ShaneLogic/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