Electronic structure and Peierls instability in graphene nanoribbons sculpted in graphane

Abstract

Graphene nanoribbons are semiconductor nanostructures with great potentials in nanoelectronics. Their realization particularly with small lateral dimensions below a few nanometers, however, remains challenging. Here we theoretically analyze zigzag graphene nanoribbons created in a graphane substrate (a fully saturated two-dimensional hydrocarbon with formula CH) and predict that they are stable down to the limit of a single carbon chain. We exploit density functional theory with B3LYP functional that accurately treats exchange and correlation effects and demonstrate that at small widths below a few chains these zigzag nanoribbons are semiconducting due to the Peierls instability similar to the case of polyacetylene. Graphene nanoribbons in graphane might represent a viable strategy for the realization of ultranarrow semiconducting graphene nanoribbons with regular edges and controlled chemical termination and open the way for the exploration of the competition between Peierls distortion and spin effects in artificial one-dimensional carbon structures.

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Electronic structure and Peierls instability in graphene
nanoribbons sculpted in graphane
Valentina Tozzini∗and Vittorio Pellegrini
Scuola Normale Superiore, NEST CNR-INFM and
IIT-Italian Institute of Technology, I-56100 Pisa, Italy
(Dated: October 31, 2009)
Abstract
Graphene nanoribbons are semiconductor nanostructures with great potentials in nanoelectron-
ics. Their realization particularly with small lateral dimensions below a few nanometers, however,
remains challenging. Here we theoretically analyze zig-zag graphene nanoribbons created in a
graphane substrate (a fully saturated two-dimensional hydrocarbon with formula CH) and predict
that they are stable down to the limit of a single carbon chain. We exploit density functional theory
with B3LYP functional that accurately treats exchange and correlation effects and demonstrate
that at small widths below a few chains these zig-zag nanoribbons are semiconducting due to the
Peierls instability similar to the case of polyacetylene. Graphene nanoribbons in graphane might
represent a viable strategy for the realization of ultra-narrow semiconducting graphene nanorib-
bons with regular edges and controlled chemical termination and open the way for the exploration
of the competition between Peierls distortion and spin effects in artificial one-dimensional carbon
structures.
PACS numbers:
∗Electronic address: tozzini@nest.sns.it
1
arXiv:0911.0060v1 [cond-mat.mes-hall] 31 Oct 2009

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Graphene is a single atomic layer of carbon atoms arranged in a honeycomb lattice. This
system is remarkably appealing both for fundamental studies and potential electronic appli-
cations since it hosts a high-mobility two-dimensional (2D) electron or hole gas displaying
unique quantum transport properties1,2,3. The unconventional properties of graphene are
the result of its peculiar band structure having zero gap and linear dispersion of conduc-
tion and valence bands at the corners of the Brillouin zone (Dirac points). Graphene-based
nanoelectronic logic devices require, however, the development of graphene nanostructures
displaying sufficiently large band gaps. This is currently achieved in narrow graphene rib-
bons (graphene nanoribbons or GNR), which represent so far one of the most promising
strategies for graphene nanoelectronics4,5. GNRs can be nowadays fabricated by exploiting
different chemical, or lithographic methods5,6,7. Although GNR field-effect transistors have
been demonstrated5,8, and much effort is being devoted to GNR fabrication, the realiza-
tion of GNRs with controllable and reproducible properties remains a challenge. This is
mainly caused by the roughness present at the physical edges of the nanoribbon and by the
large dependence of the electronic properties on the chemical edge termination and on the
nanoribbon chirality both of which cannot be finely controlled in the employed methods of
fabrication7,9,10,11. In addition, current approaches do not allow to reach ultra-narrow GNRs
with widths of just one or a few chains but are limited to values of the order of 3nm (N∼15
chains) or above.
Here we analyze a different type of GNRs that is obtained by sculpturing graphane sub-
strates. We recall that graphane, a fully saturated hydrocarbon version of graphene, is
obtained by adding hydrogen atoms to graphene with stoichiometry 1:112. Its structure,
stability and electronic properties were first theoretically predicted for symmetric configu-
rations where hydrogen atoms are bound half on one side and half on the other side of the
graphene sheet, and recently experimentally observed13. Graphane is an insulator, with a
band gap of ∼3.5eV, and its most stable conformation has the same in-plane symmetry of
graphene and their lattice parameters differ only by 4%12. This circumstance suggests that
hybrid stable graphane/graphene nanostructures with peculiar electronic properties could
be built with a designed shape by selectively removing hydrogen atoms at specific locations.
In particular GNRs defined in graphane (Graphane/Graphene nanoribbons or GGNRs) as
shown in the cartoons of Fig.1 can have in principle lateral atomic dimension down to the
single chain and well-defined chirality if, for example, an STM is used for nanoscale pattering
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by removal of the hydrogen atoms14,15. Other methods to remove hydrogen atoms could be
envisioned such as laser heating or e-beam irradiation.
In order to explore the reliability of this idea we theoretically analyze the stability and
the structural and electronic properties of zig-zag GGNRs (elongated along the y direction
as shown in Fig. 1) within the density functional theory (DFT) frame, including electronic
exchange and correlation at different levels of accuracy. We address the stability of such
nanoribbons and demonstrate they are semiconducting even in the ultra-narrow limit of
a single carbon chain. We show that the opening of the gap at very small widths is the
result of the Peierls distortion that leads to bond-length alternation (difference in lengths
of subsequent bonds or BLA) along y direction and removal of the degeneracy between the
highest occupied state (HOMO) and the lowest empty state (LUMO) as in the case of the
polyacetylene16, which has the same structure of the single chain zig-zag GNR. We show that,
as for polyacetylene, the Peierls instability is captured only including the appropriate form
of the exchange-correlation energy functional. The results predict a band-gap tunability of
such GGNRs in the range between 0.2 eV and 1.5 eV, with the latter value obtained in
the limit of a single chain with atomic lateral dimension. In the case of GNRs, the band-
gap dependence on the width is found in excellent agreement with available experimental
data. These results demonstrate that GGNRs could be exploited as building blocks for the
realization of complex electronic circuits directly sculptured in the graphene substrate and
for the exploration of the impact of Peierls instability in carbon nanostructures.
The DFT calculations here performed are based on both the extended (plane waves, PW)
and the localized (Gaussian bases, GB) wavefunction approaches. The first method has bet-
ter performances on extended systems and is therefore used for the dynamical calculations,
while the second should better reproduce the structural properties of confined systems and
is used for the study of the electronic structure and geometry optimization. Within the
PW scheme the 1s core electrons for C atoms were implicitly treated with TroullierMartins
pseudopotentials, and the valence electrons wavefunctions were expanded in plane waves
with an energy cutoff of 70 ryd (90 Ryd in selected cases). Supercell including up to six
unitary cells in the y direction were used. These calculations were performed with the
CPMD3.13 code17. Within the GB scheme all-electron calculations were performed using
the 6-31G∗basis set. The minimal unitary cell was sampled with up to 200 k points. The
GB calculations were performed with Gaussian0318. In both cases different energy func-
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tional were used: the BeckeLeeYangParr (BLYP)19exchange and correlation functional, its
hybrid version (B3LYP) including 20% of explicit exchange20, and the LSDA21,22, functional
explicitly treating the spin density. The starting graphane configuration was build using the
structural parameters given in Ref.[12] and subsequently optimized (both structure and cell
parameters) with standard local minima search algorithms. In each case the graphene rib-
bons were obtained by simply removing the hydrogen atoms in specific locations as shown in
Fig 1 and re-optimizing the structure. The molecular dynamics simulations were performed
using the BLYP functional within the Car-Parrinello approach23using the electronic mass
preconditioning scheme24and time-step of 0.193fs. We also studied GNRs with the aim of
comparing their properties with those of corresponding GGNRs. In the case of GNRs, the
edges are saturated with hydrogen, a natural termination that does not introduce states
within the gap. The GGNRs are even more naturally terminated, each dangling C- bond
being homogeneously saturated with C, although the graphane introduces some strain due
to the 4% lattice parameter mismatch between graphane and graphene. This, however, does
not significantly contribute to modify the electronic/structural properties of GGNRs with
respect to the saturated GNRs. Figure 2a shows the optimized structure of the zig-zag
of both GNRs and GGNRs. In the case of GNRs, the C-C bonds display a marked BLA
along the x direction that rapidly decreases (within ∼3 chain from the edge) merging into
the graphene-like configuration as one moves towards the center of the ribbon. A similar
behavior is seen for the ribbons embedded in graphane, although the strain induced by the
graphane lattice imposes slightly different C-C values. In addition, a similar decaying C-C
BLA can be seen within the graphane moving out from the nanoribbon. In order to address
the stability of the GGNRs, we first heated the system up to T < 600−700K and observed
no hydrogen hopping or other substantial distortions other than due to the heating itself
(data not shown). This indicates that once formed by selectively removing the hydrogen
atoms, the GGNRs are robust with respect to hopping processes of hydrogen atoms from the
graphane matrix into the graphene ribbon. In order to evaluate the energy barrier for such
a process, we performed constrained molecular dynamics on the reaction path illustrated
in the lower panel of Fig.2b. The hydrogen atom highlighted in red is forced to hop to its
nearest neighbor, without the possibility of going back. From this unstable configuration
the atom spontaneously moves to its final position yielding a configuration, representing
a terminated wire embedded in graphane, chemically consistent (i.e. no radical involved
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as in the intermediate) but nevertheless less stable than the starting one of ∼1.5 eV. The
activation barrier is of ∼5 eV. The upper panel of Fig.2b reports the evolution of the three
involved C-H distances along the reaction coordinate.
Having shown that GGNRs once formed are stable we now discuss their electronic prop-
erties. The calculated gap using the B3LYP functional for both H-passivated GNRs (filled
red dots) and GGNRs (filled blue dots) is shown in the main panel of Fig.4 as a function of
the wire width W. This approach neglects the spin polarization that is taken into account
in the LSDA calculations shown as squares (empty squares are data from Ref. [25]). The
data in Fig.4 demonstrate that the spin-induced opening of the gap vanishes at small widths
in agreement with previous calculations25,26,27. It should be noted, in addition, that the
curve extrapolated from the experimental data obtained in large nanoribbons (red line) nat-
urally merges with the experimental value of the gap of the single-chain nanoribbon, i.e. the
polyacetylene (red asterisk). This suggests significant energy gaps of around 0.5−1.5 eV at
nanoribbon widths where spin effects are negligible. These large gaps are indeed obtained by
our DFT analysis (red and blue filled dots). In this small-width regime the HOMO-LUMO
degeneracy is removed by the Peierls distortion that leads to BLA along the y direction (in
addition to BLA along the x direction) as shown in the inset to Fig.4 in agreement with
recent calculation based on a similar approach28. We found that the Peierls distortion is the
dominant mechanism responsible for the opening of the gap in both GNRs and GGNRs up
to a number of chain N = 3 − 4. For GGNRs the slightly different values of gap and BLA
compared to those in GNRs are due to the small strain induced by the graphane matrix.
For N = 4 the BLA value is very small, although it results still in non-negligible values of
the gap. For N > 4 the spin mechanism becomes important.
In order to highlight the impact of the different contributions to the band-gap, we report
in Fig. 4 the schematic structures of the π and π∗bands in zig-zag nanoribbons with variable
number N of chains as the various effects described above are included starting from the
ideal hexagonal geometry (all equal C-C bonds). Each added chain adds a couple of π?π∗
bands. The edge states start to appear at a certain width as an effect of the accumulation
of degenerate states between 2/3π/a < k < π/a. When the geometry relaxation is allowed,
even at the lower-level theory, the BLA effect in the x direction is observed, similar to
the reconstruction of the surfaces of a 3D crystal. This modifies the bands or breaks the
symmetry mainly at the k = π/a point, but the zig-zag nanoribbon remains metallic at low
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values of the width as in Refs.25,26,27,29. When the B3LYP functional with explicit exchange
is used, the gap opening at π/a at small nanoribbon width (low values of N) due to Peierls
distortion (and BLA in y direction) can be seen. The effect is still appreciable up to N=4,
where the gap opening occurs in one or more points between 2/3 · π/a and π/a. When
the spin-dependent functional is considered, the edge states degeneracy is removed. The
effect is absent for N=1 and N=2 where there are no edge states, and rapidly decrease as N
increases because the edges are less and less interacting. This behavior is also in agreement
with previous calculations25,26,27.
In summary, we have shown that graphene nanoribbons sculpted in graphane by selec-
tively removing hydrogen atoms display the stability and semiconducting properties required
for nanoelectronic applications even when their lateral dimension is within the atomic limit.
Such approach of removing hydrogen atoms from graphane might offer a promising strategy
for the realization of ultra-narrow nanoribbons with ideal edges.
Acknowledgments
We thank Fabio Beltram, Giuseppe Grosso, Giuseppe Pastori Parravicini and Paolo Gi-
annozzi for useful discussions.We aknowledge the allocation of computer resources on
CINECA national facility by means of INFM-CNR Progetto di Calcolo Parallelo 2008-2009.
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FIG. 1: (a) Schematic representation of a graphene nanoribbon (GNR) composed by four zig-
zag chains passivated with H. White balls represent H atoms, coplanar with C atoms located at
the vertices of the honeycomb lattice. The system is periodic in the y direction. (b) Schematic
representation of a graphane/graphene nanoribbon (GGNR) composed by four chains embedded
in graphane. Light grey and dark grey balls represent H atoms bonded to the C atoms of the
corresponding honeycomb lattice site and located above and below the honeycomb lattice plane,
respectively. (c) A perspective view of a GGNR with 3 chains. C and H atoms are represented as
brown and white balls, respectively.
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(a)(b)
FIG. 2: a) The cartoons correspond to the optimized structure of the zig-zag graphene/graphane
nanoribbon (GGNRs). Horizontal axis corresponds to the x direction as defined in Fig.1. The
variation of the C-C bond length along the x direction is also reported (origin at the wire edge,
negative distances are within the ribbon, from the edge to the ribbon center; positive distances
are within the graphane matrix; when BLA is present in the orthogonal direction, the points
correspond to the the average value of the C-C bond length). Solid lines, filled dots refer to
ribbons embedded in graphane; dotted lines, empty dots refer to H-passivated nanoribbons. Red,
green, cyan, blue and magenta correspond to the results of zig-zag GGNRs with widths of 1, 2, 3,
4, and 8 chains, respectively. (b) Simulation of the hopping process of a hydrogen atom from the
graphane substrate to the graphene chain. The black line shows the system energy profile along
the reaction path defined by the three black carbon atoms in the cartoon at the bottom of the
figure. As shown in the three cartoons placed along the energy profile, the hopping hydrogen (in
red) is forced to leave the graphane and hop to the graphene wire passing from a intermediate site.
The variation of the three C-H bond distances along the path is also shown in the upper part of
the figure, in red green and blue.
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FIG. 3: Main panel: energy gap versus wire width (W). Dots and squares: results of calculations,
red for H-passivated zig-zag graphene nanoribbon (GNR), blue for zig-zag graphene/graphane
nanoribbon (GGNR). Filled circles: B3LYP calculations with no spin polarization. Squares: LSDA
calculations with spin polarization (empty squares are data from Ref. [21]; the dashed line is a
guide to the eye). Solid and dotted lines display the function gap=A/(W-W∗) with A=0.2 eVnm,
fitted from experimental data (Ref.[7]) and extrapolated at small W (the experimental data range is
reported as a red band on the curve). Red asterisk with error bar corresponds to the experimental
gap of polyacetylene. Inset: Bond Length Alternation (BLA) along the y axis (see Fig.1) versus
the nanoribbon width. Colors and symbols as in the main panel. The lines are guides to the eye.
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FIG. 4: Schematic qualitative representation of π and π∗band structures in graphene zig-zag
nanoribbons of different widths (N is the number of chains) and at different levels of sophistication
of the theoretical analysis: first column corresponds to the ideal hexagonal geometry. k = 2/3π/a
corresponds to the Dirac K point of graphene Brillouin zone, where in fact the HOMO and LUMO
bands converge in the limit of large N. The second column shows the results in the case of geometric
11