose a method of rectifying spin current with a linearly-polarized electromagnetic wave in inversion-asymmetric magnetic insulators. To demonstrate the proposal, we consider quantum spin chains as a simple example; these models are mapped to fermion (spinon) models via Jordan-Wigner transformation. Using a nonlinear response theory, we find that a dc spin current is generated by the linearly-polarized waves. The spin current shows rich anisotropic behavior depending on the direction of the electromagnetic wave. This is a manifestation of the rich interplay between spins and…
The coupling of electron orbital motion and spin leads to nontrivial changes in energy-level structures, leading to various spectroscopies and applications. In atoms, such spin-orbit coupling (SOC) causes anomalous Zeeman splitting, known as the Paschen-Back (PB) effect, in the pres-ence of a strong magnetic field. In solids, SOC generates energy-band inversion or splitting, a prerequisite for topological phases or Majorana fermions, at zero or weak magnetic fields. Here, we present the first observation of PB splitting of Landau levels (LLs) in indium arsenide nan-owires in a strong-field reg…
SnTe materials are one of the most flexible material platforms for exploring the interplay of topology and different types of symmetry breaking. We study symmetry-protected topological states in SnTe nanowires in the presence of various combinations of Zeeman field, s-wave superconductivity and inversion-symmetry-breaking field. We uncover the origin of robust corner states and hinge states in the normal state. In the presence of superconductivity, we find inversion-symmetry-protected gapless bulk Majorana modes, which give rise to quantized thermal conductance in ballistic wires. By introduci…
Since the lattices of ABA-stacked graphene multilayers with an even number of layers, as well as that of monolayer graphene, satisfy spatial-inversion symmetry, their electronic bands must be spin degenerate in the presence of time-inversion symmetry. In intrinsic monolayer and bilayer graphene, when symmetry is not broken by external fields, the only spin-orbit coupling present at low energy near the corner of the Brillouin zone is the Kane-Mele term, that opens a bulk energy gap but does not break the spin degeneracy of the energy bands [C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801…
We show that in the metallic phase of a two dimensional electron gas the spin-orbit coupling due to structure inversion asymmetry leads to a characteristic anisotropy in the magnetoconductance. Within the assumption that the metallic phase can be described by a Fermi liquid, we compute the conductivity in the presence of an in-plane magnetic field. Both the spin-orbit coupling and the Zeeman coupling with the magnetic field give rise to two spin subbands, in terms of which most of the transport properties can be discussed. The strongest conductivity anisotropy occurs for Zeeman energies of the…
We describe here recent work on the electronic properties, magnetoexcitons and valley polarised electron gas in 2D crystals. Among 2D crystals, monolayer $MoS_2$ has attracted significant attention as a direct-gap 2D semiconductor analogue of graphene. The crystal structure of monolayer $MoS_2$ breaks inversion symmetry and results in K valley selection rules allowing to address individual valleys optically. Additionally, the band nesting near Q points is responsible for enhancing the optical response of $MoS_2$.We show that at low energies the electronic structure of $MoS_2$ is well approxima…
The charge and spin Hall conductance of the two-dimensional hole gas within the Luttinger model with and without inversion symmetry breaking terms in a perpendicular magnetic field are studied, and two key phenomena are predicted. The sign of the spin Hall conductance is modulated periodically by the external magnetic field, which means a possible application in the future. Furthermore, a resonant spin Hall conductance in the two-dimensional hole gas with a certain hole density at a typical magnetic field is indicated, which implies a likely way to firmly establish the intrinsic spin Hall effe…
We identify a series of topological transitions occurring in electronic spin transport when manipulating spin-guiding fields controlled by the geometric shape of mesoscopic interferometers. They manifest as distinct inversions of the interference pattern in quantum conductance experiments. We establish that Rashba square loops develop weak-(anti)localization transitions (absent in other geometries as Rashba ring loops) as an in-plane Zeeman field is applied. These transitions, boosted by non-adiabatic spin scattering, prove to have a topological interpretation in terms of winding numbers chara…
The surface of a higher order topological insulator (HOTI) comprises a two-dimensional topological insulator (TI) with broken inversion symmetry, whose mass is determined by the microscopic details of the surface such as surface potentials and termination. It hosts a helical mode pinned to selected hinges where the surface gap changes its sign. We study the effect of perturbations that break time-reversal and particle-conservation on this helical mode, such as a Zeeman field and a proximate superconductor. We find that in contrast to the helical modes of inversion symmetric TIs, which are gapp…
Skyrmions are spatially localized magnetic swirls which carry a nonzero integer topological charge. We study crystals of skyrmions in a two-dimensional ferromagnet model with chiral interactions induced by the presence of broken inversion symmetry. We show that non-linear quartic mode-coupling terms allowed by symmetry enhances the Zeeman-field range over which the skyrmion-crystal phase remains stable. Furthermore, it leads to a significant dependence of the lattice constant of this spin crystal over this wide field range. Conduction electrons coupled to such a tunable spin crystal are shown …
Novel 2D material CrI3 reveals unique combination of 2D ferromagnetism and robust excitonic response. We demonstrate that the possibility of the formation of magnetic topological defects, such as Neel skyrmions, together with large excitonic Zeeman splitting, leads to giant scattering asymmetry, which is the necessary prerequisite for the excitonic anomalous Hall effect. In addition, the diamagnetic effect breaks the inversion symmetry, and in certain cases can result in exciton localization on the skyrmion. This enables the formation of magnetoexcitonic quantum dots with tunable parameters.
The photon-assisted tunneling (PAT) through a single wall carbon nanotube quantum well (QW) under influence an external electromagnetic field for probing of the Tomonaga Luttinger liquid (TLL) state is suggested. The elementary TLL excitations inside the quantum well are density ($ρ_{\pm}$) and spin ($σ_{\pm} $) bosons. The bosons populate the quantized energy levels $ε^{ρ+}_n =Δn/ g$ and $ε^{ρ-(σ\pm)}_n = Δn$ where $Δ= h v_F /L $ is the interlevel spacing, $n$ is an integer number, $L$ is the tube length, $g$ is the TLL parameter. Since the electromagnetic field acts on the $ρ_{+}$ bosons onl…
We study the dynamical dielectric function of a two-dimensional hole gas, exemplified on [001]-GaAs and InAs quantum wells, within the Luttinger model extended to the two lowest subbands including bulk and structure inversion asymmetric terms. The plasmon dispersion shows a pronounced anisotropy for GaAs- and InAs-based systems. In GaAs this leads to a suppression of plasmons due to Landau damping in some orientations. Due to the large Rashba contribution in InAs, the lifetime of plasmons can be controlled by changing the electric field. This effect is potentially useful in plasmon field effec…
