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ECE495N-F08-Final_Exam_solution

# ECE495N-F08-Final_Exam_solution - Fall 2009 ECE 495N FINAL...

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Unformatted text preview: Fall 2009 ECE 495N FINAL EXAM CLOSED BOOK Friday, Dec.19, 7P-9P, LYNN G165 NAME: QOLUTQLOM PUID # : Please show 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. 6] 5 points Problem 6 [p. 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), D(E) = (DOLW) E and a density of modes (dimensionless), M (E ) = (M 0W)E , where DO and M0 are constants. Assuming ballistic transport, obtain an expression for the conductance per unit width at T=OK, as a function of the electron density per unit area. "WE 'ﬂwnwwamoxaﬁﬁ w E l Mm 5? all” M (:3 2i“ (Eli; mum: 5;? m Egg/W Ma file” W:Wm‘¢>m‘ﬁxﬁ3¢2q mW-wvaxmvwiw' a E? Problem 2: A channel has four energy levels all 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. Starting from the law of equilibrium, What is the average number of electrons in the channel if it is in equilibrium with chemical potential it and temperature T? Your answer should be in terms of 8, u and T. P t“ 431; “3““ “1”" ﬁt if J t Problem 3: A molecule (NOT a solid) consists of four atoms arranged at the corners of a square. Assume (1) one orbital per carbon atom as basis function ; (2) the overlap matrix [S] is a (4x4) identity matrix; and (3) the Hamiltonian matrix is given by H n ﬂ = 8 (site energy) H n ,m = t if n, m are neighboring atoms H n ,m = 0 if n, m are NOT nearest neighbors What are the four energy eigenvalues in terms of ‘ 8’ and ‘t’ and the corresponding eigenvectors. Please do not diagonalize directly: use the principle of bandstructure. Problem 4: Suppose a large two—dimensional conductor has an 805) relationship given b : .. y 8(k)= +oa/kj+ky2 Where a is a constant. Derive an expression for the density of modes, M(E). Your answer should be in terms of the energy E, a, width W and length L. ) » 1 38 k :- (vx( ) hakx Problem 5: Early in this course we showed that the maximum current through a device with a single level with escape rates 71,9’2 is given by (q/h) 7172/(71+72) using elementary arguments. Prove this result using the general matrix expressions discussed later in the course starting from a (1x1) Hamiltonian and contact self-energies [HHS] [Earth/2] , [221=—i[y2/2] °° ydE 1322+(y/2) (Useful relation. [E =27r) Problem 6: Consider a device with two spin—degenerate levels described by a O or 0 [H] = [0 8] and with one contact magnetized along +2 with [E] = [0 0]. Contact 2 is identical to contact 1, except that it is magnetized along +x instead of +2. What is the corresponding [F2]? Hint: If we use up- and down-spins along +x as our basis, [F2] would look just like [Fl]. Now transform this into the regular basis using up- and down-spins along 2. Spins along + fl 2 {C S}T, — ft I {—S* C *}T (T :transpose) where C E 003(9/2) 5W2 and s E sin(9/2) e+’."’/2 ...
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ECE495N-F08-Final_Exam_solution - Fall 2009 ECE 495N FINAL...

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