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ece340Spring04HW10Sol - ECE 340 1 Homework X Due Monday...

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ECE 340 Homework X Spring 2004 Due: Monday, March 15, 2004 1. When a prolonged diffusion or a high-energy implantation is conducted to form a p/n junction. The doping profile near the junction is usually graded, and the step-junction approach is no longer suitable to find the relationship between the width of the depletion region and the contact potential. However, the underlying principle used to establish equations 5-13 to 5-23 remains intact, and they can still be used to determine similar equations for the graded junction. Assume that the doping profile varies as N a -N d =Gx where G is 10 20 /cm 4 in a linear junction. (a) Find and plot the electric field, ε (x), for - <x<+ . (b) Determine the relationship between the width of the depletion region and contact potential for the junction at equilibrium. Solutions: (a), Please refer to section 5.6.4 and Figure 5-39 (page 220 to obtain a general understanding of the device. We are going to use depletion approximation in the junction region and quasi-neutral approximation outside junction region. The example in section 5.6.4 has N d -N a =Gx, but we have N a -N d =Gx. Therefore, in our case, the ρ , x Ε , and V will be the mirror image of those in figure 5-39 about the x- axis. After obtaining a general understanding of the device, we can solve the problem through steps similar to equations 5-13 to 5-23: Since it’s a symmetrical device, 0 0 / 2 n p x x W = = . For x outside depletion region ( | | / 2 x W > ), electrical field is 0. For x within depletion region ( | | / 2 x W < ), start with Poisson equation, ( ) ( ) ( ) ( ) Depletion d a d a approximation d x q d x q q p n N N N N Gx dx dx ε ε ε + + Ε Ε = + ⎯⎯⎯⎯⎯→ = = − The general solution to above differential equation: 2 2 x qG x Const ε Ε = − +
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