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ECE 802-604:Nanoelectronics
Prof. Virginia AyresElectrical & Computer EngineeringMichigan State Universityayresv@msu.edu
VM Ayres, ECE802-604, F13
Lecture 20, 05 Nov 13
Carbon Nanotubes and Graphene
Carbon nanotube/Graphene physical structure
Carbon bond hybridization is versatile : sp1 (Lec19), sp2, and sp3 (HW05)
More motivation for bond hybridization
CNT/Graphene electronic propertiessp2: origin of CNT/Graphene mechanical and electronic
structuressp2: electronic structure
R. Saito, G. Dresselhaus and M.S. DresselhausPhysical Properties of Carbon Nanotubes
A. Beiser, Modern Physics
E. Anderson, Quantum Mechanics
VM Ayres, ECE802-604, F13
sp1 hybridization– Use orthonormality
sp2 hybridization:– Get help from directions– Use orthonormality
VM Ayres, ECE802-604, F13
sp2 hybridization:
1
VM Ayres, ECE802-604, F13
sp2 hybridization:
1
Example: how many valence (bonding) electrons do you get in:sp1 hybridization?sp2 hybridization?
VM Ayres, ECE802-604, F13
sp2 hybridization:
Answer:
sp = sp1 hybridization:2e- per C: linear bonding
sp2 hybridization:3e- per C: trigonal bonding
sp sp spsp
sp2
sp2
sp2
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
sp2 hybridization:
VM Ayres, ECE802-604, F13
Lecture 20, 05 Nov 13
Carbon Nanotubes and Graphene
Carbon nanotube/Graphene physical structure
Carbon bond hybridization is versatile : sp1 (Lec19), sp2, and sp3 (HW05)
More motivation for bond hybridization
CNT/Graphene electronic propertiessp2: origin of CNT/Graphene mechanical and electronic
structuressp2: electronic structure
R. Saito, G. Dresselhaus and M.S. DresselhausPhysical Properties of Carbon Nanotubes
A. Beiser, Modern Physics
E. Anderson, Quantum Mechanics
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
F
Expect: 180ºGet: 180º
VM Ayres, ECE802-604, F13
O H
H
Expect: 90ºGet: 104.5º“Explanation”:Mutual repulsion of the H atoms“Plausible”:H2S, H2Se = 92º, 90ºGreater separation of H around larger S and Se atoms
VM Ayres, ECE802-604, F13
N H
H
HExpect: 90º pyramidGet: 107.5º pyramid“Explanation”:Mutual repulsion of the H atoms“Plausible”:H3P, H3As = 92º, 90ºGreater separation of H around larger P and As atoms
VM Ayres, ECE802-604, F13
C H
H
Expect: 90º CH2, like H2O
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C H
H
Expect: 90º CH2, like H2OGet: CH4 tetrahedronCH4 is impossible in this p-orbital picture.
VM Ayres, ECE802-604, F13
sp3 Hybrid orbital picture
VM Ayres, ECE802-604, F13
Lecture 20, 05 Nov 13
Carbon Nanotubes and Graphene
Carbon nanotube/Graphene physical structure
Carbon bond hybridization is versatile : sp1 (Lec19), sp2, and sp3 (HW05)
More motivation for bond hybridization
CNT/Graphene electronic propertiessp2: origin of CNT/Graphene mechanical and electronic
structuressp2: electronic structure
R. Saito, G. Dresselhaus and M.S. DresselhausPhysical Properties of Carbon Nanotubes
A. Beiser, Modern Physics
E. Anderson, Quantum Mechanics
VM Ayres, ECE802-604, F13
sp3:
Diamond
methane
alkane
VM Ayres, ECE802-604, F13
sp2:
Benzene
Graphene
CNT (curvature)
fullerene (curvature)
Polyacetylene: non-trigonal: linear
VM Ayres, ECE802-604, F13
Division of structural and electronic properties in linear sp1 that makes both effective:
VM Ayres, ECE802-604, F13
Diamond sp3: no division. Great structural strength but wide bandgap electronic properties
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Division of structural and electronic properties in trigonal sp2 makes both effective:
All
Electronic:-bonds
Structure:-bonds
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Electronic:-bonds
Structure:-bonds
Electronic:Delocalized e-
*-conduction band e-ECE, PHY-anti-bonding e-CHM
-valence band e-ECE, PHY-bonding e-CHM
VM Ayres, ECE802-604, F13
Electronic:-bonds
Structure:-bonds
Electronic:Delocalized e-
*-conduction band e-ECE, PHY-anti-bonding e-CHM
-valence band e-ECE, PHY-bonding e-CHM
VM Ayres, ECE802-604, F13
Lecture 20, 05 Nov 13
Carbon Nanotubes and Graphene
Carbon nanotube/Graphene physical structure
Carbon bond hybridization is versatile : sp1 (Lec19), sp2, and sp3 (HW05)
More motivation for bond hybridization
CNT/Graphene electronic propertiessp2: origin of CNT/Graphene mechanical and electronic
structuressp2: electronic structure:
1st: polyacetylene2nd:graphene:
R. Saito, G. Dresselhaus and M.S. DresselhausPhysical Properties of Carbon Nanotubes
A. Beiser, Modern Physics
E. Anderson, Quantum Mechanics
VM Ayres, ECE802-604, F13
Game plan:
E
VM Ayres, ECE802-604, F13
Rules for finding the electronic structure (p. 21):
2
1
3
4
Find Unit cell “a”
Find k:
Find H and S elements
Solve for E(k)
Det [H – SI] =0
VM Ayres, ECE802-604, F13
Question:We’ve been doing E(k) versus k diagrams since Datta Chp01. Why all this now? What happened to conservation of Energy as the starting point? What has changed?
VM Ayres, ECE802-604, F13
Question:We’ve been doing E(k) versus k diagrams since Datta Chp01. Why all this now? What happened to conservation of Energy as the starting point? What has changed?
Answer:Dresselhaus uses correct Bloch wave functions to describe the electrons. Datta uses travelling waves. This is discussed in Datta page 11. Therefore we must include the symmetry of the polyacetlyene, graphene, etc. lattice in our wave functions. That’s what finding the reciprocal space k is about.