  A weakly coupled system of two crossed graphene nanoribbons exhibits direct
tunneling due to the overlap of the wavefunctions of both ribbons. We apply the
Bardeen transfer Hamiltonian formalism, using atomistic band structure
calculations to account for the effect of the atomic structure on the tunneling
process. The strong quantum-size confinement of the nanoribbons is mirrored by
the one-dimensional character of the electronic structure, resulting in
properties that differ significantly from the case of inter-layer tunneling,
where tunneling occurs between bulk two-dimensional graphene sheets. The
current-voltage characteristics of the inter-ribbon tunneling structures
exhibit resonance, as well as stepwise increases in current. Both features are
caused by the energetic alignment of one-dimensional peaks in the
density-of-states of the ribbons. Resonant tunneling occurs if the sign of the
curvature of the coupled energy bands is equal, whereas a step-like increase of
the current occurs if the signs are opposite. Changing the doping modulates the
onset- voltage of the effects as well as their magnitude. Doping through
electrostatic gating makes these structures promising for application towards
steep slope switching devices. Using the atomistic Bardeen transfer Hamiltonian
method, inter-ribbon tunneling can be studied for the whole range of
two-dimensional materials, such as transition metal dichalcogenides. The
effects of resonance and of step-like increases of the current, observed in
graphene ribbons, are also expected in ribbons made from these alternative
two-dimensional materials, because these effects are manifestations of the
one-dimensional character of the density-of-states.
