Phonons drive high temperature electron tunneling in graphite and bilayer graphene

We present a new mechanism for temperature-dependent out-of-plane electron tunneling in graphite and bilayer graphene that is driven by phonon amplitude. The weak interlayer coupling in bilayer graphene and graphite permits large out-of-plane phonon amplitudes that locally enhance electron tunneling. This phenomenon cannot be described by the Bardeen tunneling theory, which has no explicit temperature dependence; furthermore, the effect cannot be captured by an electron-phonon coupling induced renormalization of electronic density of states. Instead, we develop a theory for temperature-dependent tunneling including phonon amplitudes and Fermi-Dirac distribution broadening based on the Kubo formula for conductivity. Phonon-driven tunneling gives rise to an electric conductivity that varies exponentially with temperature for a stack of infinite graphene layers. The current theory explains the anomalous temperature dependence of out-of-plane electrical conductivity in graphite at high temperatures. When applied to quantum dots, the temperature-dependent tunneling theory predicts an Arrhenius-like temperature dependency due to the band gap formed by quantum confinement.