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ECE495N-F08-Exam_1_practice - ECE 495N PRACTICE EXAM I...

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Unformatted text preview: ECE 495N PRACTICE EXAM I CLOSED BOOK, Sept. 25, 2007 Problem 1 [p. 2,] 8 points Problem 2 [p. 3] 8 points Problem 3 [p. 4] 9 points Total 25 points Problem 1 (a) 5 points Source Channel Drain A channel has a density of states as shown, namely a constant non-zero value for E 2 0 and zero for E < O, that is_._ DUE) 2 D0 6' (E ), where 6' represents the unit step function. It is connected to two contacts with the same electrochemical potential it whjich is m 0. Is there a current in the extemal circuit ? If so_._ is it in the direction shown (from the hot to the cold contact) or opposite to the direction shown (from the cold to the hot contact)? Explain your reasoning. (b) 3 points Which side (the hot side or the cold side) will be positive if we leave the tenninals open'? Problem 2 W e have seen in class that free electrons in the absence of any external potential are :1 , f 3 :3 described by (in one dimension) 1' it “if = — i— 1” W (1) :3? 2m 53-2 whose solutions can be written in the fonn gr/(XJ) : a +" h e fiEl m (2) constant . . . . '3 . With E and k related by the dispers1on relation: E : E211" ..-"2m (3) (:1) 3 points Can solutions be written in the form of (2) if a sinusoidal external potential U = U0 cos(X.-"'a) is present (a is a constant)? (1}) 5 points Can you suggest a suitable differential equation to replace (1) if you wanted the dispersion relation to look like E2 : (Inez) 2 + fracz k2 (3") (111, c, T: are all constants) instead of (3) ‘F’ Problem 3 We wish to calculate the current, I through a channel having two discrete energy levels with energy 8, but with a vely high interaction energy such that both levels CANNOT be simultaneously occupied in our voltage range of interest. One-eiectron picture i’lrfuiti-eieetron picture (a) 6 points Write rate equations in the nniltielectron picture to obtain an expression for the maximum current I that flows when a voltage V is applied with the polarity as shown. Your answer should be in terms of the couplings 3/1 and y2 for the two contacts (and fundamental constants like q and Ti). (17) 3 points Assuming 5/1 = 10 3/3, would you get more current or less current (in magnitude) if you reversed the polarity of the applied voltage? Please explain your reasoning. ...
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