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 the wire is efficient and density modulations are not visible. In the low-density, weak-confinement regime, the exchange-correlation potential induces small density modulations as the electrons are depleted from the wire. At the weakest confinements and lowest densities, the electron density splits into two rows thereby forming a pair of quantum wires that lie beneath the surface gates. An additional double-well external potential forms at very low density which enhances this row splitting phenomenon. We produce phase diagrams that show a transition between the presence of a single quantum wire in a split-gate structure and two quantum wires. We suggest that this phenomenon can be used to pattern and modulate the electron density in low-dimensional structures with particular application to systems where a proximity effect from a surface gate would be valuable.
A one-band model within the effective mass approximation is adopted to characterize the energy structure and oscillator strength of type-II semiconductor spherical core-shell quantum dots. The heteroepitaxial strain of the core-shell heterostructure is modeled by the elastic continuum approach. The model is applied to ZnTe/ZnSe core-shell, a wide band gap type-II heterostructure. The simulated absorption spectra are in fair agreement with available experimental results.
We propose a new approach to create Majorana fermions at the edge of a periodically driven semiconductor-superconductor Heterostructure. We calculate the quasi-energy spectrum of the periodically driven Heterostructure by using the Floquet's theory. When the interaction between different Brillouin zones of quasi-energy is neglected, one Majorana fermion can be created at each edge of the Heterostructure when the ratio of driven amplitude and driven frequency is larger than a minimum. Furthermore, when the interaction between the nearest Brillouin zones of quasi-energy is considered, we restric…
We study the electronic properties of a planar semiconductor-superconductor heterostructure, in which a thin ferromagnetic insulator layer lies in between and acts as a spin filtering barrier. We find that in such a system one can simultaneously enhance the strengths of all the three important induced physical quantities, i.e., Rashba spin-orbit coupling, exchange coupling, and superconducting pairing potential, for the hybrid mode by external gating. Our results show specific advantage of this stacked device geometry compared to conventional devices. We further discuss how to optimize geometr…
We present a theory that describes the appearance of circular polarization of the photoluminescence (PL) in ferromagnet-semiconductor hybrid heterostructures due to spin-dependent tunneling of photoexcited carriers from a quantum well into a magnetic layer. The theory succeeds in explaining the experimental data on time-resolved PL for heterostructures consisting of InGaAs-based quantum well (QW) and a spatially separated Mn $δ$-layer. We show that the circular polarization of the PL originates from dynamic spin polarization of electrons due to spin-dependent leakage from the QW onto Mn donor …
The phenomenon of mesoscopic Spin-Hall effect reveals in a nonequilibrium spin accumulation (driven by electric current) at the edges of a ballistic conductor or, more generally, in the regions with varying electron density. In this paper we review our recent results on spin accumulation in ballistic two-dimensional semiconductor heterostructures with Rashba/Dresselhaus spin orbit interactions, and extend the method developed previously to predict the existince of spin-Hall effect on the surface of three-dimensional topological insulators. The major difference of the new Spin-Hall effect is it…
Semiconductors in the proximity of superconductors have been proposed to support phases hosting Majorana bound states. When the systems undergo a topological phase transition towards the Majorana phase, the spectral gap closes, then reopens, and the quasiparticle band spin polarization is inverted. We focus on two paradigmatic semiconductor-superconductor heterostructures and propose an all-electrical spectroscopic probe sensitive to the spin inversion at the topological transition. Our proposal relies on the indirect coupling of a time-dependent electric field to the electronic spin due to th…
Hexagonally patterned two-dimensional $p$-type semiconductor systems are quantum simulators of graphene with strong and highly tunable spin-orbit interactions. We show that application of purely in-plane magnetic fields, in combination with the crystallographic anisotropy present in low-symmetry semiconductor interfaces, allows Chern insulating phases to emerge from an originally topologically insulating state after a quantum phase transition. These phases are characterized by pairs of co-propagating edge modes and Hall conductivities $σ_{xy} = +\frac{2 e^2}{h}, -\frac{2 e^2}{h}$ in the absenc…
We propose an implementation of external homogeneous magnetic fields in k$\cdot$p Hamiltonians for holes in heterostructures, in which we made use of the minimal coupling prior to introduce the envelope function approximation. Illustrative calculations for holes in InGaAs quantum dot molecules show that the proposed Hamiltonian outperforms standard Luttinger model [Physical Review 102, 1030 (1956)] describing the experimentally observed magnetic response. The present implementation culminates our previous proposal [Phys. Rev. B 82, 155307 (2010)].
We theoretically investigate an extrinsic spin Hall effect (SHE) in semiconductor heterostructures due to the scattering by an artificial potential created by antidot, STM tip, etc. The potential is electrically tunable. First, we formulate the SHE in terms of phase shifts in the partial wave expansion for two-dimensional electron gas. The effect is significantly enhanced by the resonant scattering when the attractive potential is properly tuned. Second, we examine a three-terminal device including an antidot, which possibly produces a spin current with polarization of more than 50%.
We present a new method for calculating electronic states in low-dimensional semiconductor heterostructures, which is based on the real-space Hamiltonian in the envelope function approximation. The numerical implementation of the method is extremely simple; all subband energy levels and envelope functions are directly obtained by a single evaluation of the heterostructure Hamiltonian matrix. We test the method in the 6- and 8-band k \cdot p models as well as in a simple parabolic one-band model and demonstrate its great accuracy. The method can be straightforwardly generalized to a general n-b…
We propose a nanodevice based on a typical planar semiconductor heterostructure with lateral confinement potential created by voltages applied to local electrodes. We show how to obtain near parabolical confinement along the nanodevice, and how to use coherent states of the harmonic oscillator for spatial separation of electron densities corresponding to opposite spin directions. In such a way, an entangled state of Schrödinger's cat type is created. We performed simulations of a realistic nanodevice model by numerical solving the time-dependent Schrödinger's equation together with simultaneou…
A new approach to the growth of diluted magnetic semiconductors with two dimensional electron gas in InAs quantum well has been developed. The method is based on molecular-beam epitaxy of coherent "hybrid" AlSb/InAs/(Zn,Mn)Te heterostructures with a III-V/II-VI interface inside. The giant Zeeman splitting of the InAs conduction band caused by exchange interaction with Mn2+ ions has been proved by measuring the microwave radiation induced spin polarized electric currents.
Van der Waals (vdW) heterostructures formed by two-dimensional magnets and semiconductors have provided a fertile ground for fundamental science and for spintronics. We present first-principles calculations finding a proximity exchange splitting of 14 meV equivalent to an effective Zeeman field of 120 T in the vdW magnet-semiconductor heterostructure MoS$_2$/CrBr$_3$, leading to a 2D spin-polarized half-metal with carrier densities ranging up to $10^{13}$ cm$^{-2}$. We consequently explore the effect of large exchange coupling on the electronic bandstructure when the magnetic layer hosts chira…
Using the Matsubara Green's function formalism we calculate the temperature dependence of the nonequilibrium spin polarization induced by an external electric field in the presence of spin-orbit coupling. The model Hamiltonian includes an isotropic k-cubed form of the Rashba spin-orbit interaction. Such a Hamiltonian captures the electronic and spin properties of two-dimensional electron (hole) gas at the surfaces or interfaces of transition metal oxides or in p-doped semiconductor heterostructures. The induced spin polarization is calculated for the nonmagnetic as well as magnetic electron/ho…
We theoretically studied the in-plane tunneling spectroscopy of the hybrid structure composed of a metal and a semiconductor with Rashba spin-orbit coupling. We found that the energy spacing between two distinct features in the conductance spectrum can be used to measure the Rashba energy of the semiconductor. We also considered the effect that varying the probability of spin-conserving and spin-flip scattering at the interface has on the overall conductance. Surprisingly, an increase in interface scattering probability can actually result in increased conductance under certain conditions. Par…
Heterostructures combining topological and non-topological materials constitute the next frontier in the effort to incorporate topological insulators (TIs) into functional electronic devices. We show that the properties of the interface states appearing at the planar boundary between a topologically-trivial semiconductor (SE) and a TI are controlled by the symmetry of the interface. In contrast to the well-studied helical Dirac surface states, SE-TI interface states exhibit elliptical contours of constant energy and complex spin textures with broken helicity. We derive a general effective Hami…
We have shown that electron spin density can be generated by a dc current flowing across a $pn$ junction with an embedded asymmetric quantum well. Spin polarization is created in the quantum well by radiative electron-hole recombination when the conduction electron momentum distribution is shifted with respect to the momentum distribution of holes in the spin split valence subbands. Spin current appears when the spin polarization is injected from the quantum well into the $n$-doped region of the $pn$ junction. The accompanied emission of circularly polarized light from the quantum well can ser…
We calculate the electronic width of quantum wires as a function of their lithographic width in analogy to experiments performed on nanostructures defined by local oxidation of Ga[Al]As heterostructures. Two--dimensional simulations of two parallel oxide lines on top of a Ga[Al]As heterostructure defining a quantum wire are carried out in the framework of Density Functional Theory in the Local Density Approximation and are found to be in agreement with measurements. Quantitative assessment of the influence of various experimental uncertainties is given. The most influential parameter turns out…
A supersolid, a counter-intuitive quantum state in which a rigid lattice of particles flows without resistance, has to date not been unambiguously realised. Here we reveal a supersolid ground state of excitons in a double-layer semiconductor heterostructure over a wide range of layer separations outside the focus of recent experiments. This supersolid conforms to the original Chester supersolid with one exciton per supersolid site, as distinct from the alternative version reported in cold-atom systems of a periodic modulation of the superfluid density. We provide the phase diagram augmented by…
The spin-orbit interaction of holes in zinc-blende semiconductors is much stronger than that of electrons. This makes the hole systems very attractive for possible spintronics applications. In three dimensions (3D) dynamics of holes is described by well known Luttinger Hamiltonian. However, most of recent spintronics applications are related to two dimensional heterostructures where dynamics in one direction is frozen due to quantum confinement. The confinement results in dimensional reduction of the Luttinger Hamiltonian, 3D ->2D. Due to interplay of the spin-orbit interaction, the external m…
The magnetization dynamics in diluted magnetic semiconductor heterostructures based on (Zn,Mn)Se and (Cd,Mn)Te has been studied experimentally by optical methods and simulated numerically. In the samples with nonhomogeneous magnetic ion distribution this dynamics is contributed by spin-lattice relaxation and spin diffusion in the Mn spin system. The spin diffusion coefficient of 7x10^(-8) cm^2/s has been evaluated for Zn(0.99)Mn(0.01)Se from comparison of experimental and numerical results. Calculations of the giant Zeeman splitting of the exciton states and the magnetization dynamics in the o…
We present a systematic construction of the probability-current operator,based on a momentum power expansion of effective Hamiltonians. The result is valid in the presence of a Rashba term and when a D'yakonov--Perel contribution is included. We propose practical tools for spin-orbit engineering of semiconductor heterostructures. We apply this formalism to a paradigmatic system, the interface between two semi-infinite media, on one side a free-electron-like material and on the other side a barrier material with spin-orbit interaction. We show that the usual boundary conditions, namely the cont…
Tunable magnetic interactions in high-mobility nonmagnetic semiconductor heterostructures are centrally important to spin-based quantum technologies. Conventionally, this requires incorporation of "magnetic impurities" within the two-dimensional (2D) electron layer of the heterostructures, which is achieved either by doping with ferromagnetic atoms, or by electrostatically printing artificial atoms or quantum dots. Here we report experimental evidence of a third, and intrinsic, source of localized spins in high-mobility GaAs/AlGaAs heterostructures, which are clearly observed in the limit of l…
We consider a two-dimensional electron gas with Rashba spin-orbit interaction (SOI) partially covered by an s-wave superconductor, where the uncovered region remains normal but is exposed to an effective Zeeman field applied perpendicular to the plane. We find analytically and numerically Andreev bound states (ABSs) formed in the normal region and show that, due to SOI and by tuning the parameters of the system deeply into the topologically trivial phase, one can reach a regime where the energy of the lowest ABS becomes pinned close to zero as a function of Zeeman field. The energy of such an …
Combining monolayers of different two-dimensional (2D) semiconductors into heterostructures opens up a wealth of possibilities for novel electronic and optical functionalities. Exploiting them hinges on accurate measurements of the band parameters and orbital hybridization in separate and stacked monolayers, many of which are only available as small samples. The recently introduced technique of angle-resolved photoemission spectroscopy with submicron spatial resolution (μ-ARPES) offers the capability to measure small samples, but the energy resolution obtained for such exfoliated samples to da…
We study the Zeeman spin-splitting in hole quantum wires oriented along the $[011]$ and $[01\bar{1}]$ crystallographic axes of a high mobility undoped (100)-oriented AlGaAs/GaAs heterostructure. Our data shows that the spin-splitting can be switched `on' (finite $g^{*}$) or `off' (zero $g^{*}$) by rotating the field from a parallel to a perpendicular orientation with respect to the wire, and the properties of the wire are identical for the two orientations with respect to the crystallographic axes. We also find that the $g$-factor in the parallel orientation decreases as the wire is narrowed. …
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…
Polar optical phonons are studied in the framework of the dielectric continuum approach for a prototypical quantum-dot/quantum-well (QD/QW) heterostructure, including the derivation of the electron-phonon interaction Hamiltonian and a discussion of the effects of this interaction on the electronic energy levels. The particular example of the CdS/HgS QD/QW is addressed and the system is modelled according to the spherical geometry, considering a core sphere of material "1" surrounded by a spherically concentric layer of material "2", while the whole structure is embedded in a host matrix assume…
The prospect of coupling a two-dimensional (2D) semiconductor heterostructure to a superconductor opens new research and technology opportunities, including fundamental problems in mesoscopic superconductivity, scalable superconducting electronics, and new topological states of matter. For instance, one route toward realizing topological matter is by coupling a 2D electron gas (2DEG) with strong spin-orbit interaction to an s-wave superconductor. Previous efforts along these lines have been hindered by interface disorder and unstable gating. Here, we report measurements on a gateable InGaAs/In…
We study the properties of heterostructures formed by two layers of diluted magnetic semiconductor separated by a nonmagnetic semiconductor layer. We find that there is a RKKY-type exchange coupling between the magnetic layers that oscilles between ferromagnetic and antiferromagnetic as a function of the different parameters in the problem. The different transport properties of these phases make that this heterostructure presents strong magnetoresistive effects. The coupling can be also modified by an electric field. We propose that it is possible to alter dramatically the electrical resistanc…
We report on microwave (mw) radiation induced electric currents in (Cd,Mn)Te/(Cd,Mg)Te and InAs/(In,Ga)As quantum wells subjected to an external in-plane magnetic field. The current generation is attributed to the spin-dependent energy relaxation of electrons heated by mw radiation. The relaxation produces equal and oppositely directed electron flows in the spin-up and spin-down subbands yielding a pure spin current. The Zeeman splitting of the subbands in the magnetic field leads to the conversion of the spin flow into a spin-polarized electric current.
Electrostatically defined nanoscale devices on two-dimensional semiconductor heterostructures are the building blocks of various quantum electrical circuits. Owing to its atomically flat interfaces and the inherent two-dimensional nature, van der Waals heterostructures hold the advantage of large-scale uniformity, flexibility and portability over the conventional bulk semiconductor heterostructures. In this letter we show the operation of a split-gate defined point contact device on a MoS2/h-BN heterostructure, a first step towards realizing electrostatically gated quantum circuits on van der …
The possibility of a novel type of semiconductor quantum dots obtained by spatially modulating the spin-orbit coupling intensity in III-V heterostructures is discussed. Using the effective mass model we predict confined one-electron states having peculiar spin properties. Furthermore, from mean field calculations (local-spin-density and Hartree-Fock) we find that even two electrons could form a bound state in these dots.
For quantum technologies based on single excitons and spins, the deterministic placement and control of a single exciton is a long-standing goal. MoSe2-WSe2 heterostructures host spatially indirect interlayer excitons (IXs) which exhibit highly tunable energies and unique spin-valley physics, making them promising candidates for quantum information processing. Previous IX trapping approaches involving moiré superlattices and nanopillars do not meet the quantum technology requirements of deterministic placement and energy tunability. Here, we use a nanopatterned graphene gate to create a sharpl…
We show that a semiconductor thin film can acquire a non-trivial spin texture due to the proximity effect induced by a topological insulator. The effect stems from coupling to the topological surface states and is present even when the insulator is doped. We propose a semiconductor/topological insulator heterostructure as a device that allows measuring interface properties and probing surface states in uncompensated samples. We also find that the topological insulator surface modes can be significantly broadened and shifted by the presence of metallic contacts.
Effect of spin-dependent tunneling and intervalley scattering in magnetic-semiconductor van der Waals heterostructures on exciton and trion polarization
In this paper, we show that HgCdTe heterostructures may support bound electronic states embedded in the continuum, such that the discrete energy spectrum overlaps the continuous spectrum. Although the proposed heterostructures are generally penetrable by an incoming electron wave, it is shown that they may support spatially localized trapped stationary states with an infinite lifetime. We discuss the possibility of a free electron being captured by the proposed open resonator, and present a detailed study of the trapping lifetime in the case of a detuned resonator.
Measurements and modeling of electron spin transport and dynamics are used to characterize hyperfine interactions in Fe/GaAs devices with $n$-GaAs channels. Ga and As nuclei are polarized by electrically injected electron spins, and the nuclear polarization is detected indirectly through the depolarization of electron spins in the hyperfine field. The dependence of the electron spin signal on injector bias and applied field direction is modeled by a coupled drift-diffusion equation, including effective fields from both the electronic and nuclear polarizations. This approach is used to determin…
We show that partially separated Andreev bound states (ps-ABSs), comprised of pairs of overlapping Majorana bound states (MBSs) emerging in quantum dot-semiconductor-superconductor heterostructures, produce robust zero bias conductance plateaus in end-of-wire charge tunneling experiments. These plateaus remain quantized at $2e^2/h$ over large ranges of experimental control parameters. In light of recent experiments reporting the observation of robust $2e^2/h$-quantized conductance plateaus in local charge tunneling experiments, we perform extensive numerical calculations to explicitly show tha…
We investigate electron spin relaxation in GaAs in the proximity of a Fe/MgO layer using spin-resolved optical pump-probe spectroscopy, revealing a strong dependence of the spin relaxation time on the strength of an exchange-driven hyperfine field. The temperature dependence of this effect reveals a strong correlation with carrier freeze out, implying that at low temperatures the free carrier spin lifetime is dominated by inhomogeneity in the local hyperfine field due to carrier localization. This result resolves a long-standing and contentious question of the origin of the spin relaxation in …
The interplay of tunneling transport and carrier-mediated ferromagnetism in narrow semiconductor multi-quantum well structures containing layers of GaMnAs is investigated within a self-consistent Green's function approach, accounting for disorder in the Mn--doped regions and unwanted spin-flips at heterointerfaces on phenomenological ground. We find that the magnetization in GaMnAs layers can be controlled by an external electric bias. The underlying mechanism is identified as spin-selective hole tunneling in and out of the Mn-doped quantum wells, whereby the applied bias determines both hole …
We study nuclear relaxation in the presence of localized electrons in a two-dimensional electron gas in a disordered delta-doped semiconductor heterostructure and show that this method can reliably probe their magnetic interactions and possible long-range order. In contrast, we argue that transport measurements, the commonly employed tool, may not sometimes distinguish between spatial disorder and long-range order. We illustrate the utility of using the nuclear relaxation method to detect long-range order by analyzing a recent proposal made on the basis of transport measurements, on the sponta…
We report a type of solar cells based on graphene/CdTe Schottky heterostructure, which can be improved by surface engineering as graphene is one-atomic thin. By coating a layer of ultrathin CdSe quantum dots onto graphene/CdTe heterostructure, the power conversion efficiency is increased from 2.08% to 3.1%. Photo-induced doping is mainly accounted for this enhancement, as evidenced by transport, photoluminescence and quantum efficiency measurements. This work demonstrates a feasible way of designing solar cells with incorporating one dimensional and two dimensional materials.
We provide a derivation of a spin Skyrmion number classification for two-dimensional topological superconductors constructed from ferromagnetic and Rashba spin-orbit coupled semiconductor-superconductor heterostructures. We show that in the non-trivial topological phase, characterized by a non-zero Chern number, there is always a topological spin texture in the occupied bands represented by a Skyrmion number. The Skyrmion number has the advantage of being both physically intuitive and directly measurable using spin-sensitive band structure imaging techniques. In addition, we show that the Skyr…
A fractal-like alignment of quantum wells is shown to accommodate resonant states with long lifetimes. For the parameters of the semiconductor heterostructure GaAs/Al$_{0.4}$Ga$_{0.6}$As with the well depth 300meV, a resonant state of the energy as high as 44meV with the lifetime as long as 2.8\{mu}s is shown to be achievable.
We investigate a heterostructure system with a spin-orbit coupled semiconductor sandwiched by an s-wave superconductor and a ferromagnetic insulator, which supports Majorana fermions (MFs) at the superconducting vortex cores. We propose a scheme of transporting and braiding the MFs, which only requires application of point-like gate voltages in a system with nano-meter patterns. By solving the time-dependent Bogoliubov-de Gennes equation numerically, we monitor the time evolutions of MF wave-functions and show that the braiding of MFs with non-Abelian statistics can be achieved by adiabatic sw…
Van der Waals heterostructures formed by two different monolayer semiconductors have emerged as a promising platform for new optoelectronic and spin/valleytronic applications. In addition to its atomically thin nature, a two-dimensional semiconductor heterostructure is distinct from its three-dimensional counterparts due to the unique coupled spin-valley physics of its constituent monolayers. Here, we report the direct observation that an optically generated spin-valley polarization in one monolayer can be transferred between layers of a two-dimensional MoSe2-WSe2 heterostructure. Using nondeg…
Based on first-principles density functional calculations, we show that SnS and SnSe layers can form mechanically rigid heterostructures with the constituent puckered or buckled monolayers. Due to the strong interlayer coupling, the electronic wavefunctions of the conduction and valence band edges are delocalized across the heterostructure. The resultant bandgap of the heterostructures reside in the infrared region. With strain engineering, the heterostructure bandgap undergoes transition from indirect to direct in the puckered phase. Our results show that there is a direct correlation between…
We develop a unified numerical approach for modeling semiconductor-superconductor heterostructures. Our approach takes into account on equal footing important key ingredients: proximity-induced superconductivity, orbital and Zeeman effect of an applied magnetic field, spin-orbit coupling as well as the electrostatic environment. As a model system, we consider indium arsenide (InAs) nanowires with epitaxial aluminum (Al) shell and demonstrate qualitative agreement of the obtained results with the existing experimental data. Finally, we characterize the topological superconducting phase emerging…
We study the unitary propagation of a two-particle one-dimensional Schrödinger equation by means of the Split-Step Fourier method, to study the coherent evolution of a spatially indirect exciton (IX) in semiconductor heterostructures. The mutual Coulomb interaction of the electron-hole pair and the electrostatic potentials generated by external gates and acting on the two particles separately are taken into account exactly in the two-particle dynamics. As relevant examples, step/downhill and barrier/well potential profiles are considered. The space- and time-dependent evolution during the scat…
We apply the quantum Singwi-Tosi-Land-Sjolander (QSTLS) theory for a study of many-body effects in the quasi-two-dimensional (Q2D) electron liquid (EL) in GaAs/AlxGa1-xAs heterojunctions. The effect of the layer thickness is included through a variational approach. We have calculated the density, spin-density static structure factors, spin-dependent pair distribution functions (PDF) and compared our results with those of two-dimensional (2D) EL given in earlier papers. Using the static structure factors (SSF) we have calculated various dynamic correlation functions such as spin-dependent local…
The propagation of charge carriers in graphene is compared to that in type II/III heterostructures for which a two-band Kane model is appropriate. In particular, conditions for a quantitative analogy between these two cases are searched for, and found to be quite restrictive. The analysis in this paper shows that the essential property of graphene is not the spinor character of its wavefunction but the linear dispersion relation, which does not hold in finite-gap two-band Kane-type semiconductors. Therefore, Kane-like and Dirac-like charge carriers behave differently, except in zero-bandgap se…
