Two-dimensional Dirac Equation Research Articles

Solvable Two-Dimensional Dirac Equation with Matrix Potential: Graphene in External Electromagnetic Field

It is known that the excitations in graphene-like materials in external electromagnetic field are described by solutions of a massless two-dimensional Dirac equation which includes both Hermitian off-diagonal matrix and scalar potentials. Up to now, such two-component wave functions were calculated for different forms of external potentials, though as a rule depending on only one spatial variable. Here, we shall find analytically the solutions for a wide class of combinations of matrix and scalar external potentials which physically correspond to applied mutually orthogonal magnetic and longitudinal electrostatic fields, both depending really on two spatial variables. The main tool for this progress is provided by supersymmetrical (SUSY) intertwining relations, specifically, by their most general—asymmetrical—form proposed recently by the authors. This SUSY-like method is applied in two steps, similar to the second order factorizable (reducible) SUSY transformations in ordinary quantum mechanics.

Improved uniform error estimates for the two-dimensional nonlinear space fractional Dirac equation with small potentials over long-time dynamics

We develop improved uniform error bounds on a second-order Strang splitting method for the long-time dynamics of the nonlinear space fractional Dirac equation (NSFDE) in two dimension (2D) with small electromagnetic potentials. First, a Strang splitting approach is implemented to discretize NSFDE in time. Afterwards the Fourier pseudospectral method is used to complete the discretization of NSFDE in space. With the aid of a second-order Strang splitting approach employed to the Dirac equation, the major local truncation error of the indicated numerical methods is established. Moreover, for the semi-discrete scheme and full-discretization, we rigorously demonstrate the improved, sharp uniform error estimates are O(ετ2) and O(h1m+h2m+ετ2) in virtue of the regularity compensation oscillation (RCO) technique. In the formulations, τ is the time step, hi(i=1,2) stands for spatial sizes in xi-directions, m is dependent on the regularity of solutions, and ε∈(0,1]. In order to verify our error bounds and to illustrate some fascinating long-time dynamical behaviors of the NSFDE with honeycomb lattice potentials for varied ε, numerical investigations are presented.

Darboux algorithms for two-dimensional Dirac equations with upper triangular potential matrix

We construct three algorithms of Darboux transformations for the two-dimensional Dirac equation with upper triangular matrix potential. Each algorithm splits in two different variants. The potential is allowed to depend arbitrarily on the two variables, and the Darboux transformations are expressed in terms of generalized Wronskians.

A new class of solvable two-dimensional scalar potentials for graphene

In the present paper, a systematic approach is presented for solution of two-dimensional massless Dirac equation with external electrostatic potential applied. This approach is based on the new - asymmetric - form of SUSY-like intertwining relations. It allows to build a wide variety of pairs of SUSY-partner external scalar potentials. If one of them is simple enough to be solvable, its partner is also solvable although it may have a non-trivial dependency on both coordinates. Physically, this kind of problems is related to the description of graphene and some other materials with external potential. Solvability obtained by means of asymmetric form of SUSY intertwining relations allows to extend the class of analytically solvable two-dimensional models.

Particlelike valleytronics in graphene

Particlelike scattering states allow for noiseless transport through quantum dots by closely retracing bundles of classical trajectories. We identify such raylike states for electron transport through graphene ribbons. Remarkably, we find that these quasiclassical scattering states can be unambiguously associated with well-defined quantum numbers of the valley degree of freedom specific to graphene. Trigonal warping---i.e., deviations of the band structure from a perfectly isotropic two-dimensional Dirac equation due to the hexagonal lattice structure---results in preferred propagation directions and scattering time delays that depend on the valley the particlelike wave travels in. By implementing a truncated mode basis of Bloch states, we achieve simulations of micrometer-sized quantum dots starting from an atomic-scale tight-binding Hamiltonian.

Multiphoton absorption and Rabi oscillations in armchair graphene nanoribbons

We present an analytical approach to the problem of the multiphoton absorption and Rabi oscillations in an armchair graphene nanoribbon (AGNR) in the presence of a time-oscillating strong electric field induced by a light wave directed parallel to the ribbon axis. The two-dimensional Dirac equation for the massless electron subject to the ribbon confinement is employed. In the resonant approximation the electron-hole pair production rate, associated with the electron transitions between the valence and conduction size-quantized subbands, the corresponding multiphoton absorption coefficient and the frequency of the Rabi oscillations are obtained in an explicit form. We trace the dependencies of the above quantities on the ribbon width and electric field strength for both the multiphoton assisted and tunneling regimes relevant to the time-oscillating and practically constant electric field, respectively. A significant enhancement effect of the oscillating character of the electric field on the intersubband transitions is encountered. Our analytical results are in qualitative agreement with those obtained for the graphene layer by numerical methods. Estimates of the expected experimental values for the typically employed AGNR and laser parameters show that both the Rabi oscillations and multiphoton absorption are accessible in the laboratory. The data relevant to the intersubband tunneling makes the AGNR a 1D condensed matter analog in which the quantum electrodynamic vacuum decay can be detected by applying an external laboratory electric field.

DFT-NEGF Study of Electron Transfer Property in Graphene Nanoribbon Double Barrier System

Current-voltage (I – V) characteristics in armchair graphene nanoribbon (AGNR) double barrier systems (DBSs) are investigated by the density functional theory (DFT) and the non-equilibrium Green’s function (NEGF) methods. The hexagonal boron nitride (h-BN) which has an analogous structure with a graphene shows wide bandgap and becomes an electrical barrier in DBSs. The heterojunction structure of AGNR and two h-BNs with one six-membered ring width has properties as a resonant tunnelling device. Due to the resonance, sharp peaks are formed in the transmission function around the Fermi energy EF . Our previous studies have explained the origins of some peaks by solving the one-dimensional Dirac equation for massless particles. Here, by using the two-dimensional Dirac equation, we show that the remaining sharp peaks depend on the transverse nodal pattern of the wavefunction as well as the anisotropy of the Fermi velocity. Moreover, in the I – V characteristics, a flat part is formed without the negative differential resistance appearing. By partial density of state (PDOS) analysis, we reveal that the flat part in the current is caused by the fact that the peak positions in the resonance region are stable against the bias voltage.

Higher-order Darboux transformations for two-dimensional Dirac systems with diagonal matrix potential

We construct the explicit form of higher-order Darboux transformations for the two-dimensional Dirac equation with diagonal matrix potential. The matrix potential entries can depend arbitrarily on the two variables. Our construction is based on results for coupled Korteweg-de Vries equations [27].

Multiphoton intersubband transitions in an armchair graphene nanoribbon

We present an analytical approach to the problem of the interband transitions in an armchair graphene nanoribbon (AGNR), exposed to the time-periodic electric field of strong light wave, polarized parallel to the ribbon axis. The two-dimensional Dirac equation for the massless electron subject to the ribbon confinement is employed. In the resonant approximation the probability of the transitions between the valence and conduction size-quantized subbands are calculated in an explicit form. We trace the dependencies of the Rabi frequency for these transitions on the ribbon width and electric field strength for both the multiphoton-assisted and tunneling regimes relevant to the fast oscillating and practically constant electric field, respectively. Estimates of the expected experimental values for the typically employed AGNR and laser technique facilities show that the Rabi oscillations can be observed under laboratory conditions. The data, corresponding to the intersubband tunneling, makes the AGNR a 1D condensed matter analog, in which the quantum electrodynamic vacuum decay can be detected by the employment of the attainable electric fields.

On the Effect of Intersection of Characteristics in a Two-Dimensional Massless Dirac Equation with Linear Potential and Localized Initial Condition

On the Effect of Intersection of Characteristics in a Two-Dimensional Massless Dirac Equation with Linear Potential and Localized Initial Condition

Closed-form representations of iterated Darboux transformations for the massless Dirac equation

It is shown that first-order Darboux transformations for the two-dimensional massless Dirac equation with scalar potential and for the Schrödinger equation are the same up to a change of coordinates. As a consequence, we obtain a closed-form representation of iterated, higher-order Darboux transformations for our Dirac equation. We use the formalism to generate several new exactly-solvable Dirac systems through higher-order Darboux transformations.

Three Discontinuous Galerkin Methods for One- and Two-Dimensional Nonlinear Dirac Equations with a Scalar Self-Interaction

Three Discontinuous Galerkin Methods for One- and Two-Dimensional Nonlinear Dirac Equations with a Scalar Self-Interaction

Fundamental solution of the stationary Dirac equation with a linear potential

We explicitly express the fundamental solution of the stationary two-dimensional massless Dirac equation with a constant electric field in terms of Fourier transforms of parabolic cylinder functions. This solution describes the flux of quasiparticles in graphene emitted by a pointlike source of electrons that are partially converted into holes (antiparticles). Using our explicit formula, we calculate its semiclassical asymptotic behavior in the hole region.

Guided modes and terahertz transitions for two-dimensional Dirac fermions in a smooth double-well potential

The double-well problem for the two-dimensional Dirac equation is solved for a family of quasi-one-dimensional potentials in terms of confluent Heun functions. We demonstrate that for a double well separated by a barrier, both the energy-level splitting associated with the wave-function overlap of well states and the gap size of the avoided crossings associated with well and barrier state repulsion can be controlled via the parameters of the potential. The transitions between the two states comprising a doublet, as well as transitions across the pseudogaps are strongly allowed, highly anisotropic, and, for realistic graphene devices, can be tuned to fall within the highly desirable terahertz frequency range.

On the Impact of Substrate Uniform Mechanical Tension on the Graphene Electronic Structure

Employing density functional theory calculations, we obtain the possibility of fine-tuning the bandgap in graphene deposited on the hexagonal boron nitride and graphitic carbon nitride substrates. We found that the graphene sheet located on these substrates possesses the semiconducting gap, and uniform biaxial mechanical deformation could provide its smooth fitting. Moreover, mechanical tension offers the ability to control the Dirac velocity in deposited graphene. We analyze the resonant scattering of charge carriers in states with zero total angular momentum using the effective two-dimensional radial Dirac equation. In particular, the dependence of the critical impurity charge on the uniform deformation of graphene on the boron nitride substrate is shown. It turned out that, under uniform stretching/compression, the critical charge decreases/increases monotonically. The elastic scattering phases of a hole by a supercritical impurity are calculated. It is found that the model of a uniform charge distribution over the small radius sphere gives sharper resonance when compared to the case of the ball of the same radius. Overall, resonant scattering by the impurity with the nearly critical charge is similar to the scattering by the potential with a low-permeable barrier in nonrelativistic quantum theory.

Higher-order Darboux transformations for the Dirac equation with position-dependent mass at nonvanishing energy

We construct Darboux transformations of arbitrary order for the two-dimensional Dirac equation within a position-dependent mass scenario. While we restrict the potential and the mass to a single dimension, the stationary energy remains arbitrary. We use our Darboux transformation to generate several new exactly solvable Dirac systems that admit bound states at nonzero energies.

Bound-state solutions of the two-dimensional Dirac equation with Aharonov–Bohm-Coulomb interaction in the presence of extended uncertainty principle

The Dirac equation in the presence of Aharonov–Bohm-Coulomb interaction is studied in the framework of the extended uncertainty principle. The equation governing the dynamics of the system is reduced to a differential equation which is solved in terms of the hypergeometric functions as the wave functions of the system. Expressions for the bound-state energies are also obtained analytically. A brief graphical evaluation of our results is also presented.

Darboux partners of Heun-class potentials for the two-dimensional massless Dirac equation

We apply the Darboux transformation to construct new exactly-solvable cases of the two-dimensional massless Dirac equation for potential classes of Lambert-W and inverse exponential type. Both of these classes originate from the Heun equation. Conditions are devised for transformed potentials to be real-valued, and to be in terms of elementary functions.

On the propagation of Dirac fermions in graphene with strain-induced inhomogeneous Fermi velocity

We consider systems described by a two-dimensional Dirac equation where the Fermi velocity is inhomogeneous as a consequence of mechanical deformations. We show that the mechanical deformations can lead to deflection and focusing of the wave packets. The analogy with known reflectionless quantum systems is pointed out. Furthermore, with the use of the qualitative spectral analysis, we discuss how inhomogeneous strains can be used to create waveguides for valley polarized transport of partially dispersionless wave packets.

Arbitrary-order Darboux transformations for two-dimensional Dirac equations with position-dependent mass

We construct higher-order Darboux transformations for Dirac equations in two dimensions that feature a position-dependent mass. Our method allows to generate closed-form expressions for both a transformed potential and a transformed position-dependent mass function.