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of Department Electrical and Computer Engineering, Cornell University ECE 407: Physics of Semiconductor and Nanostructures Spring 2009 Homework 5 ` Due on Feb. 24, 2009 at 5:00 PM Suggested Readings: a) Lecture notes Problem 5.1 (1D lattice energy bands outside the FBZ) In the last homework, in problem 4.1, you found the exact solution for an electron in a periodic 1D lattice. You will consider the same problem again here. Consider a 1D lattice of lattice constant a equal to 5 Angstroms. Suppose the potential has the form: 2 4 V (x ) = 2V1 cos x + 2V2 cos x a a Where V1 equals 0.3 eV and V2 also equals 0.3 eV. The exact solution for any wavevector k in the FBZ can be written as a superposition of plane waves in the form: k = Where: m = c k (Gm ) k + Gm = m = c (Gm ) 1 i (k + Gm ) x e L 2 a A good approximation to the exact solution can be obtained by terminating the series above at both ends, as follows: Gm = m k = m = N c k (Gm ) k + Gm N Where N is some large number, say 10. The way you hopefully solved the problem in homework 4 was to first choose a value of the wavevector k in the FBZ, then setup a matrix, and then find its three smallest eigenvalues. The question is what if one chooses a value of the wavevector k that is not in the FBZ? Would one end up with some new energy eigenvalues and new energy eigenfunctions? The goal of the problem is to explore this point. **If you are not confident of your answer in homework 4, you can download a matlab routine from the course website that will plot the first three energy bands in the FBZ** a) Choose values of the wavevector k between 3 a and + 3 a , and for each chosen value of wavevector k find the smallest three energy eigenvalues by repeating the same procedure as you followed in problem 4.1 the only difference being that now your range of wavevector values include higher order BZs. What do you find? How do the energy bands look like outside the first BZ? Do you think it makes sense to restrict the k values to the first BZ? Hand in your plots. 1 Problem 5.2 (Ethene (or Ethylene) molecule: LCAO) Two carbon atoms and four hydrogen atoms form an ethene molecule with the chemical formula C2H4. It is the most manufactured hydrocarbon in the world. The structure of the ethene molecule is shown below: H =120o C C H =120o H H a) Convince yourself that the 2p-orbitals on C atoms oriented out of the plane of the paper do not have a non-zero energy matrix element with any of the 1s-orbitals, or 2s-orbitals, or any of the 2p-orbitals that are in the plane of the paper. Explain why this is so. For parts (b) and (c) you can ignore the 2p-orbitals on C atoms that are oriented out of the plane of the paper. You are then left with 4 1s orbitals one on each H atom, 2 2s-orbitals one on each C-atom, and 4 2p-orbitals two on each C atom that are in the plane of the paper. What b) is the hybridization scheme (sp3 or sp2) for each C atom? c) Write an appropriate trial solution for the wavefunction of the electron using LCAO (remember to ignore the 2p-orbitals on C atoms that are oriented out of the plane of the paper). Use the result from part (b) to guide you. d) Using your solution in part (b), solve for the eigenenergies. Show all the eigenenergies and the corresponding degeneracies going from the atomic orbitals to the molecular orbitals in an energy level diagram. Use values of energy matrix elements given in the handouts (e.g. Vss , Vsp , etc). Now we will consider what the 2p-orbitals on C atoms oriented out of the plane of the paper do. They are going to form molecular orbitals of their own. e) Write an appropriate trial solution for the wavefunction of the electron using LCAO using only the 2porbitals on C atoms oriented out of the plane of the paper. f) Using your solution in part (e), solve for the eigenenergies. Show all the eigenenergies and the corresponding degeneracies going from the atomic orbitals to the molecular orbitals in an energy level diagram. Use values of energy matrix elements given in the handouts (e.g. Vss , Vsp , etc). What type of bond is made by the 2p-orbitals on C atoms oriented out of the plane of the paper (sigma or pi)? g) Combine your results from part (d) and part (f) and draw an energy level diagram that shows all the energies and the corresponding degeneracies going from the atomic orbitals to the molecular orbitals (and remember this time ALL atomic orbitals are being included). On the energy level diagram indicate (by black dots) the electron occupation of each molecular orbital (you have a total of 12 electrons participating in bonding). 2 Problem 5.3 (Energy bands for a simple cubic lattice) a a FBZ R ( /a, /a, /a) X ( /a, 0, 0) a Consider a 3D crystal with a simple cubic lattice of lattice constant a. At each lattice point there is an atom that has one s-orbital and three p-orbitals (px, py, and pz) that can participate in bonding and energy band formation. The orbitals and the corresponding energies are as follows: s (r ) r r px (r ) r py (r ) r pz (r ) Es Ep Ep Ep r a) Write a trial tight-binding wavefunction for the Bloch state with eigenvector k . b) Write down the matrix equation that results when the trial solution is plugged into the Schrodinger equation. Use values of energy matrix elements as given in the lecture handouts (e.g. Vss , Vsp , etc). c) Find the energies at the zone center ( -point) for all the energy bands, and remark of which atomic orbitals the corresponding Bloch states are made up of. d) Find the energies at the zone edge ( a ,0,0 ) for all the energy bands, and remark of which atomic orbitals the corresponding Bloch states are made up of. e) Find the energies at the zone edge ( a , a , a ) for all the energy bands, and remark of which atomic orbitals the corresponding Bloch states are made up of. 3

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