Massless and Massive Particle-in-a-Box States in Single- and Bi-Layer Graphene

Electron transport through short, phase-coherent metal-graphene-metal devices occurs via resonant transmission through particle-in-a-box-like states defined by the atomically-sharp metal leads. We study the spectrum of particle-in-a-box states for single- and bi-layer graphene, corresponding to massless and massive two-dimensional (2-D) fermions. The density of states D as a function of particle number n shows the expected relationships D(n) ~ n^1/2 for massless 2-D fermions (electrons in single-layer graphene) and D(n) ~ constant for massive 2-D fermions (electrons in bi-layer graphene). The single parameters of the massless and massive dispersion relations are found, namely Fermi velocity ʋ_F = 1.1 × 10⁶ m/s and effective mass m^* = 0.032 m_e, where m_e is the electron mass, in excellent agreement with theoretical expectations.