Half-integer and integer quantum-flux periods in the magnetoresistance of one-dimensional rings

A tight-binding model for a disordered ring coupled to two external leads is used to calculate the transmission coefficient T as a function of the magnetic flux φ threading through it. We found a dominant ${\phi }_0$=h/e period in two cases: (a) strongly disordered rings and (b) arbitrary disorder with weakly coupled branches. We find that the last situation occurs when the Fermi energy lies near the band edge. The phase of T, which may be equal to 0 or π, depends on energy and changes whenever the Fermi energy crosses an eigenvalue of one of the two isolated branches of the ring. Finally, resonances (peaks) and antiresonances (valleys) are found in T as a function of energy. The width and height of the resonances and antiresonances are calculated within this tight-binding model and their shapes are found to be Lorentzian. This represents an extension of the results of Azbel on resonant tunneling in the case of a tight-binding model.