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 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 (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’   Ek  Ep  Vpp f k

       f k   eik.n1  eik.n2  eik.n3

  ik 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 xk 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 Ek  Ep  Vpp f k total Lower 8 subbands will be completely full at T=0K The nanotube is a !

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 in Zigzag ik 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 ik 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 Ek  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   Ek  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 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