A schematic description of the superconductor-2DEG-superconductor junction. Two superconductors with different phases of the order parameter are connected via a two-dimensional normal electron gas, or, alternatively, a narrow ribbon of graphene. A magnetic field is applied locally between the superconductors, parallel to the plane of the 2D conductor and is taken to be negligibly weak near the SN interfaces. A similar system has been considered in Ref. 14, where the effects of Zeeman splitting and spin-orbit interaction on the Josephson current were studied.
Magnetization of a transparent S/2DEG/S junction with a harmonic (parabolic) lateral confining potential in a magnetic field , plotted as a function of the phase difference. Also shown are its asymptotes MA (Eq. (15)) and MB (Eq. (16)). For this demonstration we choose materials with the same effective masses. We assume that the superconductors are made from niobium, with a Fermi energy , and effective masses . The length of the 2DEG part of the junction is , and its width is . Results are shown for a temperature , which is much lower than the critical temperature of niobium . Note that the resonance peaks are extremely narrow—their width is approximately equal to the phase change necessary to shift the spectrum of AK levels by an amount equal to the Zeeman splitting.
Magnetization of the transparent S/2DEG/S junction in a magnetic field at several temperatures plotted vs the phase difference. The width of the junction is , and all the other parameters of the junction are as in Fig. 2.
A magnetization as a function of the width of the 2DEG part of a transparent S/2DEG junction. The length of the junction is and the temperature . The strength of the magnetic field and all other parameters are the same as in Fig. 2.
The magnetization of a transparent S/2DEG/S junction vs the phase difference , for different lengths of the 2DEG part. The width of the junction is , temperature , and the magnetic field .
The spectrum of AK levels in a S/2DEG/S junction (left) and in a S/GM/S junction (right). Both figures are schematic, that is, these figures are plotted for junctions with different geometrical parameters with the main purpose to illustrate the basic characteristic features of the energy levels in these systems. Note, that unlike the S/2DEG/S junction where the lowest AK levels of all transverse modes approach the Fermi level and become degenerate at the resonant phases , ), the lowest AK levels of the S/GM/S junction, belonging to different modes, never become degenerate at the resonant phases, and all (except one level of the lowest transverse mode) stay away from the Fermi level.
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