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_p  = Hole concentration in p-type at the edge of depletion region

                     n_n  = Electron concentration in n-type at the edge of depletion region

                     p_n  = Hole concentration in n-type at the edge of depletion region

                     n_p  = 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_J  = 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,

        P_P0  = Hole concentration in p-region

       As holes cross the junction, this concentration becomes p_n(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 p_n0 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 p_n(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 p_n(0) at the junction under forward biased condition is greater than its thermal equilibrium value p_n0. For large forward biasing p_n(0) becomes much larger compared to p_n0.

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 p_n(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

                  D_p = Diffusion constant for holes 

                  D_n = Diffusion constant for electrons

                  L_p = Diffusion length for holes 

                  L_n = 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.-

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