Width-Dependent Conductance of Graphene Nanoribbons


In the last two decades various forms of carbon nanostructures have attracted a great deal of interest due to their novel fundamental properties and possible applications in electronics. With the developments in production methods, these materials have emerged as one of the most promising materials for non-silicon electronics.

Among carbon nanostructures graphene is a flat monolayer of carbon atoms tightly packed into a 2-D honeycomb lattice and is considered as the basic building block of other graphitic materials in all dimensions. [((bibcite : geim :))]

In the past, 2-D materials were thought to be thermodynamically unstable and could not exist. In 2004, experimental discovery of graphene showed that these materials can exist in liquid suspensions or on non- crystalline substrates. Because of this fact, theoretical studies on graphite are much more than experimental ones.


Figure 1.Graphene - a 2D building block for Carbon materials which can attain the shape of 0D bucyballs, 1D nanotubes or 3D graphite

Among graphene based carbon nanostructures, graphene nanoribbons (GNRs) with widths on the order of nanometers have attracted a big attention because of their band gaps able to being designed by playing with their dimensions. They are elongated strips of graphene which can be obtained by cutting a graphene sheet along a certain direction. Since graphene has honeycomb structure, there may be two different edge shapes when they are cut, namely; armchair edge and zigzag edge GNRs.


Figure 2.Honeycomb lattice of Graphene - (a) both directions: zigzag (red) armchair (green), (b) aGNRs, (c) zGNRs

The presence of these edges in GNRs causes the states to localize at the edges of the ribbon which lead to nonzero magnetic moments on the carbon atoms. In some earlier works, it has been established that the ground state of a ZGNR is antiferromagnetic with ferromagnetically ordered spin polarization along the edges. The main contribution to the local moments is said to come from dangling bonds on the edges which are usually hydrogenated in order to stabilize the ribbon electronically and structurally.

2.Experimental Work Done

Despite numerous theoretical studies, experimental work done to study the energy band gap of graphene nanoribbons are yet very limited. Following is brief information about a recent experimental work done.

Han, Özyilmaz, Zhang & Kim (2007)[1]

This group studied the electronic transport measurements of lithographically patterned graphene nanoribbons (GNRs) where the confinement of charge carriers causes an increase of band gap. They worked with a considerable number of GNRs having different widths and which are produced from single sheets of graphene on a Si substrate. The change of band gap was realized by measuring the conductance in the nonlinear response regime at low temperatures.

The G conductance of GNRs are said to be measured by using a standard lock-in technique wit a small applied AC voltage less than 100 microVolts. On the other hand, the width and length of each GNR is reported to be measured by the aid of a scanning electron microscope (SEM). Also it is reported that by examining the differential conductance in the non linear response regime as a function of gate and bias voltage, the size of energy gap can directly be measured.

What they found is energy gaps open more for narrower ribbons. In other words, it is found that the energy gaps of GNRs are inversely proportional to the ribbon width, which makes them very versatile as one can design the desired band gap by lithographic processes.

The following figure is a Egap vs width graph of six different device sets, four of which having parallel GNRs with 15-90 nm widths and two of which having same width but different crystallographic directions.


Figure 3.Egap vs Width for the 6 device sets considered in this study

In the figure, it is clearly seen that band gaps of as high as 200 meV are achieved with approximately 15 nm width. It is thought that a narrower GNR may show even a bigger energy gap which can make it possible to use GNRs as semiconductor components in ambient conditions.

3.Computational Work Done

3.1.Son, Cohen & Louie (2006)[3]

This group calculated the electronic structure of graphene nanoribbons by using the first principles self consistent psudopotential method with the local (spin) density approximation (LDA). An energy cutoff of 400 Ry is employed and a double-ζ plus polarization basis is used for the localized basis orbitals to deal with the many atoms in GNR unit cell of various widths. They report that the electron density is obtained by integrating the density matrix with Fermi-Dirac distribution. Each GNR cell studied is fully relaxed at an order of less than 16 picoNewtons. A k point sampling of 32 k points is applied along 1D Brillouin zone.


Figure 4.(a) Schematic of a 11 aGNRs (b) Schematic of a 6 zGNR (Empty holes denoting H atoms passivating edge C atoms)

The figure above shows the schematics of armchair and zigzag shaped carbon nanoribbons used in this study. GNRs with armchair shaped edges are classified by number of dimer lines Na whereas GNRs with zigzag edges are classified by the number of zigzag chains across the ribbon width.
The group obtained such a result that the armchair nanoribbons are semiconductors with energy gaps decreasing as a function of increasing ribbon width, wa. The band structures of armchair graphene nanoribbons obtained from first principles calculations can be seen below.


Figure 5.First principles band structures of Na - aGNRs (Na equals 12,13,14 respectively)

3.2.Barone, Hod & Scuseria (2006)[8]

This group obtained carbon nanoribbons by unfolding infinite periodic carbon nanotubes and cutting/extending them into the desired width, eliminating dangling bonds by hydrogenation at the same time. Following figure shows two different types of carbon nanotubes which are unfolded and cut to obtain carbon nanoribbons. As well as their width, crystallographic direction is also an important factor in determining electronic properties of them.


Figure 6.A representative set of semiconducting hydrogen-terminated CNRs, created by “unfolding” and “cutting” different types of Carbonnanotubes (CNTs)

This group reported that they carried their calculations out by Gaussian suite of programs. Bloch functions are expanded in terms of atomic Gaussian type orbital and the Kohn-Sham equations are solved self-consistently in that basis.

Two different functional are used, namely; the PBE realization of the generalized gradient approximation and the screened exchange hybrid density functional, HSE. The latter functional has been proved to reproduce experimental band gaps accurately. Following figure shows the results of two functional. It is obvious that there is a quantitative mismatch between them however they seem to predict similar qualitative band gap oscillations as a function of carbon nanoribbon width.


Figure 7.Dependence of the band gap on the ribbon width for bare (left) and hydrogen-terminated (right) armchair CNRs

As a result of their studies, this group concluded that to be able to obtain a carbon nanoribbon with a band gap comparable to that of Ge (0.67 eV) or InN (0.7 eV) it will be necessary to go to the order of 2-3 nm and even to 1-2 nm if it is desired to be reached the band gap of Si or InP or GaAs.


To sum up, it can be said that the role of the edges is crucial for determining the values and the scaling rule for the band gaps in both armchair and zigzag GNRs.In this review we tried to give information about electronic conductivity of graphene nanoribbons and the effect of ribbon's width to the energy band gap opening. The origin of energy gaps for GNRs can be concluded to be quantum confinement and edge effects. It is clear that first principles calculations agree with the experimental results.

1. Han, M. Y. , Ozyilmaz, B. Et al. (2007). “Energy band gap engineering of graphene nanoribbons.” PHYSICAL REVIEW L 98, 206805
2. Geim, A. K., Novoselov, K. S. (2007). "The rise of graphene." NATURE MATERIALS VOL6 March 2007
3. Son, W-Y, Cohen, M. L. Et al. (2006). "Energy gaps in graphene nanoribbons." PHYSICAL REVIEW L 97,216803
4. Dubois, M., Zanolli, Z. Et al. (2009). "Electronic properties and quantum transport in graphene based nanostructures." EUROPEAN PHYSICAL JOURNAL B 00327-8
5. Şahin, H., Senger, R. T. (2008). "First principles calculations of spin dependent conductance of graphene flakes." PHYSICAL REVIEW B 78, 205423
6. Lee, G. , Cho, K. (2009). "Electronic structures of zigzag graphene nanoribbons with edge hydrogenation and oxidation." PHYSICAL REVIEW B 79, 165440
7. Krompiewski, S. (2009). "Theoretical studies of spin-dependent electronic transport in ferromagnetically contacted graphene flakes." PHYSICAL REVIEW B 80, 075433
8. Barone, V., Hod, O. Et al. (2006). "Electronic structure and stability of semiconducting graphene nanoribbons." NANO LETTERS Vol.6 No.12
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