No Arabic abstract
In a Dirac material we investigated the confining properties of massive and massless particles subjected to a potential well generated by a purely electrical potential, that is, an electric quantum dot. To achieve this in the most exhaustive way, we have worked on the aforementioned problem for charged particles with and without mass, limited to moving on a plane and whose dynamics are governed by the Dirac equation. The bound states are studied first and then the resonances, the latter by means of the Wigner time delay of the dispersion states as well as through the complex eigenvalues of the outgoing states, in order to obtain a complete picture of the confinement. One of the main results obtained and described in detail is that electric quantum dots for massless charges seem to act as sinks (or sources in the opposite direction) of resonances, while for massive particles the resonances and bound states are conserved with varying position depending on the depth of the well.
Dirac particles have been notoriously difficult to confine. Implementing a curved space Dirac equation solver based on the quantum Lattice Boltzmann method, we show that curvature in a 2-D space can confine a portion of a charged, mass-less Dirac fermion wave-packet. This is equivalent to a finite probability of confining the Dirac fermion within a curved space region. We propose a general power law expression for the probability of confinement with respect to average spatial curvature for the studied geometry.
We study excitonic effects in two-dimensional massless Dirac fermions with Coulomb interactions by solving the ladder approximation to the Bethe-Salpeter equation. It is found that the general 4-leg vertex has a power law behavior with the exponent going from real to complex as the coupling constant is increased. This change of behavior is manifested in the antisymmetric response, which displays power law behavior at small wavevectors reminiscent of a critical state, and a change in this power law from real to complex that is accompanied by poles in the response function for finite size systems, suggesting a phase transition for strong enough interactions. The density-density response is also calculated, for which no critical behavior is found. We demonstrate that exciton correlations enhance the cusp in the irreducible polarizability at $2k_F$, leading to a strong increase in the amplitude of Friedel oscillations around a charged impurity.
Motivated by the conduction properties of graphene discovered and studied in the last decades, we consider the quantum dynamics of a massless, charged, spin 1/2 relativistic particle in three dimensional space-time, in the presence of an electrostatic field in various configurations such as step or barrier potentials and generalizations of them. The field is taken as parallel to the y coordinate axis and vanishing outside of a band parallel to the x axis. The classical theory is reviewed, together with its canonical quantization leading to the Dirac equation for a 2-component spinor. Stationary solutions are numerically found for each of the field configurations considered, fromwhich we calculate the mean quantum trajectories of the particle and compare them with the corresponding classical trajectories, the latter showing a classical version of the Klein phenomenon. Transmission and reflection probabilities are also calculated, confirming the Klein phenomenon.
Effect of spin-orbit coupling (SOC) on Dirac electrons in the organic conductor $alpha$-(BETS)$_2$I$_3$ [BETS = bis(ethylenedithio)tetraselenafulvalene] has been examined by calculating electric conductivity and spin magnetic susceptibility. A tight-binding (TB) model with transfer energies consisting of real and imaginary parts is evaluated using first-principles density-functional theory calculation. The conductivity without SOC depends on both anisotropies of the velocity of the Dirac cone and the tiling of the cone. Such conductivity is suppressed by the SOC, which gives rise to the imaginary part of the transfer energy. It is shown at low temperatures that the conductivity decreases due to the SOC and the Dirac cone with linear dispersion. A nearly constant conductivity at high temperatures is obtained by an electron-phonon (e--p) scattering. Further, the property of the Dirac cone is examined for spin susceptibility, which is mainly determined by the density of states (DOS). The result is compared with the case of the organic conductor $alpha$-(BEDT-TTF)$_2$I$_3$ [BEDT-TTF=bis(ethylenedithio)tetrathiafulvalene], which provides the Dirac cone without SOC. The relevance to experiments is discussed.
We consider the Dirac equation on periodic networks (quantum graphs). The self-adjoint quasi periodic boundary conditions are derived. The secular equation allowing us to find the energy spectrum of the Dirac particles on periodic quantum graphs is obtained. Band spectra of the periodic quantum graphs of different topologies are calculated. Universality of the probability to be in the spectrum for certain graph topologies is observed.