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Unformatted text preview: ECE 495N EXAM II
CLOSED BOOK Wednesday, Nov.5, 2008 NAME: gOLUTION 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] 9 points Problem 3 [p. 6, 7] 8 points Total 25 points
Useful Relations: f (E ) = ———1——— Fermi function 1+ exp(E — M)/kT
Pa = % exp(— (Ea —— “Na )/kT) Law of equilibrium [h(/E)]= 2[Hm]exp(ii€.(&m—&n)) Bandstructure m D(E) = 2 6(E — so?» Density of States
k a h
M(E)= ENE—8%))” k ~ 1 (9802)
v"(k)_h é’kx w?)
———— Density of modes Problem 1: A channel has two energy levels with the same energy 8, but the
interaction energy is so high that no more than one of these levels can be occupied
at the same time. What is the average number of electrons in the channel if it is in equilibrium with chemical potential 14 and temperature T? Your answer should be in terms of 8 ,y and T. t) W “a“ : 26% .1, Problem 2: Benzene molecule consists of six carbon atoms arranged at the corners of a
regular hexagon of side ‘a’. Assume (1) one orbital per carbon atom as basis function ;
(2) the overlap matrix [S] is a (6x6) identity matrix; and (3) the Hamiltonian matrix is
given by H n n = a (site energy)
H n m = t if n, m are neighboring atoms
H n m = 0 if n, m are NOT nearest neighbors (a) What are the six energy eigenvalues in terms of ‘ 8’ and ‘t’? <__ a __> (b) What are the corre3ponding eigenvectors? E: 8+ 2ft 6/9955
t%[o,t1,tzizj =€+ 2t, 2+t, 5’17; 3"“7
C1) (2) U") 1 i HHHt—xr—xx—s Problem 3: Suppose a large two—dimensional conductor has an 806.) relationship given where Derive an ex
Your by:
db= AH A is a constant and k2 = k: + kyz. pression for the density of states, D(E).
answer should be in terms of the energy E, A, Width W and length L. ...
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 Spring '08
 S.Datta

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