Bag boundaries for quasispinor confinement within nanolanes on a graphene sheet

We revisit the problem of bag boundary conditions within a field-theoretic approach to study confinement of massless Dirac quasispinors in monolayer graphene. While no-flux bag boundaries have previously been used to model lattice termination sites in graphene nanoribbons, we consider a generalized setting in which the confining boundaries are envisaged as arbitrary straight lines drawn across a graphene sheet and the quasispinor currents are allowed to partially permeate (leak) through such boundaries. We specifically focus on rectangular nanolanes defined as areas confined between a pair of parallel lines at arbitrary separation on an unbounded lattice. We show that such nanolanes exhibit a considerable range of bandgap tunability depending on their widths and armchair, zigzag or intermediate orientation. The case of nanoribbons can be derived as a special limit from the nanolane model. In this case, we clarify certain inconsistencies in previous implementations of no-flux bag boundaries and show that the continuum approach reproduces the tight-binding bandgaps accurately (within just a few percent in relative deviation) even as the nanoribbon width is decreased to just a couple of lattice spacings. This accentuates the proper use of boundary conditions when field-theoretic approaches are applied to graphene systems.