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Handout 31 Carbon Nanotubes Handout 31 Carbon Nanotubes: Physics and Applications In this lecture you will learn: • Carbon nanotubes • Energy subbands in nanotubes • Device applications of nanotubes Sumio lijima Paul L. McEuen (Meijo University, Japan)) (Cornell University) Mildred Dresselhaus (MIT) ECE 407 – Spring 2009 – Farhan Rana – Cornell University Another Look at Quantum Confinement: Going to Reduced Quantum Well Dimensions by Band Slicing Quantum Wire E c L y y x z x z 2 2 2 2 2k 2 k x kz z Ec p,k x ,kz Ec1 Ep Ec p,kz Ec1 p 2me 2me 2me E E k x L k 2 x L Ec1 E1 Ec1 kz k x kz ECE 407 – Spring 2009 – Farhan Rana – Cornell University 1 Graphene and Carbon Nanotubes y a = 2.46 A a Single wall carbon nanotube (SWNT) x Multi wall carbon nanotube (MWNT) a 2a a 3 3 • 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 properties ECE 407 – Spring 2009 – Farhan Rana – Cornell University Graphene: -Energy Bands Energy Recall the energy bands of graphene: ky K K’ K’ 4 2 2 3 a 3 a M 3 a M FBZ kx K K FBZ K’ Ek Ep Vpp f k f k eik.n1 eik.n2 eik.n3 ik x k y ik.r x y n,k r e un,k r e un,k r ECE 407 – Spring 2009 – Farhan Rana – Cornell University 2 Graphene Edges Armchair edge Zigzag edge ECE 407 – Spring 2009 – Farhan Rana – Cornell University Rolling Up Graphene Zigzag nanotube Armchair nanotube ECE 407 – Spring 2009 – Farhan Rana – Cornell University 3 Zigzag Nanotubes: Crystal Momentum Quantization y Primitive L cell C L C Circumference of the zigzag nanotube: C ma m 2,3,4....... Boundary condition on the wavefunction: a i k xk y x y x n,k r e un,k r The wavefunction must be continuous along the circumference after one complete roundtrip: 3 a n,k x, y C,z n,k x,y,z ik C e y 1 2 n k n integer, range? y C The crystal momentum in the y-direction (in direction transverse to the nanotube length) Periodicity in the x-direction: 3 a has quantized values Number of atoms in the primitive cell: 4m ECE 407 – Spring 2009 – Farhan Rana – Cornell University Zigzag Nanotubes: 1D Energy Subbands k Energy y FBZ 2 Obtain all the 1D K 3 a subbands of the nanotube by taking K’ K’ 4 cross sections of the 2 2D energy band 3 a M dispersion of 3 a M kx graphene 2 K K C FBZ K’ E k E V f k kx p pp 3a 3a One will obtain two subbands (one from the conduction and one from the valence band) for each quantized value of ky But number of bands = number of orbitals per primitive cell = 4m Number of distinct quantized k y values must equal 2m 2 n k n m 1 ,....... 1,0,1,.......,m y C C ma ECE 407 – Spring 2009 – Farhan Rana – Cornell University 4 Zigzag Nanotubes: 1D Energy Subbands k Energy y FBZ 2 K 3 a K’ K’ 4 2 3 a M 3 a M kx K 2 K C FBZ K’ k 3a x 3a Suppose C = 4a (i.e. m = 4) 2 n n Bandgap! k n 3,2,1,0,1,2,3,4 y C 2a 16 1D subbands Ek Ep Vpp f k total Lower 8 subbands will be completely full at T=0K The nanotube is a semiconductor! ECE 407 – Spring 2009 – Farhan Rana – Cornell University Zigzag Nanotubes: 1D Energy Subbands ky FBZ 2 K 3 a K’ K’ 4 2 3 a M 3 a M kx 2 K K C K’ k 3a x 3a k K' The bandgap appears because the quantized y y k value is such that the “green line” misses y K' ' the K-point kx Kx Bandgap! When: R a (R = radius of nanotube) 2 v 1 E g 3R R ECE 407 – Spring 2009 – Farhan Rana – Cornell University 5 Zigzag Nanotubes: Semiconductor and Metallic Behavior ky Suppose C = 6a (i.e. m = 6) 2 n n FBZ k n 5,.... 1,0,1,......6 y C 3a K Two lines for n=4 pass through the Dirac points 4 K’ K’ 3 a M M 2 kx K C K K’ k 3a x 3a 24 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 bandgap ECE 407 – Spring 2009 – Farhan Rana – Cornell University Motion of Conduction Band Bottom Electrons in Zigzag ik x k y Nanotubes x y n,k r e un,k r ik C e y 1 2 n k n m 1 ,....... 1,0,1,.......,m y C For ky – K (K’) > 0 y • The electrons coil around the nanotube as they move forward x • The direction of coiling can be given by the right hand rule: Direction of propagation y For ky – K (K’) < 0 x or by the left hand rule ECE 407 – Spring 2009 – Farhan Rana – Cornell University 6 Armchair Nanotubes: Crystal Momentum Quantization y Primitive L cell C L C Circumference of the armchair nanotube: C m 3 a m 2,3,4....... a Boundary condition on the wavefunction: x ik x k y 3 a x y n,k r e un,k r The wavefunction must be continuous along the circumference eikxC 1 2 n k n integer, range? x C The crystal momentum in the x-direction (in direction transverse to the nanotube length) Periodicity in the y-direction: a has quantized values Number of atoms in the primitive cell: 4m ECE 407 – Spring 2009 – Farhan Rana – Cornell University Armchair Nanotubes: 1D Energy Subbands Energy FBZ ky Obtain all the 1D 2 K subbands of the 3 a nanotube by taking cross sections of the K’ K’ 2D energy band 2 dispersion of M 3 a M graphene kx K K 2 FBZ ky K’ C a a Ek Ep Vpp f k One will obtain two bands for each quantized value of kx But number of bands = number of orbitals in the primitive cell = 4m Number of distinct quantized k x values must equal 2m 2 n k n m 1 ,....... 1,0,1,.......,m x C C ma ECE 407 – Spring 2009 – Farhan Rana – Cornell University 7 Armchair Nanotubes: 1D Energy Subbands Energy ky FBZ 2 K 3 a K’ K’ M M kx K K 2 FBZ ky K’ a a C Suppose C = 4√3 a (i.e. m = 4) 2 n n k n 3,2,1,0,1,2,3,4 x C 2 3 a Ek E V f k 16 1D subbands p pp total Lower 8 subbands will be completely full at T=0K The nanotube has a zero bandgap! ECE 407 – Spring 2009 – Farhan Rana – Cornell University Armchair Nanotubes: Metallic Behavior Energy ky FBZ 2 K 3 a K’ K’ M M kx K K 2 FBZ ky K’ a a C Armchair nanotubes always have a zero bandgap Proof: Suppose C = m√3 a 2 n 2 n k n (m 1),...... 1,0,1,........,m x C m 3 a 2 For n = m : k and the line passes through the Dirac points x 3 a ECE 407 – Spring 2009 – Farhan Rana – Cornell University 8 Carbon Nanotubes: Applications CNT Nanotube PN Diode AFM Image (McEuen et. al.) CNT MEMs CNT field emission tips for electron guns ECE 407 – Spring 2009 – Farhan Rana – Cornell University Carbon Nanotubes: Applications Carbon Nanotube FET (IBM) Carbon Nanotube LEDs (IBM) Carbon Nanotube FET (Burke et. al.) ECE 407 – Spring 2009 – Farhan Rana – Cornell University 9 Carbon Nanotubes: Applications One main obstacle to making a space elevator is finding a material for the cable that is strong enough to withstand a huge amount of tension. Some scientists think that cables made from carbon nanotubes could be the answer…… Carbon Nanotube Space Elevator !! ECE 407 – Spring 2009 – Farhan Rana – Cornell University 10.
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