Time-reversal operation and transport phenomena in topological insulators

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.