Within the effective-mass approximation, we theoretically investigated the electronic and transport properties of 2D semiconductor quantum wires (QWs) with anisotropic effective masses and different orientations with respect to the anisotropic axis. The energy levels in the absence and presence of an external magnetic field are analytically calculated, showing: (i) a strong dependence on the spacing of energy levels related to the alignment QW angle and the anisotropy axis; and (ii) for non-null magnetic field, the quantum Hall edge states are significantly affected by the edge orientation. Moreover, by means of the split-operator technique, we analyzed the time evolution of wavepackets in straight and V-shaped anisotropic QWs and compared the transmission probabilities with those of isotropic systems. In the anisotropic case we found damped oscillations in the average values of velocity in both x and y directions for a symmetric Gaussian wavepacket propagating along a straight wide QW, with the oscillation being more evident as the non-collinearity between the group velocity and momentum vectors increases.
We report on a photoconductive gain in semiconductor quantum wires which are lithographically defined in an AlGaAs/GaAs quantum well via a shallow-etch technique. The effect allows resolving the one-dimensional subbands of the quantum wires as maxima in the photoresponse across the quantum wires. We interpret the results by optically induced holes in the valence band of the quantum well which shift the one-dimensional subbands of the quantum wire. Here we demonstrate that the effect persists up to a temperature of about 17 Kelvin.
We investigate the impact of electrostatics on the proximity effect between a magnetic insulator and a semiconductor wire in semiconductor-superconductor-magnetic insulator hybrid structures. By performing self-consistent Schr$\ddot{\rm o}$dinger-Poisson calculations using an effective model of the hybrid system, we find that large effective Zeeman fields consistent with the emergence of topological superconductivity emerge within a large parameter window in wires with overlapping layers of magnetic insulator and superconductor, but not in non-overlapping structures. We show that this behavior…
We investigate the spin accumulation effect in eccentric semiconductor multichannel rings with Rashba spin-orbit interaction and threaded by a magnetic flux. Due to the finite eccentricity, the spin polarization induced at the borders of the sample is anisotropic and exhibits different patterns and intensities at specific angular directions. This effect, reminiscent of the spin polarization drift induced by the application of an in plane electric field, could be used to manipulate and functionalize the spin polarization in electronic nanorings.
We study Majorana zero modes properties in cylindrical cross-section semiconductor quantum wires based on the $k \cdot p$ theory and a discretized lattice model. Within this model, the influence of disorder potentials in the wire and amplitude and phase fluctuations of the superconducting order-parameter are discussed. We find that for typical wire geometries, pairing potentials, and spin-orbit coupling strengths, coupling between quasi-one-dimensional sub-bands is weak, low-energy quasiparticles near the Fermi energy are nearly completely spin-polarized, and the number of electrons in the act…
We show that spin-flip rotation in a semiconductor quantum wire, caused by the Rashba and the Dresselhaus interactions (both of arbitrary strengths), can be suppressed by dint of an in-plane magnetic field. We found a new type of symmetry, which arises at a particular set of intensity and orientation of the magnetic field and explains this suppression. Based on our findings, we propose a transport experiment to measure the strengths of the Rashba and the Dresselhaus interactions.
In this paper, a detailed analysis of anisotropic effects on the phonon induced spin relaxation rate in III-V semiconductor quantum dots (QDs) is carried out. We show that the accidental degeneracy due to level crossing between the first and second excited states of opposite electron spin states in both isotropic and anisotropic QDs can be manipulated with the application of externally applied gate potentials. In particular, anisotropic gate potentials enhance the phonon mediated spin-flip rate and reduce the cusp-like structure to lower magnetic fields, in addition to the lower QDs radii in I…
We investigate the excitonic properties of a core-multishell semiconductor nanowire with type-II band mismatch, i.e. with spatially separated electrons and holes, under an external magnetic field. Our results demonstrate that, depending on the core wire radius, the carrier in the type-II band exhibits either a quantum dot-like or a quantum ring-like energy spectrum, corresponding to a carrier confinement in the core wire or in the outer shell, respectively. In the latter, a shell-to-core confinement transition can be induced by increasing the magnetic field intensity, which may lead to interes…
The problem of spin injection across the interface between a non-ideal ferromagnet (less than 100% spin polarization) and a semiconductor (paramagnetic) quantum wire is examined in the presence of Rashba spin orbit coupling and an axial magnetic field along the wire axis. The field is caused by the ferromagnet magnetized along the wire axis. At low temperatures and for certain injection energies, the interface can act as an ideal spin filter allowing injection only from the majority spin band of the ferromagnet. Thus, 100% spin filtering can take place even if the ferromagnet itself is less th…
We demonstrate in theory that it is possible to all-electrically manipulate the RKKY interaction in a quasi-one-dimensional electron gas embedded in a semiconductor heterostructure, in the presence of Rashba and Dresselhaus spin-orbit interaction. In an undoped semiconductor quantum wire where intermediate excitations are gapped, the interaction becomes the short-ranged Bloembergen-Rowland super-exchange interaction. Owing to the interplay of different types of spin-orbit interaction, the interaction can be controlled to realize various spin models, e.g., isotropic and anisotropic Heisenberg-l…
We develop an electron transport theory for the hybrid system of a semiconducting carbon nanotube that encapsulates a one-atom-thick metallic wire. The theory predicts Fano resonances in electron transport through the system, whereby the interaction of electrons on the wire with nanotube plasmon generated near-fields blocks some of the wire transmission channels to open up the new coherent plasmon-mediated channel in the nanotube forbidden gap outside the wire transmission band. Such a channel makes the entire hybrid system transparent in the energy domain where neither wire, nor nanotube is i…
A model of charge hopping transport that accounts for anisotropy of localized states and Coulomb interaction between charges is proposed. For the anisotropic localized states the degree of orientation relates exponentially to the ratio of conductivities in parallel and perpendicular directions, while the ratio of Seebeck coefficients stays nearly unaffected. However, the ratio of Seebeck coefficients increases if Coulomb interaction is screened stronger in a direction parallel to the predominant orientation of the localized states. This implies two different physical mechanisms responsible for…
Manipulation of Rashba effects in two-dimensional (2D) electron systems is highly desirable for controllable applications in spintronics and optoelectronics. Here, by combining first-principles investigation and model analysis, we use uniaxial stress to control BiTeI monolayer as a Rashba 2D semiconductor for useful spin and transport properties. We find that the stress-driven electron system can be described by an effective anisotropic Rashba model including all the three Pauli matrixes, and uniaxial stress allows an out-of-plane spin component. When appropriate electron carriers are introduc…
Recent numerical results for ionization of quantum dots by periodic electric field during the electric dipole spin resonance are compared with known analytical approaches. It is found that in finite-length quantum wires the numerical ionization rate is slower than the analytical one, especially for low driving field when the confinement potential strongly affects the dynamics. Still, the analytically predicted ionization times are in satisfactory correspondence with mean energy threshold times when the energy crosses the border between localized and continuum states.
We apply density functional theory, in the local density approximation, to a quasi-one-dimensional electron gas in order to quantify the effect of Coulomb and correlation effects in modulating, and therefore patterning, the charge density distribution. Our calculations are presented specifically for surface-gate-defined quasi-one-dimensional quantum wires in a GaAs-AlGaAs heterostructure but we expect our results to apply more generally for other low dimensional semiconductor systems. We show that at high densities with strong confinement, screening of electrons in the direction transverse to …
We study the exchange interaction between two hole-spin qubits in a double quantum dot setup in a silicon nanowire in the presence of magnetic and electric fields. Based on symmetry arguments we show that there exists an effective spin that is conserved even in highly anisotropic semiconductors, provided that the system has a twofold symmetry with respect to the direction of the applied magnetic field. This finding facilitates the definition of qubit basis states and simplifies the form of exchange interaction for two-qubit gates in coupled quantum dots. If the magnetic field is applied along …
Majorana fermions (MFs) are predicted to occur as zero-energy bound states in semiconductor nanowire-superconductor structures. However, in the presence of disorder or smooth confining potentials, these structures can also host non-topological nearly-zero energy states. Here, we demonstrate that the MFs and the nearly-zero topologically-trivial states have different characteristic signatures in a tunneling conductance measurement, which allows to clearly discriminate between them. We also show that low-energy non-topological states can strongly hybridize with metallic states from the leads, wh…
We investigate static charge and spin density modulation patterns along a ferromagnet/semiconductor single junction quantum wire in the presence of spin-orbit coupling. Coherent scattering theory is used to calculate the charge and spin densities in the ballistic regime. The observed oscillatory behavior is explained in terms of the symmetry of the charge and spin distributions of eigenstates in the semiconductor quantum wire. Also, we discuss the condition that these charge and spin density oscillations can be observed experimentally.
We report on the strong anisotropy of the inter-band process of impact ionization in direct-gap cubic semiconductors with either weak or strong spin-orbit coupling at low effective temperatures of electron distribution $T$, and the crossover to isotropic behavior with increasing $T$. Such anisotropy is related to specific mechanism of the impact ionization involving coupling of the electron and heavy hole states $\it via$ remote bands, which is vanishing for some high-symmetry propagation directions of an initial electron, namely $\mathrm{[100]}$ and $\mathrm{[111]}$. At room temperature impac…
We introduce the spin and momentum dependent {\em force operator} which is defined by the Hamiltonian of a {\em clean} semiconductor quantum wire with homogeneous Rashba spin-orbit (SO) coupling attached to two ideal (i.e., free of spin and charge interactions) leads. Its expectation value in the spin-polarized electronic wave packet injected through the leads explains why the center of the packet gets deflected in the transverse direction. Moreover, the corresponding {\em spin density} will be dragged along the transverse direction to generate an out-of-plane spin accumulation of opposite sig…
Electrostatic charges placed near the interface between ordinary and topological insulators induce magnetic fields, through the so-called topological magnetoelectric effect. Here, we present a numerical implementation of the associated Maxwell equations. The resulting model is simple, fast and quantitatively as accurate as the image charge method, but with the advantage of providing easy access to elaborate geometries when pursuing specific effects. The model is used to study how magnetoelectric fields are influenced by the dimensions and the shape of the most common semiconductor nanostructur…
I. Introduction (Preface, Basic properties of semiconductor nanostructures). II. Theory of Coulomb-blockade oscillations (Periodicity of the oscillations, Amplitude and lineshape). III. Experiments on Coulomb-blockade oscillations (Quantum dots, Disordered quantum wires, Relation to earlier work on disordered quantum wires). IV. Quantum Hall effect regime (The Aharonov-Bohm effect in a quantum dot, Coulomb blockade of the Aharonov-Bohm effect, Experiments on quantum dots, Experiments on disordered quantum wires).
Impact of exciton-phonon and exciton-magnon interaction on transport in anisotropic 2D magnetic semiconductors
Philipps-Universität Marburg（马尔堡菲利普斯大学）；mar.quest—Marburg Center for Quantum Materials and Sustainable Technologies（mar.quest——马尔堡量子材料与可持续技术中心）；University of Münster（明斯特大学）
The influence of the Rashba spin-orbit coupling on the electron spin dynamics is investigated for a ballistic semiconductor quantum wire with a finite width. We monitor the spin evolution using the time-dependent Schrödinger equation. The pure spin precession characteristic of the 1D limit is lost in a 2D wire with a finite lateral width. In general, the time evolution in the latter case is characterized by several frequencies and a nonrigid spin motion.
The dispersion law of plasmons running along thin wires with radius $a$ is known to be practically linear. We show that in a wire with a dielectric constant $κ$ much larger than that of its environment $κ_e$, such dispersion law crosses over to a dispersionless three-dimensional-like law when the plasmon wavelength becomes shorter than the length $(a/2) \sqrt{(κ/κ_e)\ln(κ/2κ_e)}$ at which the electric field lines of a point charge exit from the wire to the environment. This happens both in trivial semiconductor wires and wires of three-dimensional topological insulators.
The tunability of the Rashba spin-orbit coupling allows to build semiconductor heterostructures with space modulated coupling intensities. We show that a wire-shaped spin-orbit modulation in a quantum well can support propagating electronic states inside the wire only for a certain spin orientation and, therefore, it acts as an effective spin transmission guide for this particular spin orientation.
Propagating, directionally dependent, polarized spin-currents are created in an anisotropic planar semiconductor microcavity, via Rayleigh scattering of optically injected polaritons in the optical spin Hall regime. The influence of anisotropy results in the suppression or enhancement of the pseudospin precession of polaritons scattered into different directions. This is exploited to create intense spin currents by excitation on top of localized defects. A theoretical model considering the influence of the total effective magnetic field on the polariton pseudospin quantitatively reproduces the…
We use numerical simulations to investigate the spin Hall effect in quantum wires in the presence of both Rashba and Dresselhaus spin-orbit coupling. We find that the intrinsic spin Hall effect is highly anisotropic with respect to the orientation of the wire, and that the nature of this anisotropy depends strongly on the electron density and the relative strengths of the Rashba and Dresselhaus spin-orbit coupling. In particular, at low densities when only one subband of the quantum wire is occupied, the spin Hall effect is strongest for electron momentum along the $[\bar{1}10]$ axis, which is…
We introduce a new spin filter based on spin-resolved Fano resonances due to spin-split levels in a quantum ring (QR) side-coupled to a quantum wire (QW). Spin-orbit coupling inside the QR, together with external magnetic fields, induces spin splitting, and the Fano resonances due to the spin-split levels result in perfect or considerable suppression of the transport of either spin direction. Using the numerical renormalization group method, we find that the Coulomb interaction in the QR enhances the spin filter operation by widening the separation between dips in conductances for different sp…
The role of graded interfaces between materials in a cylindrical free-standing quantum wire with longitudinal heterostructures is theoretically investigated, by solving the Schrödinger equation within the effective mass approximation. Previous work on such wires with abrupt interfaces have predicted that, as the wire radius is reduced, the effective potential along the growth direction is altered and might lead to a carrier confinement at the barriers, as in a type-II system. Our results show that when graded interfaces are considered, such potential acquires a peculiar form, which presents cu…
The excitation gap above the Majorana fermion (MF) modes at the ends of 1D topological superconducting (TS) semiconductor wires scales with the bulk quasiparticle gap E_{qp}. This gap, also called minigap, facilitates experimental detection of the pristine TS state and MFs at experimentally accessible temperatures T << E_{qp}. Here we show that the linear scaling of minigap with E_{qp} can fail in quasi-1D wires with multiple confinement bands when the applied Zeeman field is greater than or equal to about half of the confinement-induced bandgap. TS states in such wires have an approximate chi…
We investigate the effect of mass anisotropy on the Wigner crystallization transition in a two-dimensional (2D) electron gas. The static and dynamical properties of a 2D Wigner crystal have been calculated for arbitrary 2D Bravais lattices in the presence of anisotropic mass, as may be obtainable in Si MOSFETs with (110) surface. By studying the stability of all possible lattices, we find significant change in the crystal structure and melting density of the electron lattice with the lowest ground state energy.
Novel bis(biphenyl)-capped polyynes have been synthesized to investigate the modulation of the electronic and optical properties of sp-hybridized carbon-atom wires (CAWs) capped with π-conjugated $sp^{2}$ endgroups. Raman and Surface Enhanced Raman spectroscopy (SERS) investigation of these systems and Density Functional Theory (DFT) calculations reveal structural changes from polyyne-like with alternating single-triple bonds towards cumulene-like with more equalized bonds as a consequence of the charge transfer occurring when wires interact with metallic nanoparticles. While polyynes have sem…
We investigate the transport properties of narrow quantum wires realized in disordered two-dimensional electron gases in the presence of k-linear Rashba and Dresselhaus spin-orbit interaction (SOI), and an applied in-plane magnetic field. Building on previous work [Scheid, et al., PRL 101, 266401 (2008)], we find that in addition to the conductance, the universal conductance fluctuations also feature anisotropy with respect to the magnetic field direction. This anisotropy can be explained solely from the symmetries exhibited by the Hamiltonian as well as the relative strengths of the Rashba an…
The carrier mobility of anisotropic two-dimensional (2D) semiconductors under longitudinal acoustic (LA) phonon scattering was theoretically studied with the deformation potential theory. Based on Boltzmann equation with relaxation time approximation, an analytic formula of intrinsic anisotropic mobility was deduced, which shows that the influence of effective mass to the mobility anisotropy is larger than that of deformation potential constant and elastic modulus. Parameters were collected for various anisotropic 2D materials (black phosphorus, Hittorf's phosphorus, BC$_2$N, MXene, TiS$_3$, G…
Control of magnetic domain wall movement by the spin-polarized current looks promising for creation of a new generation of magnetic memory devices. A necessary condition for this is the domain wall shift by a low-density current. Here I show that a strongly anisotropic exchange interaction between mobile heavy holes and localized magnetic moments enormously increases the current-induced torque on the domain wall as compared to systems with isotropic exchange. This enables one to control the domain wall motion by current density 10^4 A/cm^2 in ferromagnet/semiconductor hybrids. The experimental…
Recent observation of zero bias conductance peaks in semiconductor wire/superconductor heterostructures has generated great interest, and there is a hot debate on whether the observation is associated with Majorana fermions (MFs). Here we study the local and crossed Andreev reflections in a junction of two normal leads and a sandwiched superconductor-semiconductor wire with two spatially separated but strongly coupled MF end states. The conductance and Fano factors of such a device are sharply different from the topologically trivial case even in the presence of disorder, and can hence be used…
We extend a special kind of localized state trapped at the intersection due to the geometric confinement, first proposed in a three-terminal-opening T-shaped structure [Euro. Phys. Lett. {\bf 55}, 539 (2001)], into a ring geometry with a tangent connection to the wire. In this ring geometry, there exists one localized state trapped at the intersection with energy lying inside the lowest subband. We systematically study this localized state and the resulting Fano-type interference due to the coupling between this localized state and the continuum ones. It is found that the increase of the inner…
Motivated by a recent experimental report[1] claiming the likely observation of the Majorana mode in a semiconductor-superconductor hybrid structure[2,3,4,5], we study theoretically the dependence of the zero bias conductance peak associated with the zero-energy Majorana mode in the topological superconducting phase as a function of temperature, tunnel barrier potential, and a magnetic field tilted from the direction of the wire for realistic wires of finite lengths. We find that higher temperatures and tunnel barriers as well as a large magnetic field in the direction transverse to the wire l…
A microscopic approach is developed to compute the excitonic properties and the corresponding terahertz response for semiconductors characterized by anisotropic effective masses. The approach is illustrated for the example of germanium where it is shown that the anisotropic electron mass in the L-valley leads to two distinct terahertz absorption resonances separated by 0.8 meV.
We have investigated the anisotropic magnetoresistance (AMR) of (In,Fe)As ferromagnetic semiconductor (FMS) layers grown on semi-insulating GaAs substrates. In a 10 nm-thick (In,Fe)As layer which is insulating at low temperature, we observed crystalline AMR with two-fold and eight-fold symmetries. In a metallic 100 nm-thick (In,Fe)As layer with higher electron concentration, only two-fold symmetric crystalline AMR was observed. Our results demonstrate the macroscopic ferromagnetism in (In,Fe)As with magnetic anisotropy that depends on the electron concentration. Non-crystalline AMR is also obs…
We study the metal-insulator transition in individual self-assembled quantum wires and report optical evidences of metallic liquid condensation at low temperatures. Firstly, we observe that the temperature and power dependence of the single nanowire photoluminescence follow the evolution expected for an electron-hole liquid in one dimension. Secondly, we find novel spectral features that suggest that in this situation the expanding liquid condensate compresses the exciton gas in real space. Finally, we estimate the critical density and critical temperature of the phase transition diagram at $n…
We investigate the dynamical self-energy corrections of the electron-hole plasma due to electron-electron and electron-phonon interactions at the band edges of a quasi-one dimensional (1D) photoexcited electron-hole plasma. The leading-order $GW$ dynamical screening approximation is used in the calculation by treating electron-electron Coulomb interaction and electron-optical phonon Fröhlich interaction on an equal footing. We calculate the exchange-correlation induced band gap renormalization (BGR) as a function of the electron-hole plasma density and the quantum wire width. The calculated BG…
Electrical current manipulation of magnetization switching through spin-orbital coupling in ferromagnetic semiconductor (Ga,Mn)As Hall bar devices has been investigated. The efficiency of the current-controlled magnetization switching is found to be sensitive to the orientation of the current with respect to the crystalline axes. The dependence of the spin-orbit effective magnetic field on the direction and magnitude of the current is determined from the shifts in the magnetization switching angle. We find that the strain induced effective magnetic field is about three times as large as the Ra…
Nanoporous graphene (NPG), consisting of ordered arrays of nanopores separated by graphene nanoribbons was recently realized using a bottom-up synthesis method (Science 360(2018), 199). In this work we accordingly explored the mechanical response, thermal conductivity and electronic/optical properties of single-layer NPG using the density functional theory and molecular dynamics simulations. Along the armchair direction, NPG was found to exhibit higher tensile strength and thermal conductivity by factors of 1.6 and 2.3, respectively, in comparison with the zigzag direction. Despite of showing …
We present a detailed theoretical study of the electronic spectrum and Zeeman splitting in hole quantum wires. The spin-3/2 character of the topmost bulk-valence-band states results in a strong variation of subband-edge g factors between different subbands. We elucidate the interplay between quantum confinement and heavy-hole - light-hole mixing and identify a certain robustness displayed by low-lying hole-wire subband edges with respect to changes in the shape or strength of the wire potential. The ability to address individual subband edges in, e.g., transport or optical experiments enables …
A fourteen-fold anisotropy in the spin transport efficiency parallel and perpendicular to the charge transport is observed in a vertically-biased (Ga,Mn)As-based spin-polarized light emitting diode. The spin polarization is determined by measuring the polarization of electroluminescence from an (In,Ga)As quantum well placed a distance d (20 to 420 nm) below the p-type ferromagnetic (Ga,Mn)As contact. In addition, a monotonic increase from 0.5 to 7% in the polarization is measured as d decreases for collection parallel to the growth direction, while the in-plane polarization from the perpendicu…
A GaAs/AlGaAs based two-qubit quantum device that allows the controlled generation and straightforward detection of entanglement by measuring a stationary current-voltage characteristic is proposed. We have developed a two-particle Green's function method of open systems and calculate the properties of three-dimensional interacting entangled systems non-perturbatively. We present concrete device designs and detailed, charge self-consistent predictions. One of the qubits is an all-electric Mach-Zehnder interferometer that consists of two electrostatically defined quantum wires with coupling win…
A new two-dimensional (2D) layered material, namely, titanium trisulfide (TiS$_3$) monolayer sheet, is predicted to possess desired electronic properties for nanoelectronic applications. On basis of the first-principles calculations within the framework of density functional theory and deformation theory, we show that the TiS$_3$ 2D crystal is a direct gap semiconductor with a band gap of 1.06 eV and high carrier mobility. More remarkably, the in-plane electron mobility of the 2D TiS$_3$ is highly anisotropic, amounting to $\sim$10,000 cm$^2$V$^{-1}$s$^{-1}$ in the \emph{b} direction, which is…
Universal quantum computation (UQC) using Majorana fermions on a 2D topological superconducting (TS) medium remains an outstanding open problem. This is because the quantum gate set that can be generated by braiding of the Majorana fermions does not include \emph{any} two-qubit gate and also the single-qubit $π/8$ phase gate. In principle, it is possible to create these crucial extra gates using quantum interference of Majorana fermion currents. However, it is not clear if the motion of the various order parameter defects (vortices, domain walls, \emph{etc.}), to which the Majorana fermions ar…
We consider a single electron confined within a quantum wire in a system of two electrostatically-induced QDs defined by nearby gates. The time-varying electric field, of single GHz frequency, perpendicular to the quantum wire, is used to induce the Rashba coupling and enable spin-dependent resonant tunneling of the electron between two adjacent potential wells with fidelity over 99.5%. This effect can be used for the high fidelity all-electrical electron-spin initialization or readout in the spin-based quantum computer. In contrast to other spin initialization methods, our technique can be pe…
We propose a theoretical scheme to realize arbitrary single-qubit gates through two simple device units: one-dimensional semiconductor wires with Dresselhaus spin-orbit coupling (SOC) and Rashba SOC, separately. Qubit information coded by the electron spin can be accurately manipulated by the SOC when crossing the semiconductor wire. The different manipulative behaviors in Dresselhaus and Rashba wires enable us to make the diverse quantum logic gates. Furthermore, by connecting the Dresselhaus and Rashba units in series, we obtain a universal set of single qubit gates: Hadamard, phase, and $π/…
We show that a topologically trivial zero bias conductance peak (of height 4e^2/h) is produced in semiconductor-superconductor hybrid nanowires due to a suppressed pair potential and/or an excess Zeeman field at the ends of the heterostructure, both of which can occur in experiments. The zero bias peak (ZBP) (a) appears above a threshold parallel bulk Zeeman field, (b) is stable for a range of bulk field before splitting, (c) disappears with rotation of the bulk Zeeman field, and, (d) is robust to weak disorder fluctuations. The ZBPs from the nanowire ends are also expected to produce splittin…
