  A simple model based on the divide and conquer rule and tight-binding (TB)
approximation is employed for studying the role of finite size effect on the
electronic properties of elongated graphene nanoribbon (GNR) heterojunctions.
In our model, the GNR heterojunction is divided into three parts: a left (L)
part, middle (M) part, and right (R) part. The left part is a GNR of width
$W_{L}$, the middle part is a GNR of width $W_{M}$, and the right part is a GNR
of width $W_{R}$. We assume that the left and right parts of the GNR
heterojunction interact with the middle part only. Under this approximation,
the Hamiltonian of the system can be expressed as a block tridiagonal matrix.
The matrix elements of the tridiagonal matrix are computed using real space
nearest neighbor orthogonal TB approximation. The electronic structure of the
GNR heterojunction is analyzed by computing the density of states. We
demonstrate that for heterojunctions for which $W_{L} = W_{R}$, the band gap of
the system can be tuned continuously by varying the length of the middle part,
thus providing a new approach to band gap engineering in GNRs. Our TB results
were compared with calculations employing divide and conquer rule in
combination with density functional theory (DFT) and were found to agree
nicely.
