analyze-abrupt-step-junctions

# Analyze Abrupt Step-Junctions Use this workflow to model the electrical behavior of one-carrier abrupt step-junctions. Since these systems involve non-linear differential equations that cannot be solved in closed form, numerical integration is required. ## Procedure 1. **Establish the Governing Equations** Define the system of three simultaneous non-linear differential equations that determine the electrical behavior: - Transport/Potential Relation - Poisson's Equation - Current Continuity Equation Refer to `governing-equations.md` for the specific mathematical forms and the expanded set of four equations (Eqs. 25.6 - 25.9). 2. **Define Boundary Conditions** Apply boundary conditions assuming the "thick slices" scenario (far from the interface): - Set electron density ($n$) and electric field ($F$) to constant values at the boundaries. - Approximate charged donor density ($p_d$) as equal to total donor density ($N_d$): $p_{d1} \approx N_{d1}$ and $p_{d2} \approx N_{d2}$. - Ensure shallow, fully ionized donor assumptions are met. *Note: Six boundary conditions are required for physically meaningful solutions.* 3. **Perform Numerical Integration** - Use a numerical solver to integrate the system. - Do not attempt closed-form integration, as it is impossible for this non-linear system. 4. **Extract Results** Generate and analyze the following solution curves across the junction: - Electron density distribution $n(x)$ - Electric field distribution $F(x)$ - Potential distribution