Due to Klein tunneling in graphene only quasi-bound states are realized in graphene quantum dots by electrostatic gating. Particles in the quasi-bound states are trapped inside the dot for a finite time and they keep bouncing back and forth till they find their way out. Here we study the effect of an induced gap on the scattering problem of Dirac electrons on a circular electrostatically confined quantum dot. Introducing an energy gap inside the quantum dot enables us to distinguish three scattering regimes instead of two in the case of gapless graphene quantum dot. We will focus on these regimes and analyze the scattering efficiency as a function of the electron energy, the dot radius and the energy gap. Moreover, we will discuss how the system parameters can affect the scattering resonances inside the dot.

We study the Goos-Hänchen like shifts for Dirac fermions in graphene scattered by double barrier structures. After obtaining the solution for the energy spectrum, we use the boundary conditions to explicitly determine the Goos-Hänchen like shifts and the associated transmission probability. We analyze these two quantities at resonances by studying their main characteristics as a function of the energy and electrostatic potential parameters. To check the validity of our computations we recover previous results obtained for a single barrier under appropriate limits.

By applying the infinite-mass boundary condition, we analytically calculate the confined states and the corresponding wave functions of AA-stacked bilayer graphene quantum dots in the presence of an uniform magnetic field B. It is found that the energy spectrum shows two set of levels, which are the double copies of the energy spectrum for single layer graphene, shifted up-down by +γ and −γ, respectively. However, the obtained spectrum exhibits different symmetries between the electron and hole states as well as the intervalley symmetries. It is noticed that, the applied magnetic field breaks all symmetries, except one related to the intervalley electron-hole symmetry, i.e. E e (τ, m) = −E h (τ, m). Two different regimes of confinement are found: the first one is due to the infinite-mass barrier at weak B and the second is dominated by the magnetic field as long as B is large. We numerically investigated the basics features of the energy spectrum to show the main similarities and differences with respect to monolayer graphene, AB-stacked bilayer graphene and semiconductor quantum dots.

We present a systematic approach for the separation of variables for the two-dimensional Dirac equation in polar coordinates. The three vector potential, which couple to the Dirac spinor via minimal coupling, along with the scalar potential are chosen to have angular dependence which emanate the Dirac equation to complete separation of variables. Exact solutions are obtained for a class of solvable potentials along with their relativistic spinor wavefunctions. Particular attention is paid to the situation where the potentials are confined to a quantum dot region and are of scalar, vector and pseudo-scalar type. The study of a single charged impurity embedded in a 2D Dirac equation in the presence of a uniform magnetic field was treated as a particular case of our general study.

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-tunable graphene QD and show that our structure allow us to control the valley degeneracy. Finally, we compare our results with already published work and also discuss the possible applications of such QD.