Metastable states of graphene Dirac fermion in potential barrier

Weak quantization of the Dirac equation under a potential barrier (with height V 0 ) indicates that while the bound state of Dirac fermion (with energy E 0 ∈ ℝ and mass m ) is allowed for | E 0 | < m , the Dirac fermion is metastable for | E 0 | > m . Under the condition V 0 » m , which is well satisfied for typical graphene, it is found that the tunneling time Δ t and the lifetime τ for the metastable state are related as Δ t ≳ 2τ. As m decreases, it follows that Δ t /τ tends to increase and behaves as log(l/ m ) up to higher order logarithmic correction.