The Shockley-type boundary conditions for semiconductor p-n junctions at   medium and high injection levels
  The classical Shockley boundary conditions are used for the determination of the minority carrier concentrations at the edges of the space-charge region of semiconductor p-n junctions. They are usually employed for the calculation of the p-n junction current/ voltage characteristic. This work demonstrates that they are valid only at low injection current levels. New analytical expressions for these boundary conditions are obtained for medium and high injection levels. These expressions are verified by calculating the current/ voltage characteristics of p-n junctions operated at high current levels. 
Miron J. Cristea