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# hw9_solutions - ECE 440 Homework IX Solutions Spring 2006...

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Unformatted text preview: ECE 440 Homework IX Solutions Spring 2006 Due: Wednesday, March 08, 2006 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 3x10 20 /cm 4 in a linear junction. (a) Find and plot the electric field, ε (x), for - ∞ <x<+ ∞ . Looking at the doping profile, it can immediately be determined that the depletion width will be symmetric. That is, if we move a certain distance into the n and p side, the N d and N a concentrations (respectively) will be equal. This observation will greatly simplify calculation of the electric field. Also, when x > 0 the argument is positive, meaning that we have a majority of acceptor ions. When x < 0, the argument is negative, indicating that we have a majority of donor ions. Define the unknown depletion width as W, and the depletion width into the n and p side will follow as ±W/2. Using this description, the charge distribution can be explicitly written as a function of position. Plotted, this charge distribution has the following shape which is zero at the origin. As in homework 8, the electric field can be found from the charge distribution using Poisson’s rule. Again, the charge neutrality assumption is made so the boundary conditions can be assumed to be ε (x)=0 at the edges of the depletion region. This gives the expression and the electric field will be zero everywhere outside the depletion region (due to the charge neutrality assumption). The corresponding plot should have the shape as the plot below. (b) Determine the relationship between the width of the depletion region and contact potential for the junction at equilibrium. The two common ways to find the contact potential are to integrate the electric field across the depletion width or to use equation (5-8). Electric Field Integration Equation (5-8) Now, the question is what does one put for N a and N d ? From previous homework assignments, we know that a heavy doping stops the depletion width from growing too much. Therefore, if we can find the heaviest doping of N a in the p side and N d in the n side, we can plug those values into equation 5-8. Luckily, our doping profile is very simple, and the largest N a and N d values occur at +W/2 and –W/2 respectively. Let’s plug these values into the equation: This gives two different expressions for the contact potential. To find numerical values This gives two different expressions for the contact potential....
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hw9_solutions - ECE 440 Homework IX Solutions Spring 2006...

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