ece340Fall03HW7sol - ECE 340 Homework VII Fall 2003 Due:...

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ECE 340 Homework VII Fall 2003 Due: Friday, October 03, 2003 1. An intrinsic Si sample is doped with donors symmetrically about x o = 2 μm such that N d (x) = 10 14 + 10 17 exp [-10 9 (x-x o ) 2 ] for 1 μm x 3μm, where x is expressed in cm. (a) Assuming a uniform diffusion coefficient, D n =30 cm 2 /s at 300 K, find and plot the electron diffusion current density from x=1 μm to x=3 μm. (b) Find an expression for the built-in electric field E (x) at equilibrium over the range from x= 1μm to x=3 μm. (c) Sketch a band diagram such as in Fig. 4-15 and indicate the directions of E . Solutions: The distribution of () d Nx is plotted below for your reference: (this figure is not required for grading) (a), Assume quasi-neutrality (because di Nn ± ), then d nN = 17 9 2 9 00 19 17 9 9 2 89 2 (10 )exp[ 10 ( ) ]( 10 )(2)( ) (1.6 10 )(30)(10 )( 10 )(2)( )exp[ 10 ( ) ] 9.6 10 ( )exp[ 10 ( ) ] nn n dn Jq D q D x x x x dx xx == =− × Use your favorite computer program to draw the plot. It should look like:
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(b), At equilibrium, J = 0; Hole diffusion current can be neglected (why?), therefore Jp = 0; 0 nn n dn Jq nq D dx µ + = , therefore: n n D dn ndx Ε=− By using Einstein relation (Equation. 4-29), Dk T q = 17 9 2 9 00 14 17 9 2 0 92 24 14 17 9 2 0 (10 )exp[ 10 ( ) ]( 10 )(2)( ) 1 0.0259 0.0259 10 (10 )exp[ 10 ( ) ] exp[ 10 ( ) ]( ) 5.18 10 10 (10 )exp[ 10 ( ) ] x xx x kT dn dn qn dx n dx x x −−− =− +− −− (this figure is not required for grading) (c) The amount of bending of bands can be obtained by integrating electric fields over x. The band-diagram
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ece340Fall03HW7sol - ECE 340 Homework VII Fall 2003 Due:...

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