  Spectrum and wave function of gapless edge modes are derived analytically for
a tight-binding model of topological insulators on square lattice. Particular
attention is paid to dependence on edge geometries such as the straight (1,0)
and zigzag (1,1) edges in the thermodynamic limit. The key technique is to
identify operators that combine to annihilate the edge state in the effective
one-dimensional (1D) model with momentum along the edge. In the (1,0) edge, the
edge mode is present either around the center of 1D Brillouin zone or its
boundary, depending on location of the bulk excitation gap. In the (1,1) edge,
the edge mode is always present both at the center and near the boundary.
Depending on system parameters, however, the mode is absent in the middle of
the Brillouin zone. In this case the binding energy of the edge mode near the
boundary is extremely small; about $10^{-3}$ of the overall energy scale.
Origin of this minute energy scale is discussed.
