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Unformatted text preview: Sp 2008 EE243 HW#4 Solution Problem 1 C max = C (x=0,y ,t) = Q Dt N B = Q Dt e- x j 2 4Dt = Q 2 Dt [1 + erf(- y j 2 Dt )] or N B = C max e- x j 2 4Dt = C max [ 1 + erf(- y j 2 Dt ) 2 ] = C max [ 1 - erf( y j 2 Dt ) 2 ] x j 2 Dt = [ - ln( N B C max )] 1/2 y j 2 Dt = erf-1 [ 1 - 2N B C max ] y j x j = erf-1 [ 1 - 2N B C max ] [ - ln( N B C max )] 1/2 0.2 0.4 0.6 0.8 10-5-4-3-2-1 10 10 10 10 x x x x y j / x j log( N B C max-) Problem 2 (i) Injection of Si interstitials from the SiO2/Si interface during oxidation ( to release the mechanical stress due to oxide volume expansion). Law of Mass Action [C I C V =constant] implies more interstitial conc C I will yield less vacancy conc C V . (ii) D= D I C I +D V C V . Boron has enhanced diffusion with more C I implies its diffusion constant is affected dominantly by insterstitials. (iii) D= D I C I +D V C V . Antimony has reduced diffusion with more C I implies its diffusion constant is affected dominantly by vacancies. Problem 3 At 1000 C , n i = 1 10 19 /cm 3 D h D i + p n i At low dopant concentration <<n i , n = p= n i , and h 1. Therefore D = 1 10-14 cm 2 /sec implies N B C max y j x j 10-5 0.89 10-4 0.87 10-3 0.83 10-2 0.77 D i + = 1 10-14 cm 2 /sec At a dopant concentration of 10 20 /cm 3 , h 2 D h D i + p n i = 2 1 10-14 cm 2 /sec 10 20 10 19 = 2 10-13 cm 2 /sec Problem 4 (i) Intrinsic carrier concentration, n i ....
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- Spring '03