1ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityHandout 31Carbon Nanotubes: Physics and ApplicationsIn this lecture you will learn:• Carbon nanotubes• Energy subbands in nanotubes• Device applications of nanotubesPaul L. McEuen (Cornell University)Sumio lijima (Meijo University, Japan))Mildred Dresselhaus(MIT)ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityAnother Look at Quantum Confinement: Going to Reduced Dimensions by Band SlicingQuantum WellQuantum WirezkxkEezexpczxcmkmkEEkkpE22,,22221xyzLLkx2LkxezpczcmkEkpE2,221cEzkE1cE11EEcxyz2ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversitySingle wall carbon nanotube (SWNT)Graphene and Carbon Nanotubesaaxya = 2.46 A• Carbon nanotubes are rolled up graphene sheets• Graphene sheets can be rolled in many different ways to yield different kinds of nanotubes with very different properties32a3aMulti wall carbon nanotube (MWNT)ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityKK’KK’K’KMFBZMkfVEkEpppxkykGraphene: -Energy BandsEnergyFBZ rueruerknykxkiknrkiknyx,,.,321...nkinkinkieeekfa32a32a34Recall the energy bands of graphene:3ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityArmchair edgeZigzag edgeGraphene EdgesECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityZigzag nanotubeArmchairnanotubeRolling Up Graphene4ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityZigzag Nanotubes: Crystal Momentum QuantizationaCircumference of the zigzag nanotube:.......4,3,2 mmaCruerknykxkiknyx,,Boundary condition on the wavefunction:xyThe wavefunction must be continuous along the circumference after one complete roundtrip:range? integer,21,,,,,,nCnkezyxzCyxyCikknknyThe crystal momentum in the y-direction (in direction transverse to the nanotube length) has quantized values CLCL Periodicity in the x-direction:a3a3Primitive cellNumber of atoms in the primitive cell:m4ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityZigzag Nanotubes: 1D Energy SubbandsEnergyFBZKK’KK’K’KMFBZMxkyka32a32a34C2akax33Obtain all the 1D subbands of the nanotube by taking cross sections of the 2D energy band dispersion of graphene kfVEkEpppOne will obtain two subbands (one from the conduction and one from the valence band) for each quantized value of But number of bands = number of orbitals per primitive cell =  Number of distinct quantized values must equal 2mykm4maCykmmnCnky,.......,1,0,1,.......125ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityZigzag Nanotubes: 1D Energy SubbandsEnergyFBZK’KK’K’KMFBZMxkyka32a32a34C2akax33kfVEkEpppSuppose C = 4a (i.e. m = 4)4,3,2,1,0,1,2,322 nanCnky16 1D subbands totalLower 8 subbands will be completely full at T=0KThe nanotube is a semiconductor!Bandgap!KECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityZigzag Nanotubes: 1D Energy Subbands'yKyk'xKxkBandgap!K'The bandgap appears because the quantized kyvalue is such that the “green line” misses the K-pointWhen: (R = radius of nanotube)aRRRvEg132K’KK’K’KMFBZMxkyka32a32a34C2akax33K6ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityZigzag Nanotubes: Semiconductor and Metallic BehaviorKK’KK’K’KMFBZMxkyka34akax33Suppose C = 6a (i.e. m = 6)6,......1,0,1,....532 nanCnkyTwo lines for n=4 pass through the Dirac points24 1D subbands total, 12 lower ones will be completely filled at T=0K, and there is no bandgap!• All zigzag nanotubes for which m = 3p (p any integer) will have a zero bandgap All zigzag nanotubes with radius R = C/2= 3pa/2 (p any integer) will have a zero bandgapC2ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityMotion of Conduction Band Bottom Electrons in Zigzag Nanotubesruerknykxkiknyx,, 1Cikyex• The electrons coil around the nanotube as they move forward• The direction of coiling can be given by the right hand rule:or by the left hand ruleDirection of propagationmmnCnky,.......,1,0,1,.......12For ky– K (K’) > 0xFor ky– K (K’) < 0yy7ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityArmchair Nanotubes: Crystal Momentum QuantizationaCircumference of the armchair nanotube:.......4,3,23  mamCruerknykxkiknyx,,Boundary condition on the wavefunction:xyThe wavefunction must be continuous along the circumferencerange? integer,21nCnkexCikxThe crystal momentum in the x-direction (in direction transverse to the nanotube length) has quantized values CLCL Periodicity in the y-direction:aa3Primitive cellNumber of atoms in the primitive cell:m4ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityArmchair Nanotubes: 1D Energy SubbandsEnergyFBZKK’KK’K’KMFBZMxkyka32a32C2akayObtain all the 1D subbands of the nanotube by taking cross sections of the 2D energy band dispersion of graphene kfVEkEpppOne will obtain two bands for each quantized value of But number of bands = number of orbitals in the primitive cell =  Number of distinct quantized values must equal 2mxkm4maCxkmmnCnkx,.......,1,0,1,.......128ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityArmchair Nanotubes: 1D Energy SubbandsEnergyFBZkfVEkEpppSuppose C = 4√3 a (i.e. m = 4)4,3,2,1,0,1,2,3322 nanCnkx16 1D subbands totalLower 8 subbands will be completely full at T=0KThe nanotube has a zero bandgap!KK’KK’K’KMFBZMxkyka32C2akayECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityArmchair Nanotubes: Metallic BehaviorEnergyFBZSuppose C = m√3