Tight-binding description of patterned graphene

Semicond. Sci. Technol. 23 (2008) 075026B Gharekhanlou et alFigure 3. Showing a Dirac point in the band structure of graphene by taking first neighbors.Figure 4. Opening of an energy gap at Dirac points due to vacancies.structure of patterned graphene, E(κ)may be expanded aroundany of the Dirac points located at K asE c (κ) = + E c + ¯h2 [m∗ −12 c,x |κ x − K x | 2+ m ∗ c,y −1 |κ y − K y | 2] , (2a)E v (κ) = + E v + ¯h2 [m∗ −12 v,x |κ x − K x | 2+ m ∗ v,y −1 |κ y − K y | 2] , (2b)respectively, for conduction and valence bands. Here, E c,vcorresponds to the shifts in the peaks of conduction and valencebands due to patterning. Hence, the energy gap E g will besimply given byE g = E c + E v . (3)Please note the difference of (2a), (2b) to the bandstructure of unpatterned graphene where the energy dispersionaround the Dirac points is given by the linear expressionE(κ) =±¯hV |κ − K|, with V being the 2D Fermi velocity[22].The energy band structure (2a), (2b) of graphene wouldeventually determine not only the effective mass of electronsand holes, but also the effective velocity of Fermionic Diracparticles which normally propagate in unpatterned graphene ata speed of c/300, with c being the velocity of light in vacuum[1]. Also, as we later observe, there will be a noticeableanisotropy in the effective mass in the patterned graphene.In the tight-binding model applied here, it is assumed thatthe full Hamiltonian of the system may be approximated by theHamiltonian of an isolated atom centered at each lattice point,and only the effects of the first-nearest neighbors are involved.3

Semicond. Sci. Technol. 23 (2008) 075026B Gharekhanlou et alFigure 5. Typical patterning model of graphene with reduced symmetry (first- and second-nearest neighbors removed from the lattice).Hence, the atomic orbitals n (r), which are eigenfunctionsof the single-atom Hamiltonian H at are assumed to be verysmall at distances exceeding the lattice constant. This is whatis normally meant by the tight-binding method. We furtherassume that any corrections to the atomic potential U(r),which are required to obtain the full Hamiltonian H of thesystem, are appreciable only when the atomic orbitals aresmall. The solution to the time-independent single-electronSchrödinger equation φ(r) is then assumed to be a linearcombination of atomic orbitalsφ(r) = ∑ b n n (r).n(4)This leads to a matrix equation for the coefficients b m andBloch energies ε(κ) of the formε(κ) = E m − β m + ∑ R≠0γ m (R) exp(iκ · R)b m + ∑ R≠0α m (R) exp(iκ · R) . (5)Here, E m is the energy of the mth atomic level, and∫β m =− m ∗ (r)U(r)φ(r) d3 r, (6a)α m (R) =∫m ∗ (r)φ(r − R) d3 r, (6b)∫γ m (R) =− m ∗ (r)U(r)φ(r − R) d3 r, (6c)are the corresponding transfer and overlap integrals.Before including the vacancy defects in graphenestructure, we first calculate the band structure of grapheneup to five nearest neighbors using the described tight-bindingapproach. Then we omit the terms related to the absentatoms from the summation of atomic potential terms in theHamiltonian. Here, as is normally done in the standard tightbindingapproximation, α m and β m are neglected in (5), andjust have applied the effect of the last overlap integral. Also,we note that in graphene, each of the two nearest atoms haveatomic orbital n (r) with opposite signs for the overall atomicorbitals as ±p z . The overlapping orbital obtained by theinterference of p z orbitals corresponds to the energy bandstructure ε(κ), and in particular π and π ∗ electronic bands asshowninfigure1.In figure 2, the arrangement of carbon atoms in graphenesheet is illustrated together its unit basis vectors, and the threenearest neighbors surrounding a central atom. As can be seenhere, there are three nearest, six second-nearest, three thirdnearestneighbors, etc. In figure 3 Dirac point is clearly seen(marked by the black circle) in the side view from the bandstructure of graphene. The calculations are done here by takingonly the first-nearest neighbors. As is evident from this figure,π and π ∗ electronic bands are almost completely conical acrossthe Dirac point. We also have calculated the band structureof graphene again this time with taking five nearest neighborsinto account. Since the computational cell size is artificiallyincreased, the band structure squeezes in the reciprocal space.Nevertheless, the symmetry and degeneracy at Dirac pointsare still preserved.In figure 4 the band structure of a patterned graphene isshown, with some atoms periodically taken out of the lattice.As can be easily seen, there is no further degeneracy at thepreviously Dirac points remains. In other words, an energygap is opened in justification of the diagram in figure 1. Asthe number of defects in a cell is increased, by taking thenext set of nearest neighbors, the corresponding energy gapvaries significantly as in table 1. Here, the energy gap reachesa maximum, when two of the second-nearest neighbors aredeleted. For instance, for the structure illustrated in figure 5,the symmetry point group is reduced from the original C 6 vpoint group to its C 3 v sub-group. The new super cellis marked with the dashed hexagon, which now includes1+3× 1+6× 1 2 +6× 1 = 9 atoms in each cell.3In contrast, Zhou et al [18] have recently obtained anenergy gap of 0.26 eV, a value close to that of our structure,by removing the translational symmetry. They have achievedthis symmetry breaking through lattice-mismatched expitaxialgrowth of graphene on SiC, which results in a strainedgraphene lattice. Finally, the dependence of the Fermioneffective mass on the number of defects is also tabulated intable 1 along the principal orthogonal directions. There is a4

Semicond. Sci. Technol. 23 (2008) 075026Table 1. Energy gap and fermion effective mass along the principaldirections versus the number of defects in the graphene structure.Number ofFermion effective massneighboring Gap Energyvacancies location gap (eV) m ∗ c/m 0 m ∗ v/m 00 K 0 – –1 ƔK 0.0081 0.02 0.02012 a M 4.1094 0.9312 1.51452 b ƔK 0.0103 0.0206 0.02073 c ƔK 0 0.0534 0.05314 d ƔK 0.01 0.0206 0.02075 e ƔK 0.01 0.0205 0.02066 f ƔK 0.0104 0.0212 0.0214a Two of the nearest neighbors removed.b One of the first and one of the second-nearest neighborsremoved.c All three nearest neighbors removed.d One of the first and three of the second-nearest neighborsremoved.e One of the first and four of the second-nearest neighborsremoved.f One of the first and five of the second-nearest neighborsremoved.remarkable anisotropy in the effective mass, mainly becauseof the newly reduced point group of the structure.3. ConclusionsWe have presented the electronic band structure of an orderlypatterned graphene mono-layer, in the absence of externalelectric and magnetic fields, under the constraint that thegraphene layer is patterned in such a way that carbon atoms areperiodically replaced by vacancies. This has been shown toeither leave the symmetry point group of the two-dimensionalhoneycomb structure either unchanged or reduced to the oneof its sub-groups. We have demonstrated an adjustable energygap due to the removal of degeneracy at Dirac points, and adependence of Fermion effective mass as well as energy gapto the distance between vacancies.ReferencesB Gharekhanlou et al[1] Novoselov K S, Geim A K, Morozov S V, Jiang D,Katsnelson M I, Grigorieva I V, Dubonos S V andFirsov A A 2005 Nature 438 197[2] Berger C et al 2004 J. Phys. Chem. B 108 19912[3] Berger C et al 2006 Science 312 1191[4] Hass J, Feng R, Li T, Li X, Zong Z, de Heer W A, First P N,Conrad E H, Jeffrey C A and Berger C 2006 Appl. Phys.Lett. 89 143106[5] Son Y-W, Cohen M L and Louie S G 2006 Phys. Rev. Lett.97 216803[6] Ostrovsky P M, Gornyi I V and Mirlin A D 2006 Phys. Rev. B74 235443[7] Khveshchenko D V 2006 Phys. Rev. B 74 161402[8] Peres N M R, Castro Neto A H and Guinea F 2006 Phys. Rev. B73 241403[9] Sadowski M L, Martinez G, Potemski M, Berger C andde Heer W A 2006 Phys. Rev. Lett. 97 266405[10] Gunlycke D, Lawler H M and White C T 2007 Phys. Rev. B75 085418[11] Yao Y, Ye F, Qi X-L, Zhang S-C and Fang Z 2007 Phys. Rev. B75 041401[12] Ohta T, Bostwick A, Seyller T, Horn K and Rotenberg E 2006Science 313 951[13] Pisani L, Chan J A, Montanari B and Harrison N M 2007Phys. Rev. B 75 064418[14] Silvestrov P G and Efetov K B 2007 Phys. Rev. Lett.98 016802[15] Bunch J S, van der Zande A M, Verbridge S S, Frank I W,Tanenbaum D M, Parpia J M, Craighead H G andMcEuen P L 2007 Science 315 490[16] Bostwick A, Ohta T, Seyller T, Horn K and Rotenberg E 2007Nature Phys. 3 36 (Preprint cond-mat/0609660)[17] Novoselov K 2007 Nature Mater. 6 720[18] Zhou S Y, Gweon G-H, Fedorov A V, First P N, de Heer W A,Lee D-H, Guinea F, Castro Neto A H and Lanzara A 2007Nature Mater. 6 770[19] Tinkham M 1964 Group Theory and Quantum Mechanics(New York: McGraw-Hill)[20] Reich S, Maultzsch J, Thomsen C and Ordejón P 2002 Phys.Rev. B 66 035412[21] Pennington G and Goldsman N 2003 Phys. Rev. B 68 045426Pennington G and Goldsman N 2005 Phys. Rev. B 71 205318[22] Hwang E H and Das Sarma S 2007 Phys. Rev. B 75 205418(Preprint cond-mat/0610561)5