# hw4_solutions - ECE 440 Homework IV Due Monday Fall 2008 1...

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ECE 440 Homework IV Fall 2008 Due: Monday, September 22, 2008 1. In practice we assume that the intrinsic Fermi level, E i , coincides with the center of the band gap. In reality it is not true. Derive an expression relating the intrinsic level E i to the center of the band gap. Find the displacement of E i from the center of the band gap for both silicon and germanium at room temperature. The effective mass values are m n * = 0.55 m o for Ge and 1.1 m o for Si; m p * = 0.37 m o for Ge and 0.56m o for Si where m o is the free electron mass. We know that at thermal equilibrium, the intrinsic carrier concentrations for electrons and holes are equal. By setting equations 3-15 & 3-19 equal to one another, a relationship between the intrinsic Fermi Level and temperature can be found. At this stage, it is a good idea to draw an energy band diagram to visualize what these quantities physically mean. What we are saying here is that our intrinsic Fermi Level is simply half the energy band gap plus some offset term. You could have also defined the intrinsic Fermi Level as half the band gap minus some energy offset as well. Thus we have: Also, if we use the common convention that Ev = 0, we can say that:

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By combining these three equations (plus some simplification) we can find an expression for the displacement energy with respect to the middle of the gap. Now we have to plug and chug to find the actual displacements for both Ge and Si at room temperature (T=300K): Clearly, the “real” intrinsic Fermi level lies below mid-gap for both Ge and Si at room temperature. 2. The electron concentration in a silicon material is 8x10 2 /cm 3 at room temperature under equilibrium conditions. (a) What is the hole concentration? (b) Where is E F positioned relative to E i ? (c) Draw the energy band diagram for the material. Repeat for parts (a), (b) and (c) for the same sample if the temperature is 400 K.
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