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Band Structures in Zigzag Graphene Nanoribbons and Armchair Carbon Nanotubes

​
compare Brillouin zone
center
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
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.

Details

In this Demonstration, the width index
w
of a
ZGNR(w)
and the chiral index
n
of an
ACNT(n,n)
can be varied independently to allow investigating band structures for the tubes and ribbons with different lateral sizes. As discussed in[1], band structures that match at the center
k=0
and at the edge
k=π
of the dimensionless Brillouin zone can be obtained when
n=w+1
or
w
, respectively. Choose the "full" comparison mode to overlay the two electronic band structures. In this mode, it is seen that, unlike the case of armchair graphene nanoribbons and zigzag carbon nanotubes[2], for a zigzag ribbon and armchair tube the band matching cannot be achieved in the whole Brillouin zone.
In all modes of comparison, the bands are explicitly numbered as proposed in[1]. You can choose "inverse" or "direct" band enumeration. Since some of the bands for ACNTs are double degenerate, they are labeled with two numbers, with one number enclosed in parentheses.
For ZGNRs, the transition points
k
t
and
k
t
'
are defined via the width index
w
as
±2arccos
w
2(w+1)
.
The energy band matching presented here for armchair tubes and zigzag ribbons has been predicted to result in peak correlation of the optical absorption spectra in the tubes and ribbons[1]. This prediction may soon be tested experimentally, since atomically precise zigzag edges have recently been produced through surface-assisted polymerization and cyclodehydrogenation of specifically designed precursor monomers[3].
Snapshot 1:
ZGNR(5)
and
ACNT(4,4)
energy band matching at the Brillouin zone center
Snapshot 2:
ZGNR(4)
and
ACNT(4,4)
energy band matching at the Brillouin zone edge
Snapshot 3: comparison of
ZGNR(2)
and
ACNT(3,3)
energy bands in the "full" mode

References

[1] V. A. Saroka, M. V. Shuba and M. E. Portnoi, "Optical Selection Rules of Zigzag Graphene Nanoribbons," Physical Review B, 95(15), 2017 155438. doi:10.1103/PhysRevB.95.155438.
[2] C. T. White, J. Li, D. Gunlycke and J. W. Mintmire, "Hidden One-Electron Interactions in Carbon Nanotubes Revealed in Graphene Nanostrips," Nano Letters, 7(3), 2007 pp. 825–830. doi:10.1021/nl0627745.
[3] P. Ruffieux, S. Wang, B. Yang, C. Sánchez-Sánchez, J. Liu, T. Dienel, L. Talirz, P. Shinde, C. A. Pignedoli, D. Passerone, T. Dumslaff, X. Feng, K. Müllen and R. Fasel, "On-Surface Synthesis of Graphene Nanoribbons with Zigzag Edge Topology," Nature, 531(7595), 2016 pp. 489–492. doi:10.1038/nature17151.

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