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Summary Notes for p-n Junctions1.TerminologyNa: net acceptor concentration on the p-side.Nd: net donor concentration on the n-side.2.p-n Junction ElectrostaticsA region depleted of mobile charge carriers exists in the vicinity of the metallurgical junction(the boundary separating the p- and n- regions), due to mobile charge carrier redistribution (holediffusion into the n-type region from the p-side and electron diffusion into the p-type region from the n-side, initially).This region (called the space-charge region or the depletion region), extends a distance xpinto the p-sideand a distance xninto the n-side.Depletion Approximation”: n, p <<|Nd– Na|in depletion regionCharge densityρ(x) =-eNafor -xpx <0eNdfor 0< x≤xn0 for x < -xpand x > xnFor a step junction, the charge-density distribution looks like:Note that the metallurgical junction is at the boundary between n- and p-regions where x=0.Also note that xpNa= xnNdfor charge neutrality. The depletion region extends further into theside with the lower dopant concentration.
Summary Notes for p-n JunctionsThe charge distribution in the depletion region gives rise to a non-zero electric field distribution,which can be obtained using Poisson’s equation:𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑=𝜌𝜌𝜖𝜖𝑠𝑠𝑜𝑜𝑜𝑜𝐸𝐸=𝜌𝜌𝜖𝜖𝑠𝑠𝑑𝑑𝑑𝑑.For a step junction, the electric-field distribution looks like:Note E0 in the quasi-neutral p- and n-type regions (x < -xpand x > xn).The electric field serves to counteract the diffusion of mobile charge carriers.At equilibrium (Va= 0), there is no net current flow across the junction.The peak E-field is|𝐸𝐸𝑚𝑚𝑚𝑚𝑑𝑑(𝑑𝑑= 0)| =2𝑒𝑒(𝑉𝑉𝑏𝑏𝑏𝑏−𝑉𝑉𝑎𝑎)𝜀𝜀𝑠𝑠𝑁𝑁𝑎𝑎𝑁𝑁𝑑𝑑𝑁𝑁𝑎𝑎+𝑁𝑁𝑑𝑑=−2(𝑉𝑉𝑏𝑏𝑏𝑏−𝑉𝑉𝑎𝑎)𝑊𝑊The electric potential distribution is obtained by integrating the electric field distribution (𝑉𝑉=− ∫ 𝐸𝐸𝑑𝑑𝑑𝑑).For a step junction, the electric-potential distribution looks like:Note that the area underneath the E(x) curve is equal to the voltage dropped across the depletionregion, and that most of this voltage is dropped across the more lightly doped side.
Summary Notes for p-n JunctionsIn equilibrium (Va= 0), the total voltage dropped across the depletion region is equal to the built-inpotential, Vbi, which is due to the work function (energy difference between the vacuum level and theFermi energy) difference between the p-type and n-type regions:eVbi= (EFi– EF)p-side+ (EF– EFi)n-sidewhere (EFi– EF)p-side=Eg/2 for a degenerate p-sidekTln(Na/ni) for a non-degenerate p-side(EF– EFi)n-side=Eg/2 for a degenerate p-sidekTln(Nd/ni) for a non-degenerate p-sideIf both sides are non-degenerately doped then𝑉𝑉𝑏𝑏𝑏𝑏=𝑘𝑘𝑘𝑘𝑒𝑒𝑙𝑙𝑙𝑙 �𝑁𝑁𝑎𝑎𝑁𝑁𝑑𝑑𝑛𝑛𝑏𝑏2When an external bias voltage Va0 is applied, the total voltage dropped across the junction is Vbi-Va.

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