The junction contact potential

The built-in voltage can also be written in terms of the intrinsic carrier concentration ni. The result after some algebra is  

The electric field Eq of the junction is found by integration of the charge Q across either side of the depletion region. For example, on the p-type side  

The variables Na, W and Xp in Equation 3.16 are not independent. Specifically, W = x +Xp. Furthermore, it is necessary to balance the charge accumulated on the two sides of the junction, from which XpNA= x Nd- Therefore, W = Xp(l + Na/Nq). Solving for Xp gives 

Substituting for Vk from Equation 3.14 and solving for W leads eventually to  

Examination of this equation shows that most of the depletion lies on the lightly-doped side of the junction. 

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