### Derivation of V-I Characteristics of p-n Junction Diode

Let us study the derivation of the mathematical expression for the current through a diode, which gives its V-I characteristics.
Let         p= Hole concentration in p-type at the edge of depletion region
n= Electron concentration in n-type at the edge of depletion region
p= Hole concentration in n-type at the edge of depletion region
n= Electron concentration in p-type at the edge of depletion region
Note : Note that in the symbol basic letter indicates type of charge carrier concentration, hole (p) or electron (n). The base indicates type of material in which exists.
Under unbiased condition, when holes move from p-side to n-side due to diffusion, their concentration behaves exponentially. This is mathematically expressed as,
where     V= Barrier potential or junction potential
Now consider forward biased diode as shown in the Fig. 1. The junction is at x = 0. Fig. 1  p-n junction diode

Thought the proportion of holes and electrons in constituting a current through the p-region is changing the hole concentration throughout the entire p-region is constant and denoted as,
PP0  = Hole concentration in p-region
As holes cross the junction, this concentration becomes pn(0) which is concentration of holes on n-side just near the junction. This further behaves exponential as given in the equation (1). From equation (1) we can write,
Note : The term V1 becomes V1 -V as the forward biased voltage V opposes the barrier potential. So net voltage across the junction becomes V1 - V.
The equation (2) can be written for open circuited unbiased p-n junction diode by putting V = 0 as,
where pn0 is the concentration of holes on n-side just near the junction when diode is open circuited i.e. at thermal equilibrium and hence different than pn(0).
As the concentration of holes in entire p-region is constant equating equations (2) and (3) we get,
Note : This equation represents boundary condition and called law of junction.
This indicates that the hole concentration pn(0) at the junction under forward biased condition is greater than its thermal equilibrium value pn0. For large forward biasing pn(0) becomes much larger compared to pn0.
Note : The discussion is equally applicable for the electron concentration on the p-side.
Now the difference between two concentrations at the junction under unbiased and biased condition is called injected or excess concentration denoted as pn(0).
Using equation (4) in equation (6),
The hole current crossing the junction from p-side to n-side is given by,
While an electron current crossing the junction from n-side to p-side is given by,
where  A = Area of cross-section of junction
Dp = Diffusion constant for holes
Dn = Diffusion constant for electrons
Lp = Diffusion length for holes
Ln = Diffusion length for electrons
Using equations (7), (8) in equations (9), (10), the total current I at the junction is given by,
The equation (11) is the required expression for diode current.
Note : In the derivation, the generation and recombination in the depletion region is neglected. To consider its effect, which is dominant is Si diodes, the factor η is introduced in the equation.
The value of η = 1 for Ge diodes and η = 2 for Si diodes.-