Superconductor-normal junction in a strong magnetic field: the double quantum Hall effect

The integer quantum Hall effect with a superconducting contact is analyzed. It is shown that the conductance plateaus occur at σxy = 2ne /h which is double the usual value. Non-Andreev scattering at the interface and flux penetration into the superconductor are shown to not affect the result. The effect should be observable in InAs quantum wells with Nb a contact. Typeset using REVTEX 1 This paper will examine the integer quantum Hall effect (IQHE) [1] in the presence of a superconductor-normal (SN) junction [2,3] at one of the leads. The IQHE occurs in a two dimensional electron gas (2DEG) in a strong magnetic field as quantized conductance plateaus with the value σxy = ne /h where n is the number of filled Landau levels. SN junctions have received considerable attention because of the transport properties caused by Andreev reflection, which takes place when an electron (hole) incident from the normal side has an energy lying within the superconducting gap of the superconductor. Because current is carried in the superconductor as Cooper pairs, a single electron (hole) at the Fermi level cannot enter the superconductor. For a Cooper pair to be injected into the superconductor a hole (electron) must be reflected back into the normal region. SN junctions in magnetic fields have been studied previously [4], although the magnetic field has usually been assumed small enough that Landau quantization is not significant. The system to be analyzed is a non-interacting Lx × Ly 2D electron gas (2DEG) that is periodic in the ŷ direction, with a uniform perpendicular magnetic field B = Bẑ. At x = Lx there is an infinite barrier, and the semi-infinite superconductor is located at x < 0. The superconductor is placed on only one side of the 2DEG to avoid the Josephson effect and to simplify some of the later analysis. The magnetic field and superconducting gap are assumed to change abruptly at the SN interface and are otherwise constant throughout the two different regions. Let us first briefly review Laughlin’s derivation of the IQHE without a superconductor [5,6] in which the periodic 2DEG encircles a magnetic flux Φ. The flux is adiabatically increased from 0 to one flux quantum, Φ0 = hc/e, and then removed by a gauge transformation. The addition of ∆Φ = Φ0 increases the energy of the 2DEG, but the gauge transformation returns the Hamiltonian to its original form. Therefore, the net result is an excited state of the original system. Examination of the electron wave functions in the Landau gauge shows that adding one flux quantum transfers a single electron per Landau level from one edge of the sample to the other. Since the current is related to the change in energy of the 2DEG by Iy = c∆U/∆Φ, we may equate ∆U to the energy for transporting 2 one electron per Landau level across a potential difference V to obtain Iy = ne V/h. Now consider what happens when the superconductor is present. Assume Φ0 is added such that the electrons in the 2DEG are shifted towards the superconductor. (Φ0 corresponds to two flux quanta for the superconductor, so it may still be gauge transformed away.) To be removed from the sample, an electron must enter the superconductor. However at the Fermi level there are no quasiparticle states available in the superconductor and therefore another charge is needed to make a Cooper pair. This extra charge comes from an Andreev reflected hole which, because of its charge, is shifted in the opposite direction, away from the superconductor. Hence, adiabatic addition of one flux quantum transports a charge 2e across the sample leading to Iy = 2ne V/h, (1) or the double quantum Hall effect (2QHE). Clearly the superconductor does not change the role of impurities in providing localized states that lead to the conductance plateaus [5,6]. To examine the 2QHE in more detail, we will compute the states of the system, and explicitly compute Iy using the edges states. The system will be described by the Bogoluibovde Genne (BdG) equation [7], Hψ(r) = Eψ(r) (2) H = 