Let us study the derivation of the mathematical expression for the
current through a diode, which gives its V-I characteristics.
Let pp = Hole concentration in p-type at the edge of depletion region
nn = Electron concentration in n-type at the edge of depletion region
pn = Hole concentration in n-type at the edge of depletion region
np = 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 VJ = 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.-
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