Project Overview
This project models the quantum mechanical phenomenon known as the Josephson effect, where a supercurrent flows between two superconductors separated by a thin normal metal region. The effect is mediated by Cooper pairs tunneling through the barrier without resistance, governed by a phase difference between the superconductors.
We solve the Usadel equations numerically to study how superconducting correlations penetrate into the normal metal and give rise to measurable currents. These equations describe diffusive superconducting systems and are widely used in mesoscopic physics.
🔬 Key Physics Concepts
Josephson Effect: A phenomenon where electrical current flows between two superconductors through a non-superconducting barrier.
Cooper Pairs: Bound pairs of electrons that enable superconductivity by moving through the material without resistance.
Proximity Effect: The penetration of superconducting correlations into adjacent normal metals.
Usadel Equation: Describes the behavior of superconducting correlations in diffusive systems.
🎯 Project Objectives
- Solve boundary value problems using shooting methods
- Simulate Green functions and density of states
- Analyze supercurrents as functions of energy and phase
- Understand minigaps and proximity effects
🔧 Numerical Methods
- Runge-Kutta method of order 3(2) with adaptive step size
- Secant method for nonlinear boundary problems
- Complex matrices as real-valued vectors
- scipy.integrate.solve_bvp for boundary value problems
📊 Key Results
- Current integrand vs energy for different phase differences
- Supercurrent vs phase difference relationship
- Density of states showing minigap formation
- Analysis of proximity effect in normal metals