  We investigate the edge state of a two-dimensional topological insulator
based on the Kane-Mele model. Using complex wave numbers of the Bloch wave
function, we derive an analytical expression for the edge state localized near
the edge of a semi-infinite honeycomb lattice with a straight edge. For the
comparison of the edge type effects, two types of the edges are considered in
this calculation; one is a zigzag edge and the other is an armchair edge. The
complex wave numbers and the boundary condition give the analytic equations for
the energies and the wave functions of the edge states. The numerical solutions
of the equations reveal the intriguing spatial behaviors of the edge state. We
define an edge-state width for analyzing the spatial variation of the
edge-state wave function. Our results show that the edge-state width can be
easily controlled by a couple of parameters such as the spin-orbit coupling and
the sublattice potential. The parameter dependences of the edge-state width
show substantial differences depending on the edge types. These demonstrate
that, even if the edge states are protected by the topological property of the
bulk, their detailed properties are still discriminated by their edges. This
edge dependence can be crucial in manufacturing small-sized devices since the
length scale of the edge state is highly subject to the edges.
