Materials with Dirac electronic spectra ("Dirac materials") have attracted much interest since the first successful electronic transport measurement in graphene in 2004. Dirac quasiparticles have novel physical properties such as absence of backscattering and Klein tunneling. Topological insulators are a more recently discovered class of materials that have a bulk band gap and gapless edge/surface states. The surface state in 3D topological insulators has a Dirac electronic spectrum like graphene, but is singly spin-degenerate, with spin-momentum locking. This thesis will describe electronic transport experiments in graphene and in Bi₂Se₃ ultrathin films, which are predicted to be either 2D topological insulators or conventional insulators.

The basic quantum physics of a particle confined in a box is demonstrated using electrons in single and bilayer graphene as examples of massless and massive 2D Fermions, respectively. Ballistic metal-graphene-metal devices act as Fabry- Pérot cavities for electrons, and resonant states of the Fabry-Pérot cavity observed in electronic transport are used to measure the density of states as a function of particle number for massless and massive 2D Fermions. Nonlocal spin-valve experiments are demonstrated up to room temperature in mesoscopic graphene contacted by ferromagnetic electrodes. At low temperature the spin-valve signal shows changes in magnitude and sign with back-gate voltage, which may also result from quantum-coherent transport through Fabry Pérot cavities.

The temperature- and magnetic-field-dependent longitudinal (ρ_xx) and Hall(ρ_xy) components of the resistivity of graphene were measured. Near the minimum conductivity point ρxx(H) is strongly enhanced and ρ_xy(H) is suppressed, indicating nearly equal electron and hole contributions to the current. The data are inconsistent with the standard two-fluid model, but consistent with the prediction for inhomogeneously distributed electron and hole regions of equal mobility.

Ultrathin three quintuple layer (3QL) Bi₂Se₃ field effect transistors (FETs) were fabricated by mechanical exfoliation on 300 nm SiO₂/Si susbtrates. Temperature and gate-voltage-dependent conductance measurements show a clear OFF state at negative gate voltage, with activated temperature-dependent conductance and energy barriers up to 250 meV, implying that 3QL-Bi2Se3 films are conventional insulators rather than 2D topological insulators, likely due to coupling of the topological surface states through the thin bulk.