ECE 802-604: Nanoelectronics Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu.edu
Lecture 21, 07 Nov 13 Carbon Nanotubes and Graphene CNT/Graphene electronic properties sp2: electronic structure Find E-k relationship/graph for polyacetylene Find E-k relationship/graph for graphene R. Saito, G. Dresselhaus and M.S. Dresselhaus Physical Properties of Carbon Nanotubes VM Ayres, ECE802-604, F13
Goal: polyacetylene: VM Ayres, ECE802-604, F13
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All Structure: s-bonds Electronic: p-bonds Lec 20: Division of structural and electronic properties in sp2: This is for graphene but it shows you the valence and conduction energy states from both ss and ps All Structure: s-bonds Electronic: p-bonds VM Ayres, ECE802-604, F13
All Structure: s-bonds Electronic: p-bonds Lec 20: Division of structural and electronic properties in sp2: All Structure: s-bonds p valence and conduction energy levels for polyacetylene Electronic: p-bonds VM Ayres, ECE802-604, F13
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-1 VM Ayres, ECE802-604, F13
Will come back to this step VM Ayres, ECE802-604, F13
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Plot E+ and E- versus k. What is the range of k? Symmetric: VM Ayres, ECE802-604, F13
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Can graph this way if you acknowledge that you have two inequivalent carbons: go up using a1 to get to “A” types and down by a2 to get to “B” types a H H H “A” c c c c c “B” H H VM Ayres, ECE802-604, F13
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TYPO CORRECT: VM Ayres, ECE802-604, F13
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Back to this step Transfer matrix Overlap matrix VM Ayres, ECE802-604, F13
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Quick review of what the F are: Bloch wavefunctions: “a” here is the distance between identical situations. VM Ayres, ECE802-604, F13
“a” here is the distance between identical situations. From previous slide 30, for polyacetylene: “a” is the size of our Unit cell. VM Ayres, ECE802-604, F13
Because: polyacetylene has inequivalent C atoms VM Ayres, ECE802-604, F13
a H H H “A” c c c c c “B” More simply: where is C-atom “B” relative to C-atom “A”? H H VM Ayres, ECE802-604, F13
a H H H “A” c c c c c “B” Question: where is C-atom “B” relative to C-atom “A”? Answer: +a/2 H H VM Ayres, ECE802-604, F13
a H H H “A” c c c c c “B” Where are the nearest neighbor C-atoms “B” relative to “A”? H H VM Ayres, ECE802-604, F13
a H H H “A” c c c c c “B” Where are the nearest neighbor C-atoms “B” relative to “A”? Answer: +a/2 and –a/2 H H VM Ayres, ECE802-604, F13
Where are the nearest neighbor “A” to “A” C-atoms? VM Ayres, ECE802-604, F13
This is the summation from “A” to all “A” type carbon atoms: Where are the nearest neighbor “A” to “A” C-atoms? Answer: +a and -a This is the summation from “A” to all “A” type carbon atoms: a a H H H H “A” c c c c c c c “B” H H H -a +a VM Ayres, ECE802-604, F13
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✔ ✔ VM Ayres, ECE802-604, F13
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✔ ✔ ✔ ✔ ✔ ✔ VM Ayres, ECE802-604, F13
Need to model the wavefunctions: |2px> Model |p> orbital ECE 802-604: Use this result, p.24: t = -1.0 s = +0.2 e2p = 0.0 VM Ayres, ECE802-604, F13
1 Find Unit cell “a” 2 Find k: 3 Find H and S elements 4 Rules for finding the electronic structure (p. 21): 1 Find Unit cell “a” 2 Find k: 3 Find H and S elements Det [H – SI] =0 4 Solve for E(k) VM Ayres, ECE802-604, F13