Proximity effect in the presence of Coulomb interaction and magnetic field

TL;DR

This paper develops a self-consistent theory to analyze how Coulomb interactions and magnetic fields influence the proximity effect in a metallic grain coupled to a superconductor, revealing phase transitions and distinct energy gaps.

Contribution

It introduces a comprehensive model that captures the interplay of proximity, charging, and magnetic effects in small superconducting grains, including phase diagram and transition mechanisms.

Findings

Magnetic field suppresses the proximity-induced minigap in an unusual manner.

Two distinct states with different minigap behaviors exist, separated by a first-order phase transition.

Energy dependence of the tunneling density of states shows Coulomb and proximity gaps, which can be separated at high magnetic fields.

Abstract

We consider a small metallic grain coupled to a superconductor by a tunnel contact. We study the interplay between proximity and charging effects in the presence of the external magnetic field. Employing the adiabatic approximation we develop a self-consistent theory valid for an arbitrary ratio of proximity and Coulomb strength. The magnetic field suppresses the proximity-induced minigap in an unusual way. We find the phase diagram of the grain in the charging energy - magnetic field plane. Two distinct states exist with different values and magnetic field dependences of the minigap. The first-order phase transition occurs between these two minigapped states. The transition to the gapless state may occur by the first- or second-order mechanism depending on the charging energy. We also calculate the tunneling density of states in the grain. The energy dependence of this quantity demonstrates two different gaps corresponding to the Coulomb and proximity effects. These gaps may be separated in sufficiently high magnetic field.