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Unformatted text preview: ECE 495N EXAM I
CLOSED BOOK Wednesday, Oct.1, 2008 NAME : PUID # : Please show all work and write your answers clearly. This exam should have seven pages. Problem 1 [11. 2, 3] 8 points
Problem 2 [p. 4, 5] 8 points
Problem 3 [11. 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 2g rlrz
1=— dEDE—U — E—
h l ( )y1+y2[f1( ) f2(E)] 1 1 where W and = 8(E—nzﬂkT +1 AlsoU : UL + U0(N —No). Assume U0 = 0 and the Laplace potential UL to be a
fraction 0: of the drain potential VD (the source potential is assumed zero): U L = — q OEVD, a being a constant between 0 and 1. A channel has a density of states
as shown, namely a constant non
zero value for E2 EC and zero for
E< EC. Assume that the
equilibrium electrochemical
potential a 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 = O) and
(b) assumes a value halfway
between the source and drain
potentials (a = 0.5), Expkxin year reasoning clearly. (a) Channel potential ﬁxed with respect to source: (I = 0. I (13) Channel potential halfway between source and drain: a = 0.5 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
3! 2m 3x2
whose solutions can be written in the form wag) = Q g “b 3 45m (2)
man:
with E and it related by the dispersion relation: E = 52kg am (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 = T127: 2 I 2mL2 (4) (a) Can you suggest a suitable differential equation to replace (1) if you wanted the
dispersion relation to look like E = Ak‘ (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? Problem 3: M1 A box has four degenerate energy levels all having energy s. We know that for nor:
interacting electrons the maximum current under bias is Use the multielectl‘on picture to derive the correct expression for the maximum current if
the electronelectron interaction energyr 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

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