Quasiparticles for quantum dot array in graphene and the associated Magnetoplasmons

We calculate the low-frequency magnetoplasmon excitation spectrum for a square array of quantum dots on a two-dimensional (2D) graphene layer. The confining potential is linear in the displacement from the center of the quantum dot. Consequently, the corresponding Klein-Gordon equation may be solved analytically for the single-particle eigenstates since they are given by a simple harmonic oscillator operator. The electron eigenstates in a magnetic field and confining potential are mapped onto a 2D plane of electron-hole pairs in an effective magnetic field without any confinement. The tight-binding model for the array of quantum dots leads to a wavefunction with inter-dot mixing of the quantum numbers associated with an isolated quantum dot. It is verified that our tight-binding wave function obeys the Bloch-Peierls condition in the Landau gauge, For chosen confinement, magnetic field, wave vector and frequency, we plot the dispersion equation as a function of the period $d$ of the lattice. We obtain those values of $d$ which yield collective plasma excitations. For the allowed transitions between the valence and conduction bands in our calculations, we obtain plasmons when $d < 100 {\AA} $.