Coursehero >> New York >> Cornell >> ECE 4070

Course Hero has millions of student submitted documents similar to the one below including study guides, homework solutions, papers, exam answer keys and textbook solutions.

of Department Electrical and Computer Engineering, Cornell University ECE 407: Physics of Semiconductor and Nanostructures Spring 2009 Exam 1 ` Feb. 26, 2009 INSTRUCTIONS: Every problem must be done in the blue booklet Only work done on the blue exam booklets will be graded do not attach your own sheets to the exam booklets under any circumstances To get partial credit you must show all the relevant work Correct answers with wrong reasoning will not get points All questions do not carry equal points All questions do not have the same level of difficulty DO NOT WRITE IN THIS SPACE 1 Problem 1 (2D lattice) 30 points Consider the following lattice consisting of three different kinds of atoms: A-atom (black-filled), B-atom (gray-filled), and C-atom (black-unfilled). y a a x A B C a a 2 a r r a) Find and give expression for the primitive lattice vectors a1 and a2 and draw them on the figure and submit the figure with your exam copy. b) Sketch the Wigner-Seitz primitive cell for the lattice on the figure and submit the figure with your exam copy. c) How many A-toms, how many B-toms, and how many C-atoms are in one Wigner-Seitz primitive cell? d) Sketch the reciprocal lattice on the square grid provided below in the figure by black dots and label the dimensions where question marks are shown. Submit this figure with your exam copy. ky ? kx 2 Problem 2 (2D lattice) 40 points Consider the following crystal. y r a1 x r a2 a a On each lattice site there is an atom that has one s-orbital with energy Es, and one px-orbital and one pyorbital and one pz-orbital all with energies Ep. The tight binding solution for this crystal can be written as: r rr e i k . Rm (r ) = c1 s r R m N m rr [ ( ) + c2 px r R m + c 3 py r R m + c 4 pz r R m (r r ) (r r ) (r r The )] resulting 4x4 matrix can be written as: H11 H12 H 21 H 22 H 31 H 32 H 41 H 42 H13 H 23 H 33 H 43 H14 H 24 H 34 H 44 c1 c1 r c 2 c 2 = E k c 3 c 3 c 4 c 4 () In answers to questions below, use the standard expressions for the matrix elements (i.e. Vss , V pp , etc). a) Find the 4 diagonal elements of the matrix: H11 , H 22 , H 33 , H 44 b) Find these off-diagonal components: H12 , H 23 , H 23 . c) Find these off-diagonal components: H14 , H 24 , H 34 3 Problem 3 (Misc) 30 points PART I: The chemical formula for ethyne (acetylene) is C2H2. Each carbon atom in ethyne is sp-hybridized (a scheme in which a s-orbital and a p-orbital get superposed with equal weights to generate two new sporbitals that point in opposite directions). a) Draw the structure of the ethyne molecule indicating with labels all the sigma and pi bonds. You can draw a bond with a dash and label it appropriately (e.g. s-s- , p-p- , p-p- , s-sp- , s-sp- , etc) PART II: Consider a one-dimensional crystal shown below. x a At each lattice site there is one atom that has a single s-orbital that contributes to bonding. The resulting energy band dispersion is: r E k = E s 2Vss cos(k x a ) () b) Find the 1D density of states g1D (E ) that will let you convert an integral over k-space for this energy band to an integral over energy-space and plot g1D (E ) as a function of the energy E. PART III: In a certain 2D crystal the energy band dispersion of the electrons is given by the formula: r Ek =hv () 2 2 kx + ky c) Find the 2D density of states g 2D (E ) that will let you convert an integral over k-space for this energy band to an integral over energy-space and plot g 2D (E ) as a function of the energy E. 4

Find millions of documents here - Study Guides, Homework Solutions, Papers, Exam Answer Keys and more. Course Hero has millions of course related materials that will enable you to learn better, faster and get an A in all your courses.
Below is a small sample set of documents: