Study of excitons in gapped graphene and of the influence of hole geometry on the band gap of graphene antidot lattices
Author
Petersen, René
Term
4. term
Education
Publication year
2009
Pages
62
Abstract
Grafen-antidotgitre er grafenflager perforeret med et regelmæssigt mønster af huller. De kan indeholde excitoner—bundne par af en elektron og et hul. Denne afhandling beregner excitoners bindingsenergier i sådanne gitre ved hjælp af Wannier-modellen. Først, inden for den effektive masse-approksimation, bestemmes skærmningen af elektron–hul-interaktionen ved at løse Poisson-ligningen for stakken luft–grafen–SiO2 og ved at beregne grafenlagets dielektriske funktion med en to-bånds model for grafen med båndgab. Herfra udledes et udtryk for bindingsenergien som funktion af grafenlagets tykkelse, og det viser, at bindingsenergien er næsten uafhængig af tykkelsen. Hvis lagtykkelsen sættes til grafens gitterkonstant a_cc = 1,42 Å, bliver bindingsenergierne omkring 2,6 gange mindre end i en tidligere, enklere model. Dernæst beregnes bindingsenergier med en Wannier-model med lineære bånd, som bedre beskriver grafens båndstruktur nær Dirac-punktet. Denne lineære båndmodel øger bindingsenergien, hvilket betyder, at elektron og hul er mere tæt bundet. I nogle tilfælde divergerer bindingsenergien, sandsynligvis på grund af begrænsninger i den anvendte variationsmetode. Arbejdet undersøger også antidotgitre med huller placeret i rektangulære kontra hexagonale gitterstrukturer. Om der opstår et båndgab afhænger stærkt af geometrien. For rektangulære gitre optræder et markant båndgab kun, når enhedscellens bredde i armchair-retningen opfylder L_y = 3 + 2n (n = 1, 2, ...). I hexagonale gitre er båndgabet altid placeret i Γ-punktet (centrum) i Brillouin-zonen, mens det i rektangulære gitre kan flytte sig rundt i Brillouin-zonen og endda ligge mellem to høj-symmetripunkter.
Graphene antidot lattices are graphene sheets patterned with a regular array of holes. They can host excitons—bound pairs of an electron and a hole. This thesis calculates exciton binding energies in such lattices using the Wannier model. First, within the effective-mass approximation, the screening of the electron–hole interaction is obtained by solving the Poisson equation for the air–graphene–SiO2 stack and by computing the dielectric function of the graphene layer with a two-band model of gapped graphene. From this, an expression for the binding energy as a function of graphene layer thickness is derived, and the result shows the binding energy is almost independent of thickness. If the layer thickness is set to the graphene lattice constant a_cc = 1.42 Å, the binding energies are reduced by a factor of about 2.6 compared with an earlier, simpler model. Next, exciton binding energies are evaluated using a Wannier model with linear bands, which better represent graphene’s band structure near the Dirac point. This linear-band model increases the binding energy, meaning the electron and hole are more tightly bound. In some cases the binding energy diverges, likely reflecting limitations of the variational approximation used. The work also examines antidot lattices with holes arranged on rectangular versus hexagonal grids. Whether a band gap forms depends strongly on structural details. For rectangular lattices, only unit cells whose width along the armchair direction satisfies L_y = 3 + 2n (n = 1, 2, ...) show a sizable gap. In hexagonal lattices the gap is always located at the Γ point (the center) of the Brillouin zone, while in rectangular lattices the gap position shifts within the Brillouin zone and can even lie between two high-symmetry points.
[This abstract was generated with the help of AI]
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