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Unformatted text preview: Fall 2009 ECE 495N FINAL EXAM CLOSED BOOK Friday, Dee.19, 7P9P, LYNN G165 NAME : PUID # : Please showr all work and write your answers clearly. This exam should have eight pages + 1 page of notes. Problem 1 [p. 2} 5 points
Problem 2 [p. 3] 5 points
Problem 3 [p. 4] 5 points
Problem 4 [p. ‘5] 5 points
Problem 5 [p. E] 5 points
Problem 6 [11. 7,8] 5 points Total 30 points Problem 1: Suppose: a 2—D conductor of length L and width W has a density of states (per
unit energy), DIIE) = (DDLW) E am! a density of modes (dimensionless),
M(E)= (MHW)E, where Dr: and M:] are constants. Assuming ballistic transport,
obtain an expression for the couductance per unit width at T=ﬂK, as a function of the. electron density per unit ma. Problem 2: A channel has four energ},r levels all with the same energy a, but the
interaction energyr is so high that no more than one of these levels can be occupied at the
same time. Starting from the law of equilibrium. what is the average number of electrons
in the ehatmel if it is in equilibrium with chemical potential u and temperature T? Your
answer should be in terms of e, u and T. Problem 3. A melecule (NOT a solid) consists of four atoms arranged at the comers of a
square. Assume (1) one orbital per carbon atom as basis ﬁmetion ; (2) the overlap matrix
[S] is a {4x4} idﬂl'ltit}? matrix; and (3) the Hamiitenian matrix is given by H” n =3 (site energy)
Hum =r ifn, mareneighberingatems Ham =0 if 11, m are NOT nearest neighbors What are the four energy eigenvalues in terms of ‘ 8' and ‘t’ and the corresponding eigemrectors. Please do net diagonaﬂze direelb’: use Ike principle
efbeudstmcmre. Problem 4: Suppose a large twodimensional conductor has an £(E) relationship given W 50:”) = +121 (k: +kf where a: isamnstant. Deﬁveanexpmssion for the densityr (3me M03}. Your
answer should be in tenns of the enemyIr E, a, width W and length L. (v,(E)=§§: ) Problem 5: Early in this course we showed that the maximum current threngh a device
with a single level with escape rates r1, 2 is given by (11”!) Tlf'zf'iiﬁ +31%} using
elementary arguments. Prove this result using the general matrix expressions discussed later in the ccmse starting ﬂmn a (1x1) Hamiltonian and contact selfenergies
{RHe] [211:4 [3’1le . [Earth/2m] . " Y435 U ml Jana : —= i ”f m " iguana? —w 25:) Probiem 6: Consider a device with two spin—degenerate lcvcls described by E 0 at D
[H ]= 1: 0 a] and with cnc ccntacl magnetized along +2 with [I1]: [0 0]. Contact 2 is identical to contact 1, cxccp‘t mat it is magnetized along +x instead of +2. What is the
concspunding [T2]? Hint: If we use up and dawnspins along +x as our basic, [F2] would look just like
[Fl]. Now transfarm this into the regular basis using up and dawmspins airing 2. 5pm ailing + F: :{c 3?, — Fr, : {——3* If? *}T (Tﬂmnspose) where c E cos{l9f2) 3"” and SE si11(fil'2).c+””I ...
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 Spring '08
 S.Datta

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