The transfer matrix approach to circular graphene quantum dots

Abstract

We adapt the transfer matrix (T-matrix) method originally designed for one-dimensional quantum mechanical problems to solve the circularly symmetric two-dimensional problem of graphene quantum dots. In similarity to one-dimensional problems, we show that the generalized T-matrix contains rich information about the physical properties of these quantum dots. In particular, it is shown that the spectral equations for bound states as well as quasi-bound states of a circular graphene quantum dot and related quantities such as the local density of states and the scattering coefficients are all expressed exactly in terms of the T-matrix for the radial confinement potential.As an example, we use the developed formalism to analyse physical aspects of a graphene quantum dot induced by a trapezoidal radial potential. Among the obtained results, it is in particular suggested that the thermal fluctuations and electrostatic disorders may appear as an obstacle to controlling the valley polarization of Dirac electrons.

Quantitative study of electronic whispering gallery modes in electrostatic-potential induced circular graphene junctions

“…Given U (r), we computed LDOSs for the studied resonator, using the approach suggested in Refs. [19,20](see Supplementary Materials). Shortly, the computing procedure is as following: (i) solving the Dirac equation of Hamiltonian (1) to calculate LDOSs with a given angular momentum j -the partial LDOSs (S4); (ii) taking the sum of partial LDOSs over all possible j provides LDOS ρ(E, r) (S3) that depends on the energy E and the distance r; and (iii) integrating ρ(E, r) over r provides the total density of states (TDOS) ρ T (E) (S9).…”

“…( 2) and (3) also lies in their simplicity so that the Hamiltonian of eq. ( 1) could be exactly solved [19,23].…”

On the density of states of circular graphene quantum dots

“…For the same QBSs of j = 1 2 , 3 2 , 5 2 , and 7 2 of the studied CGQD, we also calculate (− Im E) of the QBS complex energies E using our T -matrix approach suggested in Ref. [22]. For a given QBS, the quantity (− Im E) should provide a direct measure of the resonance width.…”

“…This approach equally applies to practically any structure created by axially symmetric electrostatic potentials on a continuous graphene sheet. It can be easily extended to include a mass term in the Hamiltonian (1) [22]. Under an external magnetic field, the current formulation does not however apply directly and further studies are needed; see [7,14,15,24] for alternative approaches.…”

“…( 3) can be solved. The case when the potential can be considered to be flat near the origin, namely, U (r) = U i for r ≤ r i , has been studied using the T -matrix method [22]. In this case, the eigenfunction of eq.…”