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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 currentvoltage (IV) 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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 Spring '08
 S.Datta
 Electron, Energy, Potential Energy, Fundamental physics concepts, dispersion relation

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