The properties of a two-dimensional electron are investigated in the presence of a circular step magnetic field profile. Both electrons with parabolic dispersion as well as Dirac electrons with linear dispersion are studied. We found that in such a magnetic quantum dot no electrons can be confined. Nevertheless close to the Landau levels quasi-bound states can exist with a rather long life time.
At an interface between contacts and graphene, the mathematical equation that governs the propagation of electrons transforms from the Schrodinger to the Dirac equation. The condition of current probability conservation at such an interface does not determine uniquely the boundary conditions for the quantum wavefunction. We discuss the possible form of boundary conditions, determine its influence on the transmission coefficient of a contacted graphene structure and suggest that optical experiments on photonic crystals with Dirac points can help identifying, under certain circumstances, the pro…
Broken Generalized Kohn Theorem in Harmonic Dot Lattices due to Coulomb Interaction between the Dots: Exact solution of the Schrödinger equation in the dipole approximation
We predict that a strong nonreciprocity in the resonance spectra of Dirac quantum dots can be induced by the Berry phase. The nonreciprocity arises in relatively weak magnetic fields and is manifest in anomalously large field-induced splittings of quantum dot resonances which are degenerate at $B=0$ due to time-reversal symmetry. This exotic behavior, which is governed by field-induced jumps in the Berry phase of confined electronic states, is unique to quantum dots in Dirac materials and is absent in conventional quantum dots. The effect is strong for gapless Dirac particles and can overwhelm…
A Schrödinger-cat state is a coherent superposition of macroscopically distinguishable quantum states, in quantum optics usually realized as superposition of coherent states. Protocols to prepare photonic cats have been presented for atomic systems. Here, we investigate in what manner and how well the preparation protocols can be transferred to a solid state platform, namely a semiconductor quantum-dot--cavity system. In quantum-dot--cavity systems there are many disruptive influences like cavity losses, the radiative decay of the quantum dot, and the pure-dephasing type coupling to longitudin…
The RKKY interaction between two impurities on a curved surface is investigated. Practical curved sample parameters are considered for investigating curvatures, similar to those observed experimentally in available samples such as graphene. For this type of two-dimensional curved systems and for two types of Schrödinger and Dirac carriers, the RKKY interaction and local density of states (LDOS) are calculated. Results show that the strength of the RKKY interaction on a curved surface depends significantly on the nature of particles embedded on this surface, indicating that the geometry has a d…
We study the interplay between Dirac and Schrödinger fermions in the polarization properties of a two-dimensional electron gas (2DEG). Specifically, we analyze the low-energy sector of narrow-gap semiconductors described by a two-band Kane model. In the context of quantum spin Hall insulators, particularly, in Hg(Cd)Te quantum wells, this model is named Bernevig-Hughes-Zhang model. Interestingly, it describes electrons with intermediate properties between Dirac and Schrödinger fermions. We calculate the dynamical dielectric function of such a model at zero temperature within random phase appro…
A spatially modulated Dirac gap in a graphene sheet leads to charge confinement, thus enabling a graphene quantum dot to be formed without the application of external electric and magnetic fields [Appl. Phys. Lett. \textbf{97}, 243106 (2010)]. This can be achieved provided the Dirac gap has a local minimum in which the states become localised. In this work, the physics of such a gap-induced dot is investigated in the continuum limit by solving the Dirac equation. It is shown that gap-induced confined states couple to the states introduced by an electrostatic quantum well potential. Hence the r…
In neutral graphene dots the Fermi level coincides with the Dirac points. We have investigated in the presence of a magnetic field several unusual properties of single electron states near the Fermi level of such a rectangular-shaped graphene dot with two zigzag and two armchair edges. We find that a quasi-degenerate level forms near zero energy and the number of states in this level can be tuned by the magnetic field. The wavefunctions of states in this level are all peaked on the zigzag edges with or without some weight inside the dot. Some of these states are magnetic field-independent surf…
The Dirac equation is solved for triangular and hexagonal graphene quantum dots for different boundary conditions in the presence of a perpendicular magnetic field. We analyze the influence of the dot size and its geometry on their energy spectrum. A comparison between the results obtained for graphene dots with zigzag and armchair edges, as well as for infinite-mass boundary condition, is presented and our results show that the type of graphene dot edge and the choice of the appropriate boundary conditions have a very important influence on the energy spectrum. The single particle energy leve…
Electrostatic confinement of charge carriers in graphene is governed by Klein tunneling, a relativistic quantum process in which particle-hole transmutation leads to unusual anisotropic transmission at pn junction boundaries. Reflection and transmission at these novel potential barriers should affect the quantum interference of electronic wavefunctions near these boundaries. Here we report the use of scanning tunneling microscopy (STM) to map the electronic structure of Dirac fermions confined by circular graphene pn junctions. These effective quantum dots were fabricated using a new technique…
We present complete analytical and numerical results that demonstrate the anomalous universal fluctuations of the spin-Hall conductance in chiral materials such as graphene and topological insulators. We investigated both the corresponding fluctuations, the Universal Fractionated and the Universal Quantized, and also the open channels orbital number crossover between the two regimes. In particular, we show that the Wigner-Dyson symmetries does not properly describe such conductances and the preponderant role of the chiral classes on the Dirac quantum dots. The results are analytical and solve …
Electronic transport through a material depends on the response to local perturbations induced by defects or impurities in the material. The scattering processes can be described in terms of phase shifts and corresponding cross sections. The multiorbital nature of the spinor states in transition metal dichalcogenides would naturally suggest the consideration of a massive Dirac equation to describe the problem, while the parabolic dispersion of its conduction and valence bands would invite a simpler Schrödinger equation description. Here, we contrast the scattering of massive Dirac particles an…
We show that the conductivity of a two-dimensional electron gas can be intrinsically anisotropic despite isotropic Fermi surface, energy dispersion, and disorder configuration. In the model we study, the anisotropy stems from the interplay between Dirac and Schrödinger features combined in a special two-band Hamiltonian describing the quasiparticles similar to the low-energy excitations in phosphorene. As a result, even scalar isotropic disorder scattering alters the nature of the carriers and results in anisotropic transport. Solving the Boltzmann equation exactly for such carriers with point…
Graphene flakes placed on hexagonal boron nitride feature in the presence of a magnetic field a complex electronic structure due to a hexagonal moiré potential resulting from the van der Waals interaction with the substrate. The slight lattice mismatch gives rise to a periodic supercell potential. Zone folding is expected to create replica of the original Dirac cone and Hofstadter butterflies. Our large-scale tight binding simulation reveals an unexpected coexistence of a relativistic and non-relativistic Landau level structure. The presence of the zeroth Landau level and its associated butter…
We present a description of vacuum polarization in a circular Dirac quantum dot in two spatial dimensions assuming $α$ - the relative strength of the Coulomb interaction small enough to render an approximation with a single electron (hole) lowest energy level relevant. Applying this approximation, we find that for $ α_c \approx 1.05$ the lowest level is half-filled irrespective of the number of flavors that are present. The ground state can be represented as a superposition of particular (even number) excitonic states which constitute an excitonic cloud that evolves in a crossover manner. The …
A simple and reliable finite difference approach is presented for solution of the Dirac equation eigenproblem for states confined in rotationally symmetric systems. The method sets the boundary condition for the spinor wave function components at the external edge of the system and then sweeps the radial mesh in search for the energies for which the boundary conditions are met inside the flake. The sweep that is performed from the edge of the system towards the origin allows for application of a two-point finite difference quotient of the first derivative, which prevents the fermion doubling p…
We study $N$ interacting massless Dirac fermions confined in a two-dimensional quantum dot. Physical realizations of this problem include a graphene monolayer and the surface state of a strong topological insulator. We consider both a magnetic confinement and an infinite mass confinement. The ground state energy is computed as a function of the effective interaction parameter $α$ from the Hartree-Fock approximation and, alternatively, by employing the Müller exchange functional. For N=2, we compare those approximations to exact diagonalization results. The Hartree-Fock energies are highly accu…
An electrostatic quantum dot cannot be formed in monolayer graphene, because of the Klein tunnelling. However, a dot can be formed with the help of a uniform magnetic field. As shown here, a spatial modulation of the Dirac gap leads to confined states with discrete energy levels, thus defining a dot, without applying external electric and magnetic fields. Gap-induced dot states can coexist and couple with states introduced by an electrostatic potential. This property allows the region in which the resulting states are localized to be tuned with the potential.
Graphene p-n junctions provide an ideal platform for investigating novel behavior at the boundary between electronics and optics that arise from massless Dirac fermions, such as whispering gallery modes and Veselago lensing. Bilayer graphene also hosts Dirac fermions, but they differ from single-layer graphene charge carriers because they are massive, can be gapped by an applied perpendicular electric field, and have very different pseudospin selection rules across a p-n junction. Novel phenomena predicted for these massive Dirac fermions at p-n junctions include anti-Klein tunneling, oscillat…
Analyzing the scattering and conversion process between photons and phonons coupled via radiation pressure in a circular quantum dot on a honeycomb array of optomechanical cells, we demonstrate the emergence of optomechanical Dirac physics. Specifically we prove the formation of polaritonic quasi-bound states inside the dot, and angle-dependent Klein tunneling of light and emission of sound, depending on the energy of the incident photon, the photon-phonon interaction strength, and the radius of the dot. We furthermore demonstrate that forward scattering of light or sound can almost switched o…
We consider a quantum dot described by a cylindrically symmetric 2D Dirac equation. The potentials representing the quantum dot are taken to be of different types of potential configuration, scalar, vector and pseudo-scalar to enable us to enrich our study. Using various potential configurations, we found that in the presence of a mass term an electrostatically confined quantum dot can accommodate true bound states, which is in agreement with previous work. The differential cross section associated with one specific potential configuration has been computed and discussed as function of the var…
We study the localization of dressed Dirac electrons in a cylindrical quantum dot (QD) formed on monolayer and bilayer graphene by spatially different potential profiles. Short lived excitonic states which are too broad to be resolved in linear spectroscopy are revealed by cross peaks in the photon-echo nonlinear technique. Signatures of the dynamic gap in the two-dimensional spectra are discussed. The effect of the Coulomb induced exciton-exciton scattering and the formation of biexciton molecules are demonstrated.
We investigate the conductance and thermopower of massless Dirac fermions through a quantum dot using a pseudogap Anderson model in the non-crossing approximation. When the Fermi level is at the Dirac point, the conductance has a cusp where the thermopower changes its sign. When the Fermi level is away from the Dirac point, the Kondo temperature illustrates a quantum impurity transition between an asymmetric strong coupling Kondo state and a localized moment state. The conductance shows a peak near this transition and reaches the unitary limit at low temperatures. The magnitude of the thermopo…
We show that in bilayer graphene it is possible to achieve a very restrictive confinement of the massless Dirac fermions zero-modes by using inhomogeneous magnetic fields. Specifically, we show that, using a suitable nonuniform magnetic fields, the wave function may be restricted to a specific region of the space, being forbidden all transmission probability to the contiguous regions. This allows to construct mesoscopic structures in bilayer graphene by magnetic fields configurations.
Within an effective Dirac-Weyl theory we solve the scattering problem for massless chiral fermions impinging on a cylindrical time-dependent potential barrier. The set-up we consider can be used to model the electron propagation in a monolayer of graphene with harmonically driven quantum dots. For static small-sized quantum dots scattering resonances enable particle confinement and interference effects may switch forward scattering on and off. An oscillating dot may cause inelastic scattering by excitation of states with energies shifted by integer multiples of the oscillation frequency, which…
We develop the stub model for the Dirac Quantum Dot, an electron confining device on a grapheme surface. Analytical results for the average conductance and the correlation functions are obtained and found in agreement with those found previously using semiclassical calculation. Comparison with available data are presented. The results reported here demonstrate the applicability of Random Matrix Theory in the case of Dirac electrons confined in a stadium.
We study the confinement of Dirac fermions in armchair graphene nanoribbons by means of a quantum-dot-type electrostatic potential. With the use of specific projection operators, we find exact solutions for some bound states that satisfy appropriate boundary conditions. We show that the energies of these bound states belong either to the gap of valence and conducting bands or they represent BIC's (bound states in the continuum) whose energies are embedded in the continuous spectrum.
We report on a theoretical study on the rise of radiation-induced magnetoresistance oscillations in two-dimensional systems of massive Dirac fermions. We study the bilayer system of monolayer graphene and hexagonal boron nitride (h-BN/graphene) and the trilayer system of hexagonal boron nitride encapsulated graphene (h-BN/graphene/h-BN). We extend the radiation-driven electron orbit model that was previously devised to study the same oscillations in two-dimensional systems of Schrödinger electrons (GaAs/AlGaAS heterostructure) to the case of massive Dirac fermions. In the simulations we obtain…
Graphene nanoribbon quantum dot qubits have been proposed as promising candidates for quantum computing applications to overcome the spin-decoherence problems associated with typical semiconductor (e.g. GaAs) quantum dot qubits. We perform theoretical studies of the electronic structures of graphene nanoribbon quantum dots by solving the Dirac equation with appropriate boundary conditions. We then evaluate the exchange splitting based on an unrestricted Hartree-Fock method for the Dirac particles. The electronic wave function and long-range exchange coupling due to the Klein tunneling and the …
The dynamics of low energy charge carriers in a graphene quantum dot subjected to a time-dependent local field is investigated numerically. In particular, we study a configuration where a Coulomb electric field is provided by an ion traversing the graphene sample. A Galerkin-like numerical scheme is introduced to solve the massless Dirac equation describing charge carriers subjected to space- and time-dependent electromagnetic potentials and is used to evaluate the field induced interband transitions. It is demonstrated that as the ion goes through graphene, electron-hole pairs are generated d…
We discuss a geometric approach to confining a Dirac neutral particle with a permanent magnetic dipole moment interacting with external fields to a hard-wall confining potential in the Minkowski spacetime through noninertial effects. We discuss the behaviour of external fields induced by noninertial effects, and a case where relativistic bound states can be achieved in analogous way to having a Dirac particle confined to a quantum dot. We show that this confinement of a Dirac neutral particle analogous to a quantum dot arises from noninertial effects that give rise to the geometry of the manif…
We study the Dirac electron scattering problem on a potential barrier in a circular quantum dot of Monolayer Molybdenum Disulfide quantum dot MoS2 subjected to magnetic flux. By solving Dirac's equation, we formulate analytical expressions for the eigenstates, scattering coefficients, scattering efficiency, and the reflected current's radial component. We show that the scattering coefficients, the scattering efficiency, and the radial component of the reflected current depend explicitly on the magnetic flux. The magnetic flux may cause a slight shift in the oscillation position of the scatteri…
We demonstrate quantum dot (QD) formation in three-dimensional Dirac semimetal Cd$_{3}$As$_{2}$ nanowires using two electrostatically tuned p$-$n junctions with a gate and magnetic fields. The linear conductance measured as a function of gate voltage under high magnetic fields is strongly suppressed at the Dirac point close to zero conductance, showing strong conductance oscillations. Remarkably, in this regime, the Cd$_{3}$As$_{2}$ nanowire device exhibits Coulomb diamond features, indicating that a clean single QD forms in the Dirac semimetal nanowire. Our results show that a p$-$type QD can…
We obtain the solution of the Dirac equation in (2+1) dimensions in the presence of a constant magnetic field normal to the plane together with a two-dimensional Dirac-oscillator potential coupling. We study the energy spectrum of graphene quantum dot (QD) defined by electrostatic gates. We give discussions of our results based on different physical settings, whether the cyclotron frequency is similar or larger/smaller compared to the oscillator frequency. This defines an effective magnetic field that produces the effective quantized Landau levels. We study analytically such field in gate-tuna…
Coulomb interaction is of central importance in localized energy levels (bound states) or electronic flat bands and could result in many exotic quantum phases, such as magnetic, superconducting, and topological phases in graphene systems1-14. In graphene monolayer, the relativistic massless Dirac fermion nature of the charge carriers enables us to realize unprecedented quasibound states, which are trapped temporarily via whispering-gallery modes (WGMs) with the lifetime (trapping time) of ~ 10 fs, in circular graphene quantum dots (GQDs)15-20. Here we show that Coulomb interaction still plays …
We investigate the propagation of waves in one-dimensional systems with Lévy-type disorder. We perform a complete analysis of non-relativistic and relativistic wave transmission submitted to potential barriers whose width, separation or both follow Lévy distributions characterized by an exponent $0 < α<1$. For the first two cases, where one of the parameters is fixed, non-relativistic and relativistic waves present anomalous localization, $\langle T \rangle \propto L^{-α}$. However, for the latter case, in which both parameters follow a Lévy distribution, non-relativistic and relativistic wave…
The dynamics of Dirac-Weyl spin-polarized wavepackets driven by periodic electric field is considered for the electrons in a mesoscopic quantum dot formed at the edge of two-dimensional HgTe/CdTe topological insulator with Dirac-Weyl massless energy spectra, where the motion of carriers is less sensitive to disorder and impurity potentials. It was observed that the interplay of strongly coupled spin and charge degrees of freedom creates the regimes of irregular dynamics both in coordinate and spin channels. The border between the regular and irregular regimes determined by the strength and fre…
We show how it is possible to trap two-dimensional massless Dirac fermions in spatially inhomogeneous magnetic fields, as long as the formed magnetic quantum dot (or ring) is of a slowly decaying nature. It is found that a modulation of the depth of the magnetic quantum dot leads to successive confinement-deconfinement transitions of vortexlike states with a certain angular momentum, until a regime is reached where only states with one sign of angular momentum are supported. We illustrate these characteristics with both exact solutions and a hitherto unknown quasi-exactly solvable model utiliz…
Generalized solutions of the standard gauge transformation equations are presented and discussed in physical terms. They go beyond the usual Dirac phase factors and they exhibit nonlocal quantal behavior, with the well-known Relativistic Causality of classical fields affecting directly the phases of wavefunctions in the Schrödinger Picture. These nonlocal phase behaviors, apparently overlooked in path-integral approaches, give a natural account of the dynamical nonlocality character of the various (even static) Aharonov-Bohm phenomena, while at the same time they seem to respect Causality. For…
