Theoretical Investigations On The Manipulation Of Electronic Properties In Graphene

Graphene, an atomically thin graphitic sheet, has been widely investigated inphysics and relevant subjects since its experimental acquirement in2004. Thereare two inequivalent band-touching points in the Brillouin zone of thistwo-dimensional honeycomb lattice, called the Dirac points. In the low-energyregion relative to the Dirac points, the electron or hole follows a linear dispersionrelation. Thus, the behavior of the electron in the vicinity of two Dirac points canbe described by the massless Dirac-equation method. Such an electron structure isresponsible for most interesting electronic properties unique to graphene. Forexample, the anomalous half-integer quantum Hall effect, weak antilocalization,Klein tunneling, minimal conductivity and Zitterbewegung effect etc. From theviewpoint of application, the high electron mobility at room temperature and theplane structure make graphene an ideal material for nanoelectronics. Therefore,graphene has also drawn much attention for the research community of electronicmaterial science, in addition to the condensed matter physics.In this thesis we theoretically study the band structures and the electronictransport properties of graphene tuned, respectively, by the lattice edge, staggeredpotential from a substrate, external periodic potential, magnetic field and latticelinear defects. These results can provide some new physical models for theapplication of graphene in nanoelectronics. We analysis and calculate the bandstructures by two kinds of theoretical frameworks, i.e. the tight-bindingapproximation and the Dirac-equation approaches. As for the studies on electronictransport properties, we use the Landauer-Büttiker formula to calculate theconductance of graphene nanoribbons, while we employ the Kubo formula tostudy the quantum transport properties of bulk graphene materials. This thesisconsists of the following four parts.Firstly, by means of the transfer-matrix technique, we present an analyticalsolution of the edge states localized at the lateral zigzag edge of a semi-infinite graphene nanoribbon. The electric field tuning on the energy level, the localizedlength, and the local electron probability distribution of an edge state is thenstudied in detail. The dependence of the edge state on the size of the ribbon, thepresence of impurities, and the structural variation in the lateral edge is discussed.In contrast to the natural boundary condition, we find that the periodic boundarycondition employed in some previous work is inappropriate to describe the edgestates of the graphene nanoribbons of small size. The physical natures of someprevious numerical conclusions about the edge state are clarified. For example, itwas previously expected that any edge state cannot survive while the width of thegraphene nanoribbon becomes smaller than three times of the lattice constant andwhenever such a width increases by triple lattice constants, one more edge state isadded. With the same theoretical framework, we derive the analytical expressionabout the edge Green’s functions which are desirable for the study on theelectronic transport through some graphene nanostructures. In addition, we alsofind that a localized edge states can be eliminated by exerting a gate voltage at thelateral zigzag edge. The impurities in the region near the zigzag edge have theeffect to relax the energy degeneracy of the edge states. Noting that the localizededge states are responsible to the spontaneous magnetic ordering of manygraphene nanostructures, our findings concerning the gate tuning and impurityeffect on the edge states will influence the magnetic properties of graphenenanostructures to some extent.Secondly, we note that the epitaxial growth of a graphene monolayer onsome substrates can open a gap between the conduction and valence bands. Basedon the Kubo formula, we have studied the electron transport properties of agapped graphene in the presence of a strong magnetic field. By solving the Diracequation, we find that the Landau level spectra in two valleys differ from eachother in that the n=0level in the K valley is located at top of the valence band,whereas it is at the bottom of the conduction band in the K’ valley. Thus, in anindividual valley, the symmetry between conduction and valence bands is brokenby the presence of a magnetic field. By using the self-consistent Bornapproximation to treat the long range potential scattering, we formulate thediagonal and the Hall conductivities in terms of the Green’s function. To performthe numerical calculation, we find that a large bandgap can suppress the quantum Hall effect, owing to the enhancement of the bandgap squeezing the spacingbetween the low-lying Landau levels. On the other hand, if the bandgap is notvery large, the odd-integer quantum Hall effect experimentally, observed in thegapless graphene, remains in the gapped one. However, such a result does notindicate the half-integer quantum Hall effect in an individual valley of the gappedgraphene. This is because the heights of the Hall plateaux in either valley can becontinuously tuned by the variation of the bandgap. More interestingly, we findthat the height of the diagonal conductivity peak corresponding to the n=0Landaulevel is independent of the bandgap if the scattering is not very strong. In the weakscattering limit, we demonstrate analytically that such a peak takes a universalvalue e2/(hπ), regardless of the bandgap.Thirdly, we calculate the electronic transport properties of a graphene sheetsubject to a one-dimensional cosinusoidal potential. We first focus on the opticalfield, in which the optical conductivity presents strong anisotropy and can betuned by the strength of the superlattice potential. When this strength is within acertain range the terahertz conductivity can be enhanced by two orders ofmagnitude. Based on such a feature, this kind of graphene superlattice can beconsidered as a promising candidate for optronics application such as sensor forterahertz frequencies. Then, the eigenenergy and the conductivity of the graphenesuperlattice with three Dirac points in the presence of a magnetic field arecalculated. Such a graphene superlattice presents three distinct magnetic minibandstructures as the magnetic field increases. They are, respectively: the triplydegenerate Landau level spectrum, the nondegenerate minibands with finitedispersion and the same Landau level spectrum with the pristine graphene. Theratio of the magnetic length to the period of the potential function is thecharacteristic quantity to determine the electron structure of the superlattice.Corresponding to these distinct electron structures, the diagonal conductivitypresents very strong anisotropy in the weak and moderate magnetic field cases,while the predominant magnetotransport orientation changes from the transverseto the longitudinal direction of the superlattice. More interestingly, in the weakmagnetic field case, the superlattice exhibits half-integer quantum Hall effect, butwith a large jump between the Hall plateaux. Thus, it is different from the one ofthe pristine graphene. At last, we note that an extended linear defect of millimeter scale in anepitaxial layer of graphene is successfully fabricated recently. The topologicalgeometry of the linear defect consists of the periodic repetition of one octagonalplus two pentagonal rings along one direction. We develop a tight-binding theoryto study the electronic transport through the extended linear defect in monolayergraphene. After establishing an analytical expression of the transmissionprobability, we clarify the following issues concerning the valley polarization inthe electronic transport process. We find that the valley polarization is robust inthe total linear dispersion region. More interestingly, the lattice deformationaround the linear defect plays an important role in tuning the incident angle forcomplete transmission. Then we indicate that next nearest neighbor interactiononly causes a small suppression to the valley polarization. Finally, by establishingan appropriate connection condition for the spinor wave function across the lineardefect, we find that the massless Dirac equation is still a valid theoretical model todescribe low-energy electronic properties of the linear defect embedded graphenestructure. To check the validity of the wave-function connection condition, wetake two kinds of linear defect embedded graphene structures as examples tostudy the low-energy electronic states by solving the Dirac equation. First, for alinear defect embedded zigzag-edged graphene nanoribbon, we obtain analyticalresults about the subband dispersion and eigenwave function, which coincide wellwith the numerical results from the tight-binding approach. Then, for atwo-dimensional graphene embedded with an extended linear defect, we get anexact expression about the valley polarized electronic transmission probability,which demonstrates the simple result estimated previously in the zero-energy limit.More interestingly, our analytical result indicates that in such a two-dimensionalgraphene structure a quasi-one-dimensional electronic state occurs along the lineardefect. And the electronic group velocity in this quasi-one-dimensional electronicstate can be readily modulated by applying a strain field around the linear defect.