  We prove that any forms of electric current measurements within a gapped or
gapless quantum system necessarily violate the time reversal symmetry (TRS). We
then use the ionization energy theory to unequivocally show that the metallic
surface states of a topological insulator consist of some special energy level
crossings, not due to TRS induced Kramers degeneracy because there is a finite
energy gap due to different wavefunctions. We use this special crossings and
derive the electron-ion scattering rate required to explain the resistivity and
carrier-type transition in (Bi1-xSbx)2Te3 and Pb1-xSnxSe topological
insulators.
