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Unformatted text preview: ECE 440 Homework XII Spring 2008 Solution to be posted by Friday, April 04, 2008 1. Assume that a p + -n diode with a uniform cross section area, A, is built with an n region width l smaller than a hole diffusion length ( l < L p ). This is the so-called narrow base diode. Since for this case holes are injected into a short n region under forward bias, we cannot use the assumption p(x n = ) = 0 in Eq. 4-35. Instead, we must use as a boundary condition the fact that p =0 at x n = l . (a) Solve the diffusion equation to obtain p x n ( 29 = p n e l- x n ( 29 L p- e x n- l ( 29 L p e l L p- e- l L p (b) Show that the current in the diode is I = qAD p p n L p ctnh l L p e qV kT- 1 (c ) If the n-region is relatively short compared to the diffusion length, the excess hole p(x n ) can be approximated as a straight line, i.e. it varies linearly from p n at x n =0 to zero at the ohmic contact (x n = l ). Find the steady-state total excess charges, Q p in the n-region and determine the percentage of error comparing the total excess holes in the n-region obtained from part (a) with that from the straight-line approximation for l /L p = 0.05, 0.1, 0.5, 1 and 5. (d) Calculate the current due to recombination in the n region. (a) Start from the diffusion equation, (4-34b) in Streetman: 2 2 2 ) ( ) ( p n n n L x p dx x p d = From differential equations we know the general solution to this equation is of the form: p n p n L x L x n Be Ae x p / / ) (- + = (*) To find the solution, we apply the boundary conditions and solve for A and B....
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