exam1sol - skectch the See 4 2 sin 2 2 2 2 4 2 2 sin 2 2 2...

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1 ECE407 Exam 1 Solutions (By Farhan Rana) Problem 1 (2D lattice) – 30 points a) Note that the B atoms (or the C atoms) form a centered rectangular Bravais lattice. The primitive vectors are as shown: y a x a a y a x a a ˆ 2 ˆ ˆ 2 ˆ 2 1 = + = r r b) Wigner-Seitz primitive cell is shown in the figure (done in two different ways). c) There are one B atom, one C atom, and two A toms in each primitive cell. d) See below. Problem 2 (2D lattice) – 40 points a) ( ) ( ) [ ] () ( ) [] () ( ) [] () ( ) [] 2 1 44 2 1 2 1 33 22 2 1 11 . cos . cos 2 . cos . cos . cos . cos . cos . cos 2 a k a k V E H a k a k V a k a k V E H H a k a k V E H pp p pp pp p ss s r r r r r r r r r r r r r r r r + = + + + = = + = π σ 2 a x y a a a A B C a 1 a r 2 a r x k y k a
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2 b) ( ) ( ) [ ] () ( ) [] () ( ) [] () ( ) [] 2 1 2 1 23 2 1 13 2 1 12 . cos . cos . cos . cos . sin . sin 2 . sin . sin 2 a k a k V a k a k V H a k a k V i H a k a k V i H pp pp sp sp r r r r r r r r r r r r r r r r + = = + = π σ c) 0 34 24 14 = = = H H H Problem 3 (Misc) – 30 points a) b) () () () () () () above
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Unformatted text preview: skectch the See 4 2 sin 2 2 2 2 4 2 2 sin 2 2 2 1 1 2 2 s ss D D x ss V E V E x a x a a x x ss x E E V a E g E g dE a k dE aV dE dE dk dk dk a k aV dk dE ss s ss s − − = ∫ → ∫ → ∫ × → ∫ × → ∫ × ⇒ = ∞ ∞ + − − c) ( ) ( ) ( ) ( ) ( ) ( ) skectch the See 1 2 2 2 2 2 2 2 2 2 v E E g E g dE dE dE dk k kdk k d v dk dE k v k E D D h r h h = ∫ → ∫ × → ∫ × → ∫ × ⇒ = = ∞ ∞ ∞ C C H H sp-s-σ π s-sp-σ sp-sp-σ p-p-π p-p-π E g 1D E s 4V ss E g 2D...
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This note was uploaded on 04/04/2009 for the course ECE 4070 taught by Professor Rana during the Spring '08 term at Cornell.

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exam1sol - skectch the See 4 2 sin 2 2 2 2 4 2 2 sin 2 2 2...

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