Peierls Transition Of Graphene Nanoribbons:
Crossover From Polyacetylenes To Graphene
Tatsuo Suzuki
Tokyo Metropolitan College of Industrial Technology, 8-17-1, Minami-Senju, Arakawa-ku, Tokyo 116-8523, JapanAbstract. Graphene nanoribbons (GNRs) are stripes of single layered graphenes with a finite width. The electronic
states of GNRs are different by the edge conditions (armchair edges or zigzag edges). The results of tight-binding
approximations are that the hydrogen terminated zigzag GNRs are metallic, and the hydrogen terminated armchair
GNRs are metallic or semiconducting. But the narrowest hydrogen terminated GNRs are polyacethlenes, that aresemiconducting. The reason of this contradiction is that the structures of polyacethlenes are changed by the Peierls
transition. Therefore the effects of the Peierls transition of GNRs from polyacetylenes to graphene were examined
systematically using density functional theory with structural optimizations.
Keywords: gaphene, nanoribbon, edge, armchair, zigzag, polyacetylene, Peierls transition, bond length
PACS: 73.22.-f, 72.80.Vp, 71.30.+h, 72.80.Le, 71.20.Rv, 36.20.Kd

INTRODUCTION
Graphene nanoribbons (GNRs) are stripes of single
layered graphenes with a finite width. The electronic
states of GNRs are different by the edge conditions,those are armchair edges or zigzag edges (Fig. 1). The
results of tight-binding approximations [1] are that thehydrogen terminated zigzag GNRs are metallic, andthe hydrogen terminated armchair GNRs are metallic
or semiconducting. But the narrowest hydrogenterminated GNRs are polyacethlenes, those aresemiconducting. The reason of this contradiction isthat the structures of polyacethlenes are changed by
the Peierls transition. First-principles calculations [25] show that the hydrogen terminated armchair GNRs
are semiconducting, but the energy gaps of thesecalculations do not agree one another. Therefore inthis paper, more accurate first-principles calculations
were done, and the effects of the Peierls transition ofGNRs from polyacetylenes to graphene wereexamined systematically.

COMPUTATIONAL METHOD
The electronic states of GNRs were calculatedusing density functional theory with structural
optimizations. All calculations were performed using
Gaussian 03 package [6] with periodic boundary
conditions. These calculations did not include spinpolarized effects. I wanted to calculate the electronicstates using the hybrid B3LYP exchange andcorrelation functions combined with 6-311+G(d,p)
basis set, but unfortunately these calculations did notconverge. Therefore armchair GNRs (N=2, 3) andzigzag GNRs (N=1, 2) were calculated using
B3LYP/6-311G(d,p); armchair GNRs (N=4, 5, 6, 7, 8,9, 10) and zigzag GNRs (N=3, 4, 6) were B3LYP/631G(d); a zigzag GNR (N=5) was B3LYP/6-31G; andzigzag GNRs (N=7, 8, 9) were B3LYP/3-21G.

RESULTS AND DISCUSSION
Calculated band structures are shown in Fig. 2.
And the energy gaps are shown in Fig. 3, which
include the results of other first-principles calculations[2-5] and the experiment values of polyacetylenes [7].
The calculated values of this work agree well with theexperiment values. The reason of this agreement is thatthe hybrid B3LYP exchange and correlation functions
are used in this work. The results of tight-binding
approximations [1] are that armchair GNRs aremetallic when N=3M-1 (M is an integer), but thecalculations of this work show that all armchair GNRs
are semiconducting. This result is coincident withother first-principles calculations [2-5] qualitatively.
The calculated bond lengths between carbon atomsof armchair GNRs are shown in the top of Fig. 4,
which include the experiment value of cispolyacetylenes [8] and the bond length of bulk
graphene calculated using B3LYP/3-21G.
Theidentification number of bond of the horizontal axis isexplained in Fig. 1. The bond lengths of b series agreecompletely with the bond lengths of c series. Thebond lengths between carbon atoms connected with ahydrogen atom (a1 and aN) are much shorter thanother bond lengths. The bond lengths of dimer lines ofcis-polyacetylene (a1 and a2 of N=2) are the same, so
the effect of the Peierls transition does not appear as
the difference among bond lengths in armchair GNRs.
The inner bond lengths of armchair GNRs agree withthe bond length of bulk graphene.
The calculated bond lengths between carbon atomsof zigzag GNRs are shown in the bottom of Fig. 4,which include the experiment values of transpolyacetylenes [8] and the bond length of bulkgraphene.
The bond lengths of carbon atoms
connected with a hydrogen atom (d1, e1, dN, and eN)
are shorter, but the bond lengths of outer verticalbonds (f1 and FN-1) are longer than other bondlengths. The bond lenghs of trans-polyacetylene (d1and e1 of N=1) are not the same, which is theappearance of the Peierls transition. The bond lengthsof the width N=2 (d1, e1, d2, and e2 of N=2) arealmost the same, so the effect of the Peierls transition
is not clear. But the bond lengths of the width N=3(d2 and e2 of N=3) does not agree with each other, so
this is the appearance of the Peierls transition also. Inthe wider zigzag GNRs, the bond lengths of d series
agree with the bond lengths of e series. Therefore theeffects of the Peierls transition vanish rapidly
increasing the ribbon width. The inner bond lengths ofzigzag GNRs does not agree with the bond length ofbulk graphene. The bond lengths of zigzag chains (dseries and e series) are shorter than the bond length ofbulk graphene, and the bond lengths of vertical bonds(f series) are longer than the bond length of bulkgraphene.

.
FIGURE 1. (Left top) cis-polyacetylene. (Left bottom)
armchair GNR with N dimer lines. (Right top) transpolyacetylene. (Right bottom) zigzag GNR with N zigzag
chains.

FIGURE 3. The energy gaps of GNRs. Round solid marks
are calculated values of this work. Red is armchair, and blueis zigzag. Red open marks are other first-principles
calculations of armchair GNRs [2-5]. Black cross marks arethe experiment values of polyacetylenes [7].

FIGURE 4. The calculated bond lengths between carbon
atoms. of armchair GNRs. Black line is the bond length ofbulk graphene calculated using B3LYP/3-21G.
(Top)
Armchair GNRs. Black mark is the experiment value of cispolyacetylenes [8]. (Bottom) Zigzag GNRs. Black marks
are the experiment values of trans-polyacetylenes [8].

REFERENCES
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2.
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4.
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FIGURE 2. The calculated band structures. (Left) armchairGNRs. (Right) zigzag GNRs.

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