  The tight-binding method is employed to investigate the effects of three
typical in-plane electric fields on the electronic structure of a triangular
zigzag graphene quantum dot. The calculation shows that the single-electron
eigenstates evolute independently in two subspaces no matter how the electric
fields change. The electric field with fixed-geometry gates chooses several
scattered parts of the zero-energy eigenspace as the new zero-energy
eigenstates, regardless of the field strength. Moreover, the new zero-energy
eigenstates remain unchanged and the associated levels are linear as the field
strength. In contrast, the new nonzero-energy eigenstates mix mutually and the
associated levels are nonlinear as the field strength. By comparing the effects
of three electric fields, we demonstrate that the degeneracy of the zero-energy
eigenstates accounts for the linearity of the associated levels.
