ECE495N-F08-Exam_1_solution - ECE 495N EXAM I CLOSED BOOK...

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Unformatted text preview: ECE 495N EXAM I CLOSED BOOK Wednesday, Oct.1, 2008 NAME: SOLUTION PUID # : Please show all work and write your answers clearly. This exam should have seven pages. Problem 1 [p. 2, 3] 8 points Problem 2 [p. 4, 5] 8 points Problem 3 [p. 6, 7] 9 points Total 25 points Problem 1: We have seen in class that the current-voltage (I-V) characteristics of a nanoscale device can be calculated from 2 Y 7 1=—‘1 I dE D(E—U) 1 2 [f1(E)—f2(E)] h yl +y2 l 1 where = e(E—/.i1)/kT +1 and = 6(E—u2)/kT +1 AlsoU = UL + U0(N—N0) . Assume U0 = 0 and the Laplace potential UL to be a fraction a of the drain potential VD (the source potential is assumed zero): U L = — q OCVD , a being a constant between 0 and 1. A channel has a density of states as shown, namely a constant non- zero value for EZEC and zero for E < EC . Assume that the equilibrium electrochemical potential y is located above EC as shown. Sketch the current versus drain voltage assuming that the electrostatic potential of the [J channel (a) remains fixed with respect to the source (a = 0) and (b) assumes a value halfway between the source and drain potentials (a = 0.5), Explain your reasoning clearly. (a) Channel potential fixed with respect to source: a = 0. my: Ado‘T'b'iorht Current .— —. _ - ---- f Eqmlibrmm Chan'th New“ Problem 2: We have seen in class that free electrons in the absence of any external potential are described by (in one dimension) 2 2 m 9K = _ Lil/I (1) 8t 2m 3x2 whose solutions can be written in the form w(x,t) = g e “k” e "mm (2) constant with E and k related by the dispersion relation: E = 71sz /2m (3) We have also seen that if the electrons are confined in a box of length L, the energy levels become discrete with the lowest energy given by E1 = 7’127z:2/2mL2 (4) (a) Can you suggest a suitable differential equation to replace (1) if you wanted the dispersion relation to look like E = Ak4 (3’) (A being a constant) instead of (3) ? (b) If a system of electrons with a dispersion relation given by (3’) were confined in a box of length L, how would the expression for the lowest energy given in (4) be modified? . h (a) JPN: 24 “{3x4’ w) 4' (M) 4 gr ll Problem 3: F1 A box has four degenerate energy levels all having energy 8. We know that for non- interacting electrons the maximum current under bias is [:1 47172 hn+h Use the multielectron picture to derive the correct expression for the maximum current if the electron—electron interaction energy is so high that no more than one electron can be inside the box at the same time. ...
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ECE495N-F08-Exam_1_solution - ECE 495N EXAM I CLOSED BOOK...

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