Band Structures in Zigzag Graphene Nanoribbons and Armchair Carbon Nanotubes

compare Brillouin zone

cen

ter

edge

full

band enumeration

direct

inverse

ribbon width index, w

1

2

3

4

5

6

tube chiral index, n

1

2

3

4

5

6

0

π/2

π

3 π/2

2 π

-3

-2

-1

0

1

2

3

k

E/γ

ACNT(6,6) versus ZGNR(5)

J

12

1(11)

2(10)

3(9)

4(8)

5(7)

6

12

1(11)

2(10)

3(9)

4(8)

5(7)

6

J

1

2

3

4

5

1

2

3

4

5

K

′

k

t

'

1

2

3

4

5

This Demonstration compares electronic band structures of zigzag graphene nanoribbons (ZGNRs) with those of armchair carbon nanotubes (ACNTs) [1]. In the nearest-neighbor tight-binding model, the dimensionless electron energy

E/γ

, where

γ

is the hopping integral, is plotted against the dimensionless electron wavenumber

k∈(-π,π)

. This plot is supplemented by the tube and ribbon atomic structures. Positions of the

k

points corresponding to the Dirac points in the graphene Brillouin zone are marked by red vertical lines labeled

K

and

K'

. For ZGNRs, the gray vertical lines designate positions of the transition points

k

t

and

k

t

'

, where ribbon energy bands sneak outside the light blue area corresponding to the region of the graphene band structure. For both ZGNRs and ACNTs, the band indices

J

are specified for all their bands.