Dirac-Weyl Equation Research Articles

A note on degeneracy of excited energy levels in massless Dirac fermions

We propose a mechanism to construct the eigenvalues and eigenfunctions of the massless Dirac-Weyl equation in the presences of magnetic flux Φ localized in a restricted region of the plane. Using this mechanism we analyze the degeneracy of the existed energy levels. We find that the zero and first energy level has the same N+1 degeneracy, where N is the integer part of Φ2π. In addition, and contrary to what is described in the literature regarding graphene, we show that higher energy levels are N+m degenerate, being m the level of energy. In other words, this implies an indefinite growth of degenerate states as the energy level grows.

On the Majorana representation of the optical Dirac equation

We consider the representations of the optical Dirac equation, especially ones where the Hamiltonian is purely real-valued. This is equivalent, for Maxwell’s equations, to the Majorana representation of the massless Dirac (Weyl) equation. We draw analogies between the Dirac, chiral and Majorana representations of the Dirac and optical Dirac equations, and derive two new optical Majorana representations. Just as the Dirac and chiral representations are related to optical spin and helicity states, these Majorana representations of the optical Dirac equation are associated with the linear polarization of light. This provides a means to compare electron and electromagnetic wave equations in the context of classical field theory.

Graphene Dirac fermions in symmetric electric and magnetic fields: the case of an electric square well

In this paper, a simple method is proposed to get analytical solutions (or with the help of a few numerical calculations) of the Dirac-Weyl equation for low energy electrons in graphene in the presence of certain electric and magnetic fields. In order to decouple the Dirac-Weyl equation we have assumed a displacement symmetry of the system along a direction and some conditions on the magnetic and electric fields. The resulting equations have the natural form to apply the technique of supersymmetric quantum mechanics. The example of an electric well with square profile is worked out in detail to illustrate some of the most interesting features of this procedure.

Electronic Mach-Zehnder interference in a bipolar hybrid monolayer-bilayer graphene junction

Graphene matter in a strong magnetic field, realizing one-dimensional quantum Hall channels, provides a unique platform for studying electron interference. Here, using the Landauer-Büttiker formalism along with the tight-binding model, we investigate the quantum Hall (QH) effects in unipolar and bipolar monolayer-bilayer graphene (MLG-BLG) junctions. We find that a Hall bar made of an armchair MLG-BLG junction in the bipolar regime results in valley-polarized edgechannel interferences and can operate a fully tunable Mach-Zehnder (MZ) interferometer device. Investigation of the bar-width and magnetic-field dependence of the conductance oscillations shows that the MZ interference in such structures can be drastically affected by the type of (zigzag) edge termination of the second layer in the BLG region [composed of vertical dimer or non-dimer atoms]. Our findings reveal that both interfaces exhibit a double set of Aharonov-Bohm interferences, with the one between two oppositely valley-polarized edge channels dominating and causing a large amplitude conductance oscillation ranging from 0 to 2e2/h. We explain and analyze our findings by analytically solving the Dirac-Weyl equation for a gated semi-infinite MLG-BLG junction.

Isolated and hybrid bilayer graphene quantum rings

Using the continuum model, we investigate the electronic properties of two types of bilayer graphene (BLG) quantum ring (QR) geometries: (i) an isolated BLG QR and (ii) a monolayer graphene (MLG) with a QR put on top of an infinite graphene sheet (hybrid BLG QR). Solving the Dirac-Weyl equation in the presence of a perpendicular magnetic field and applying the infinite-mass boundary condition at the ring boundaries, we obtain analytical results for the energy levels and corresponding wave spinors for both structures. In the case of isolated BLG QR, we observe a sizeable and magnetically tunable band gap which agrees with the tight-binding transport simulations. Our analytical results also show the intervalley symmetry $ E^K_e (m) = -E^{K'}_h(m) $ between the electron (e) and hole (h) states ($ m $ being the angular momentum quantum number) for the energy spectrum of the isolated BLG QR. The presence of interface boundary in a hybrid BLG QR modifies drastically the energy levels as compared to that of an isolated BLG QR. Its energy levels are tunable from MLG dot, to isolated BLG QR, and to MLG Landau energy levels as magnetic field is varied. Our predictions can be verified experimentally using different techniques such as by magnetotransport measurements.

Electronic Mach-Zehnder Interference in a Bipolar Hybrid Monolayer-Bilayer Graphene Junction

Graphene matter in a strong magnetic field, realizing one-dimensional quantum Hall channels, provides a unique platform for studying electron interference. Here, using the Landauer-B\"uttiker formalism along with the tight-binding model, we investigate the quantum Hall (QH) effects in unipolar and bipolar monolayer-bilayer graphene (MLG-BLG) junctions. We find that a Hall bar made of an armchair MLG-BLG junction in the bipolar regime results in valley-polarized edge-channel interferences and can operate a fully tunable Mach-Zehnder (MZ) interferometer device. Investigation of the bar-width and magnetic-field dependence of the conductance oscillations shows that the MZ interference in such structures can be drastically affected by the type of (zigzag) edge termination of the second layer in the BLG region [composed of vertical dimer or non-dimer atoms]. Our findings reveal that both interfaces exhibit a double set of Aharonov-Bohm interferences, with the one between two oppositely valley-polarized edge channels dominating and causing a large-amplitude conductance oscillation ranging from 0 to $ 2 e^2 / h$. We explain and analyze our findings by analytically solving the Dirac-Weyl equation for a gated semi-infinite MLG-BLG junction.

Electronic properties and quasi-zero-energy states of graphene quantum dots

In this paper, research has been carried out on the electronic properties of nanostructured graphene. We focus our attention on trapped states of the proposed systems such as spherical and toroidal graphene quantum dots (GQDs). Using a continuum model, by solving the Dirac-Weyl equation, and applying periodic boundary conditions of two types, i.e., either with zigzag edges only or with both armchair and zigzag edges, we obtain analytical results for energy levels yielding self-similar energy bands located subsequently one after another on the energy scale. Only for the toroidal quantum dot (owing to the lack of curvature) the distribution of electron density is like Bohr atomic orbitals. However, although the quasi-zero-energy band exists for both spherical and toroidal quantum dots, no electron density is present on this band for the toroidal quantum dot. This causes the formation of a pseudogap between the hole and electron bands because of the absence of the electron density at the quantum dot center, like in the case of an ordinary atom. Conversely, the confinement of the charge-carrier density is observed for both geometries of GQDs.

Thermodynamic properties of massless Dirac–Weyl fermions under the generalized uncertainty principle* *Project supported by the National Natural Science Foundation of China (Grant No. 11565009).

The Dirac–Weyl equation characterized quasi-particles in the T3 lattice are studied under external magnetic field using the generalized uncertainty principle (GUP). The energy spectrum of the quasi-particles is found by the Nikiforov–Uvarov method. Based on the energy spectrum obtained, the thermodynamic properties are given, and the influence of the GUP on the statistical properties of systems is discussed. The results show that the energy and thermodynamic functions of massless Dirac–Weyl fermions in the T3 lattice depend on the variation of the GUP parameter.

The solutions of Dirac equation on the hyperboloid under perpendicular magnetic fields

In this study, firstly it is reviewed how the solutions of the Dirac-Weyl equation for a massless charge on the hyperboloid under perpendicular magnetic fields are obtained by using supersymmetric (SUSY) quantum mechanics methods. Then, the solutions of the Dirac equation for a massive charge under magnetic fields have been computed in terms of the solutions which were found before for the Dirac-Weyl equation. As an example, the case of a constant magnetic field on the hyperbolic surface for massless and massive charges has been worked out.

A polynomial approach to the spectrum of Dirac–Weyl polygonal Billiards

The Schrödinger equation in a square or rectangle with hard walls is solved in every introductory quantum mechanics course. Solutions for other polygonal enclosures only exist in a very restricted class of polygons, and are all based on a result obtained by Lamé in 1852. Any enclosure can, of course, be addressed by finite element methods for partial differential equations. In this paper, we present a variational method to approximate the low-energy spectrum and wave-functions for arbitrary convex polygonal enclosures, developed initially for the study of vibrational modes of plates. In view of the recent interest in the spectrum of quantum dots of two dimensional materials, described by effective models with massless electrons, we extend the method to the Dirac–Weyl equation for a spin-1/2 fermion confined in a quantum billiard of polygonal shape, with different types of boundary conditions. We illustrate the method’s convergence in cases where the spectrum is known exactly, and apply it to cases where no exact solution exists.

Coherent states for graphene under the interaction of crossed electric and magnetic fields

We construct the coherent states for charge carriers in a graphene layer immersed in crossed external electric and magnetic fields. For that purpose, we solve the Dirac–Weyl equation in a Landau-like gauge avoiding applying techniques of special relativity, and thus we identify the appropriate raising and lowering operators associated to the system. We explicitly construct the coherent states as eigenstates of a matrix annihilation operator with complex eigenvalues. In order to describe the effects of both fields on these states, we obtain the probability and current densities, the Heisenberg uncertainty relation and the mean energy as functions of the parameter β=cE∕(vFB). In particular, these quantities are investigated for magnetic and electric fields near the condition of the Landau levels collapse (β→1).

Quantum dots in AA-stacked bilayer graphene

While electrostatic confinement in single-layer graphene and AA-stacked bilayer graphene is precluded by Klein tunneling and the gapless energy spectrum, we theoretically show that a circular domain wall that separates domains of single-layer graphene and AA-stacked bilayer graphene can provide bound states. Solving the Dirac-Weyl equation in the presence of a global mass potential and a local electrostatic potential, we obtain the energy spectrum of these states and the corresponding radial probability densities. Depending on the mass potential profile, regular bound states can exist inside the quantum dot and topological bound states at the domain wall. Controlling the electrostatic potential inside the quantum dot enables the simultaneous presence of both types of states. We find that the number of nodes of the radial wave function of the regular bound states inside the quantum dot is equal to the radial quantum number. The energy spectra of the bound states display anticrossings, reflecting coupling of electron- and holelike states.

Orbital magnetization in axially symmetric two-dimensional carbon allotrope: influence of electric field and geometry

In this paper, we use supersymmetry formalism to study (2 + 1) dimensional Dirac–Weyl equation describing the motion of electrons in different axially symmetric carbon allotrope surfaces under the influence of crossed electromagnetic fields. In particular, we consider three carbon allotrope with different geometrical shapes and properties: open, half-open and closed geometries. The effect of the electric field and geometry on Landau levels and orbital magnetization has been examined. It has been shown that at a critical electric field the orbital magnetization exhibits a discontinuity which is associated with the collapse of Landau levels.

Dirac-Weyl equation on a hyperbolic graphene surface under magnetic fields

In this paper the Dirac-Weyl equation on a hyperbolic surface of graphene under magnetic fields is considered. In order to solve this equation analytically for some cases, we will deal with vector potentials symmetric under rotations around the z axis. Instead of using tetrads we will get this equation from a more intuitive point of view by restriction from the Dirac-Weyl equation of an ambient space. The eigenvalues and corresponding eigenfunctions for some magnetic fields are found by means of the factorization method. The existence of a zero energy ground level and its degeneracy is also analysed in relation to the Aharonov-Casher theorem valid for flat graphene.

Deformation induced pseudomagnetic fields in complex carbon architectures

We show that the physics of deformation in $\alpha$-, $\beta$-, and $6,6,12$-graphyne is, despite their significantly more complex lattice structures, remarkably close to that of graphene, with inhomogeneously strained graphyne described at low energies by an emergent Dirac-Weyl equation augmented by strain induced electric and pseudo-magnetic fields. To show this we develop two continuum theories of deformation in these materials: one that describes the low energy degrees of freedom of the conical intersection, and is spinor valued as in graphene, and one describing the full sub-lattice space. The spinor valued continuum theory agrees very well with the full continuum theory at low energies, showing that the remarkable physics of deformation in graphene generalizes to these more complex carbon architectures. In particular, we find that deformation induced pseudospin polarization and valley current loops, key phenomena in the deformation physics of graphene, both have their counterpart in these more complex carbon materials.

Enhancing the energy spectrum of graphene quantum dot with external magnetic and Aharonov-Bohm flux fields

In this paper, we have to apply the Dirac-Weyl equation to find the analytical energy eigenvalues of the graphene quantum dot interacting in the presence of AB-flux field and external magnetic field. We find that the energy eigenvalue of the graphene quantum dot decreases with both magnetic and AB-flux field but the effect of AB-flux field is more dominant. By ameliorating the intensity of the AB-flux field and keeping the magnetic field constant, the quantum-dot states entangled to produce Landau Levels. We show that besides using the graphene sheet and external magnetic field, the Aharonov-Bohm AB-flux field could as well be used to manipulate the carriers state energies in graphene.

Atomic collapse in pseudospin-1 systems

We investigate atomic collapse in pseudospin-1 Dirac material systems whose energy band structure constitutes a pair of Dirac cones and a flat band through the conic intersecting point. We obtain analytic solutions of the Dirac-Weyl equation for the three-component spinor in the presence of a Coulomb impurity and derive a general criterion for the occurrence of atomic collapse in terms of the normalized strength of Coulomb interaction and the angular momentum quantum number. In particular, for the lowest angular momentum state, the solution coincides with that for pseudospin-1/2 systems, but with a reduction in the density of resonance peaks. For higher angular momentum states, the underlying pseudospin-1 wave functions exhibit a singularity at the point of zero kinetic energy. Divergence of the local density of states associated with the flat band leads to an inverse square type of singularity in the conductivity. These results provide insights into the physics of the two-body problem for relativistic quantum pseudospin-1 quasiparticle systems.

Comment on “Effective of the q-deformed pseudoscalar magnetic field on the charge carriers in graphene” [J. Math. Phys. 57, 082105 (2016)

We point out a misleading treatment in a recent paper published in this journal [M. Eshgi and H. Mehraban, J. Math. Phys. 57, 082105 (2016)] concerning solutions for the two-dimensional Dirac-Weyl equation with a q-deformed pseudoscalar magnetic barrier. The authors misunderstood the full meaning of the potential and made erroneous calculations, and this fact jeopardizes the main results in this system.

Straintronics beyond homogeneous deformation

We present a continuum theory of graphene treating on an equal footing both homogeneous Cauchy-Born (CB) deformation, as well as the microscopic degrees of freedom associated with the two sublattices. While our theory recovers all extant results from homogeneous continuum theory, the Dirac-Weyl equation is found to be augmented by new pseudo-gauge and chiral fields fundamentally different from those that result from homogeneous deformation. We elucidate three striking electronic consequences: (i) non-CB deformations allow for the transport of valley polarized charge over arbitrarily long distances e.g. along a designed ridge; (ii) the triaxial deformations required to generate an approximately uniform magnetic field are unnecessary with non-CB deformation; and finally (iii) the vanishing of the effects of a one dimensional corrugation seen in \emph{ab-initio} calculation upon lattice relaxation are explained as a compensation of CB and non-CB deformation.

Solutions of the Dirac-Weyl equation in graphene under magnetic fields in the Cartesian coordinate system

In this paper, we obtain the exact analytical solutions of the Dirac-Weyl equation in the Cartesian coordinate system in various magnetic fields. We obtain the eigenfunctions and the energy spectrum by the Nikoforov-Uvarov (NU) methods.