electronic properties of nano-graphene sheets calculated ... · electronic properties of...

Available online at www.sciencedirect.com

ARTICLE IN PRESS

www.elsevier.com/locate/commatsci

Computational Materials Science xxx (2008) xxx–xxx

Electronic properties of nano-graphene sheets calculatedusing quantum chemical DFT

Sangam Banerjee a, Dhananjay Bhattacharyya b,*

a Surface Physics Division, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064, Indiab Biophysics Division and Center for Applied Mathematics and Computational Science, Saha Institute of Nuclear Physics,

1/AF Bidhannagar, Kolkata 700064, India

Abstract

Electronic properties of nano-graphene with hydrogen terminated edges are significantly different from infinite graphene sheets with-out any defined edge. Structure of edges of 2D graphene sheets are either in trans (zig-zag) or cis (arm-chair) forms and these edges areknown to spontaneously reduced by hydrogen to neutralize the valancies of all the carbons at the edges. Recent experiments revealeddifferent electronic properties on these edges as measured by conducting tip atomic force microscope (AFM) and scanning tunnelingmicroscopy (STM). We shall present here some theoretical understanding of these edges using molecular orbital calculations basedon density functional theory with B3LYP functional. We have shown here that electron density around a mono-vacancy in a graphenesheet is high, which further leads to bending of the graphene sheet. This reduced vacancy can be considered equivalent to H3 complexwhich may lead to magnetism. We also find that HOMO–LUMO gap of graphene sheet varies as a function of its size and spatial var-iation of the electron density across the nano-graphene sheet depends on the sheet separation.� 2008 Elsevier B.V. All rights reserved.

PACS: 71.15.Mb; 73.22�f; 81.05.Uw

Keywords: Graphene; DFT calculation; Electronic structure; Vacancy in nano-graphene; Edge effect

1. Introduction

A sheet of carbon atoms arranged in a two-dimensionalhexagonal lattice is known as graphene. These graphenesheets can be transformed into various dimensional carbonmaterials by self-assembly and is thus the (2D) buildingblock for many carbon materials. For example:- 2D graph-ene sheets can be periodically stacked to form 3D graphite,it can be rolled to form 1D nanotubes, it can be wrappedinto 0D fullerenes (buckyballs), nanocones etc., and allthese formations depend on the growth conditions [1–4].Recently, electronic properties of these graphene nano-structures have attracted much attention from the pointof view of basic research and applications [5]. It has been

0927-0256/$ - see front matter � 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.commatsci.2008.01.044

* Corresponding author. Tel.: +91 33 2337 5346; fax: +91 33 2337 4637.E-mail addresses: [email protected] (S. Banerjee), dhanan-

[email protected] (D. Bhattacharyya).

Please cite this article in press as: S. Banerjee, D. Bhattacharyya, C

noted that graphene can act as field effect transistors(FET) [6], its conducting property can be changed undercertain conditions [8,9]. Graphene sheet can also showmagnetic moment [7] and all these can be explained by elec-tronic modification of graphene. It is now well establishedthat properties of most nano-cluster materials are quite dif-ferent from bulk due to confinement effect and this is alsotrue for nano-graphene.

Most of the earlier calculations of graphene were carriedout using tight binding approach utilizing translationalperiodicity of infinite 2D sheet. Due to the recent enhancedinterest on graphene and other nano-materials, many quan-tum chemical analysis of graphene have been done recently,mostly using DFT studies following plane wave pseudo-potential method [10–12] or semi-empirical methods [13].In order to use the intrinsic translational symmetry of boththe method and the molecule, the edges were generallyignored in most of the studies. A few recent studies, how-

omput. Mater. Sci. (2008), doi:10.1016/j.commatsci.2008.01.044

mailto:[email protected]



2 S. Banerjee, D. Bhattacharyya / Computational Materials Science xxx (2008) xxx–xxx

ARTICLE IN PRESS

ever, have pointed out the need of terminal hydrogens insuch calculations [14]. However, those models also havesome inherent translational symmetry on some of the direc-tions and hence a true electronic picture of a nano-metersize sheet is still unknown. We have therefore attemptedto study properties of real finite size graphene sheet witha number of possible defects and edge effects and comparethe results with available experimental findings. In order tocalculate electronic properties of small graphene sheet, wehave adopted the methodology mostly used to measureproperties of small chemical compounds in gas phase, i.e.Hartree–Fock (HF) or density functional theory (DFT)with generalised gradient approximation (GGA). Now weshall try to understand the following phenomena usingmolecular orbital calculations based on density functionaltheory i.e., (1) Observation of high electron density alongthe zig-zag edge and low electron density at the arm-chairedge. (2) Spatial variation of the electron density in nano-graphene as a function of interlayer separation. (3) Varia-tion of HOMO–LUMO gap of graphene sheet as a functionof its size. (4) Density of electron orbitals around a mono-vacant site. Phenomena 1 and 2 has been observed experi-mentally [8,9] while phenomenon 3 and 4 are importantfor device fabrication using nano-graphene.

2. Methodology

Atomic coordinates of all the carbon atoms arranged inan hexagonal lattice having zig-zag and arm-chair edgeswith the edges terminated by hydrogen atoms, were gener-ated by the molecular modeling software MOLDEN [15]

Fig. 1. Structures of three nano-graphene sheets of different dimension along ware located at the corners while the trans edges line up along the sides of themodel.


using standard bond lengths (b(C–C) = 1.421 A and b(C–H) = 1.009 A) and all angles equal to 120�. The edges wereterminated by hydrogens to form C–H bonds for neutraliz-ing valencies of all the carbon atoms. All the electroniccalculations were performed by the free softwareGAMESS-US [16] using density functional theory (DFT)with B3LYP/6-31G(d,p) method [17]. Three graphenesheets were generated with trans (zig-zag) and cis (arm-chair) edges for different sizes:- (i) smallest graphene sheet(a pyrene molecule) with 16 carbon atoms and 10 hydrogenatoms, (ii) intermediate model with 32 carbons and 14hydrogens and (iii) largest model with 73 carbons and 21hydrogens (Fig. 1). Another sheet was generated with allbonds in arm-chair (cis) orientation in one side with 68 car-bons and 22 hydrogens (see Fig. 2). We also generated asheet with a vacancy at the center (Fig. 3), while the valen-cies of the carbons at the vacant site were fulfilled byadding three hydrogens. This molecule has 72 carbonsand 24 hydrogens. We have energy minimized structuresof all these molecules after initial model building by Har-tree–Fock method with RHF/4-21G basis set byGAMESS-US. All the molecular orbitals were visualizedby MOLDEN software.

3. Results and discussion

Graphene is generally infinite two-dimensional sheet,hence quantum chemical calculations using plane wavepseudo-potential and making use of crystalline transla-tional periodicity is quite suitable to understand itselectronic properties. However we are interested in

ith molecular orbitals corresponding to HOMO are shown. The cis edgessheets (a) smallest model (pyrene), (b) intermediate model and (c) largest


Fig. 2. Total partial charges on the carbon atoms along the edges of (a) largest model and (b) the model containing only cis edge on one side. The numberon the edges indicates electron population subtracted from the nuclear charge and the color representation gives indication of the inner carbon atoms alsowhere the positive value/red color represents electron deficiency and the negative value/blue color are representative of excess electron accumulation.Electronic charges were interpolated at finer grid spacing (using matlab) from partial charges of all the atoms of the molecules. (For interpretation of thereferences in color in this figure legend, the reader is referred to the web version of this article.)

S. Banerjee, D. Bhattacharyya / Computational Materials Science xxx (2008) xxx–xxx 3

ARTICLE IN PRESS

understanding structure and electronic properties of theedges or its defects due to vacancy creation in finite sizegraphene sheets of nano-meter dimension. Hence we haveadopted the hybrid-DFT formalism, which is suitable forsmall molecular systems without periodicity. The computa-tional resource requirement in this implementation of DFTis however proportional to n4 where n is the number of elec-trons, as compared to n3 in other DFT codes utilizing planewave pseudo-potential [18]. Hence we had to restrict thesize of the molecules so as to suite our computationalresources.

We have performed geometry optimizations of threeperfect graphene models with both cis and trans edges, of


different size as shown in Fig. 1 using HF method and com-pared their orbitals calculated by DFT procedures. Thethree models of graphene sheets of different sizes have106 electron in 53 doubly filled orbitals for pyrene (smallestmodel), 206 electrons in 103 doubly filled orbitals for theintermediate model and we used 460 paired electrons in230 doubly filled orbitals for the largest model. We used290, 508 and 1200 basis functions for DFT calculation todescribe the electron distribution in the three models con-sisting of trans and cis edges. This clearly suggests thatthorough geometry optimization using DFT is almostimpossible. Comparison of energy Eigen values for thehighest occupied molecular orbitals (HOMO) or valence


Fig. 3. Structure of a nano-graphene with cis and trans edges and areduced mono-vacancy at the center along with the molecular orbital ofthe HOMO is shown.

4 S. Banerjee, D. Bhattacharyya / Computational Materials Science xxx (2008) xxx–xxx

ARTICLE IN PRESS

band (VB) and lowest unoccupied molecular orbitals(LUMO) or conduction band (CB), which provides directestimation of band gap, indicates that band gap of pyrene(the smallest model) to be 0.14880H (4.05 eV), the interme-diate model has band gap of 0.0944 H (2.57 eV) whichreduces to 0.0680 H (1.85 eV) for the largest model. Thisindicates that finite size graphene will have a band gapwhich decreases on increasing the dimension of the sheetto near metallic situation (band gap vanishes) [19,20].

Computer graphics representation of the highest occu-pied molecular orbitals (HOMO) for the smallest, interme-diate and largest models with trans and cis edges are shownin Fig. 1. We can clearly see that the molecular orbitalslocated at pure trans edges are more prominent than theorbitals at the cis edges. A very similar property was notedby AFM [8,9] and STM [21] studies. In order to character-ize total electronic population on all the atoms we have cal-culated the Mulliken charges [22] on each of the atoms andtheir values for carbons at the edges are shown in Fig. 2aand b. In Fig. 2b we show a nano-graphene with arm-chaircarbons at a stretch on one side unlike the three modelsconsidered earlier. In Fig. 2b we also show the partial elec-tronic (Mulliken) charges on the carbon atoms at the edgesto compare it with the orbitals of the largest model shownin Fig. 1. Negative values indicates electron density andpositive values are the hole density indicating they are elec-tron deficient. These absolute negative values are clearlymuch smaller on the carbon atoms at the arm-chair edgesthan at the zig-zag edges.

The graphene sheet with a vacancy at its center, under-goes significant structural alteration from planarity toaccommodate three extra hydrogens at the place of onecarbon atom. Electronic population analysis of the HOMO(Fig. 3) clearly shows that significantly more electroniccrowd is located near the vacant site. Many groups [7,23–25] have recently studied magnetisation of graphene intro-duced by edge defect or cavity creation. It was noted that


irradiation of graphene sheet by proton (hydrogen ion)gives rise to magnetic moment while those irradiated byalpha particle does not show magnetism [23]. Hence it givesan indication that reduction of carbon by hydrogen aroundthe vacancy giving rise to extra electron occupation aroundthe vacancy may be responsible for observed magnetism ingraphite. We can argue with a conjecture that when twohydrogen atoms come close together to form a H2 moleculethe total spin goes to singlet state, whereas when three Hatoms come close to form a complex, then the orbital stateis antisymmetric and the spin state is symmetric to makethe total wave function antisymmetric. Hence, this complexwould presumably give rise to high moment state leading tomagnetism. Thus, one can guess why proton (H) irradiatedsamples only shows magnetism and not the alpha particleirradiated samples.

We have further attempted to understand the role ofinterlayer separation between two graphene sheets in termsof electron accumulation. We considered two graphene lay-ers, each having 54 carbons and 18 hydrogen atoms in twomodels: (1) with 3.4 A separation representative of stacked3D graphite and (2) with 10.0 A separation emulating prac-tically unstacked graphene layers. The lower layer is shiftedin plane so that the carbons of the lower layer are juxta-posed to the centers of the top carbon hexagons. Werequired 1800 basis functions to represent wave functionof all the electrons, which is close to the limiting numberof basis functions in the software adopted. In order tounderstand the enhancement of electrical conductivity asa function of sheet separation, the above two systems wereconsidered. We compared electronic populations ofHOMOs of these two systems. It is found that average pop-ulation of electrons considering only the inner carbons onthe closely separated layers (3.4 A) is 0.0123 (std =0.011), while that on the well separated layers (10 A) is0.0243 (std = 0.0198). Considering the small values of par-tial charges, we have also calculated these by a bettermethod namely electrostatic surface potential fit (ESP fit)by MOLDEN. Average value of the ESP fit charges are�0.00083 (std = 0.036) and �0.06993 (std = 0.006) forthe 3.4 A and 10.0 A layers, respectively. Average valueof charge, even smaller than its standard deviation, possi-bly indicates effectively zero charge density on the interiorcarbon atoms for the closely spaced layers. These resultsfurther indicates that there is a net increase in electron pop-ulation in the inner region of the sheet as a function ofsheet separation. This will lead to increase in the electricalconductivity and this has been observed experimentally[8,9]. This charge distribution on the sheet as a functionof sheet separation arises as a result of localization/confine-ment of electron on transformation from 3D graphite to2D graphene sheet.

4. Conclusion

To summarize, it appears that quantum chemical calcu-lation by using density functional theory by hybrid B3LYP


S. Banerjee, D. Bhattacharyya / Computational Materials Science xxx (2008) xxx–xxx 5

ARTICLE IN PRESS

functional to consider exchange correlation energy giveselectronic properties of nano-graphene as observed experi-mentally. This method is also suitable for characterizingnano-meter size graphene sheet as periodic boundary con-dition or translational periodicity of the molecule is notassumed in this method. In our investigation we observedhigh electron density along the zig-zag edge and low den-sity at the arm-chair edge. We noted increase in the elec-tron density at the central region of the sheet giving riseto spatial variation of the electron density across the sheetin nano-graphene as the interlayer separation is increased.We also observed decrease in the HOMO–LUMO gap ofgraphene sheet as the size increases. Large enhancementof density of electron orbitals around a monovacant sitewas also observed.

References

[1] For review see R. Setton, P. Bernier, S. Lefrant (Eds.), CarbonMolecules and Materials, Taylor and Francis, New York, 2002.

[2] V.N. Popov, Mater. Sci. Eng. R. 43 (2004) 61.[3] S.P. Jordan, V.H. Crespi, Phys. Rev. Lett. 93 (2004) 255504.[4] R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of

Carbon Nanotubes, Imperial College Press., London, 1998.[5] K.S. Novoselov et al., Nature Phys. 2 (2006) 177.[6] K.S. Novoselov et al., Nature (London) 438 (2005) 197;

Y. Zhang et al., Nature (London) 438 (2005) 201.[7] P. Esquinazi, A. Setzer, R. Hohne, C. Semmelhack, Y. Kopelevich,

D. Spemann, T. Butz, B. Kohlstrunk, M. Losche, Phys. Rev. B 66(2002) 024429;P. Esquinazi, D. Spemann, R. Hohne, A. Setzer, K.-H. Han, T. Butz,Phys. Rev. Lett. 91 (2003) 227201.

[8] S. Banerjee, M. Sardar, N. Gayathri, A.K. Tyagi, Baldev Raj, Phys.Rev. B 72 (2005) 075418.


[9] S. Banerjee, M. Sardar, N. Gayathri, A.K. Tyagi, B. Raj, Appl. Phys.Lett. 88 (2006) 062111.

[10] H. Lee, N. Park, Y.-W. Son, S. Han, J. Yu, Chem. Phys. Lett. 398(2004) 207211.

[11] H. Lee, Y.-W. Son, N. Park, S. Han, J. Yu, Phys. Rev. B 72 (2005)174431.

[12] A. Montoya, T.N. Truong, A.F. Sarofim, J. Phys. Chem. A 104(2000) 6108.

[13] H. Sabzyan, M. Babajani, J. Mol. Struct.: Theochem 76 (2005) 155.[14] K. Nakada, M. Fujita, G. Dresselhaus, M.S. Dresselhaus, Phys. Rev.

B 54 (1996) 17954;Frauenheim, S. Suhai, G. Seifert, Phys. Rev. B 58 (1998) 7260.

[15] G. Schaftenaar, J.H. Noordik, J. Comput. Aid. Mol. Des. 14 (2000)123–134.

[16] M.W. Schmidt, K.K. Baldridge, J.A. Boatz, S.T. Elbert, M.S.Gordon, J.J. Jenson, S. Koseki, N. Matsunaga, K.A. Nguyen, S.Su, T.L. Windus, M. Dupuis, J.A. Montgomery Jr., J. Comput.Chem. 14 (1993) 1347–1363.

[17] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785–789;A.D. Becke, J. Chem. Phys. 98 (1993) 5642–5648.

[18] C.J. Cramer, Essentials of Computational Chemistry, John Wiley &Sons Ltd, West Sussesx, UK, 2002.

[19] B. Obradovic, R. Kotlyar, F. Heinz, P. Matagne, T. Rakshit, M.D.Giles, M.A. Stettler, D.E. Nikonov, Appl. Phys. Lett. 88 (2006)142102.

[20] L. Yang, C.-H. Park, Y.-W. Son, M.L. Cohen, S.G. Louic, Cond-mat0706.1589v1 (2007).

[21] Y. Niimi, T. Matsui, H. Kambara, K. Tagami, M. Tsukada, H.Fukuyama, Phys. Rev. B 73 (2006) 085421.

[22] R.J. Mulliken, J. Chem. Phys. 23 (1955) 1833.[23] P.O. Lehtinen, A.S. Foster, Yuchen Ma, A.V. Krasheninnikov, R.M.

Nieminen, Phys. Rev. Lett. 93 (2004) 187202.[24] Yuchen Ma, P.O. Lehtinen, A.S. Foster, R.M. Nieminen, New J.

Phys. 6 (2004) 68.[25] K. Kusakabe, M. Maruyama, Phys. Rev. B 67 (2003) 092406.


electronic properties of nano-graphene sheets calculated ... · electronic properties of...

Documents