Potential Variation in a Continuously Graded Semiconductor

       Consider a p-type continuously graded bar i.e. nonuniformly doped bar as shown in the Fig. 1.
Fig. 1 Continuously graded p-type bar
       No external voltage is applied to the bar. The bar is open circuited. As no external voltage is applied and it is open circuited net current through the bar is zero.
       But due to the nonuniform doping there exists a diffusion current as holes move from high concentration to low concentration area. Hence there exists a diffusion current density of,
       But as bar is open circuited, net current through it is zero. This means there exists one more internal current which is equal to diffusion current but in opposite direction to it. This is a drift current flowing in the bar in opposite direction to that of diffusion current. The current density of this current is,
       But drift current can not exist without a potential difference and applied voltage to the bar is zero. So externally E is zero. This indicates that the E required for the circulation of drift current gets generated internally.
       This indicates that nonuniform doping of bar results in the induced voltage.
       And as net current through the bar is zero we can write for open circuited bar,
1.1 Expression for the Potential Difference
       To derive the expression for the potential difference between any two points of a nonuniformly doped bar, consider the two points at a distance of x =  x1 and x = x2 as shown in the Fig. 2.
Fig. 2  Graph of Concentration against distance x
       Let concentration of holes at x = x1 is p = p1
       and concentration of holes at x = x2 is p = p2
       This is due to nonuniform doping and is indicated in the graph shown in the Fig. 2.
       There exists a potential difference between x1 and x2 which is responsible to circulate drift current equal and opposite to diffusion current.
       Let          potential at x1 = V1
       And         potential at x2 = V2
       From the equation (3) which indicates the the net current through the bar is zero, we can write,
       According to Einstein'S relation, Dpp is VT.
       Now E is the electric field intensity generated internally. From the definition of electric field intensity we can write,
       E = -dV/dx
       Hence equation (5) modifies to,
       To get the voltage generated between x1 and x2, integrate the equation (6),
        where V21 is the potential difference between the two concentrations p1 and p2.
Note : The potential difference depends on the concentrations and not on the distance between x1 and x2.
       From the equation (7) we can write,
       Similarly for n-type semiconductor bar which is nonuniformly doped, if and are the concentrations of electrons at two different points, we can write,
Note : There is change of sign of exponential term in equation (9a) and (9b) as compared to exponential term in equations (8a) and (8b). This is because diffusion current density in n-type bar is q dn/dx which is positive.
       Multiplying equations (8a) and (9a) we get,
       The product of the concentrations of electrons and holes is always constant. This proves the law of mass action.

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