hw8_solutions

hw8_solutions - ECE 440 Homework VIII Solutions Spring 2008...

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ECE 440 Homework VIII Solutions Spring 2008 1. An abrupt Si p-n junction is formed by alloying a uniformly doped n-type silicon bar of which N d = 4x10 16 /cm 3 . During the alloying process, a uniform counter doping of acceptors of N a = 1.5x10 17 /cm 3 is introduced in the region for x<0. For x>0, the doping remains to be N d = 4x10 16 /cm 3 . So, x<0 is the p-side and x>0 is the n-side. (a) Calculate the Fermi level positions at 300 K in the p and n regions. Treat each side separately (as if they were unconnected) to calculate the Fermi level positions far from the junction. p-side : eV n N N kT n p kT E E i d a i f i 409 . 0 10 5 . 1 10 4 10 5 . 1 ln 0259 . 0 ln ln 10 16 17 0 = × × - × = - = = - n-side : eV n N kT n n kT E E i d i i f 383 . 0 10 5 . 1 10 4 ln 0259 . 0 ln ln 10 16 0 = × × = = = - (b) Draw an equilibrium band diagram for the junction and determine the contact potential V o from the diagram. Looking at the band diagram we can calculate qV 0 geometrically. The contact potential is the difference between the Fermi level positions on the n and p sides of the junction. eV eV eV F E E F F F qV p i i n p n 792 . 383 . 0 409 . 0 ) ( ) ( 0 = + = - + - = - = (c) Compare the results of part (b) with V o as calculated from Eq. (5-8).
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eV kT n N N kT qV i d a 792 . ) 10 5 . 1 ( ) 10 4 ( ) 10 4 10 5 . 1 ( ln ln 2 10 16 16 17 2 0 = × × × - × = = In fact, equation (5-8) can easily be derived from the expression in part (b). (d) Using Eq. (5-8), calculate and plot V o versus temperature ranging from 250 K to 500 K. Looking at equation (5-8), there is a linear dependence between contact potential and temperature. However, the intrinsic carrier concentration, n i (T), also has a temperature dependence (see equation (3-17)). Either using an approximation for n i (T) such as an
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hw8_solutions - ECE 440 Homework VIII Solutions Spring 2008...

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