By John R. Wallbank
The final decade has witnessed the invention of, and dramatic development in knowing the physics of graphene and comparable two-dimensional fabrics. the improvement of equipment for production and aligning fine quality two-dimensional crystals has facilitated the construction of a brand new iteration of fabrics: the heterostructures of graphene with hexagonal crystals, within which the graphene electrons gather new, qualitatively varied houses. This thesis presents a accomplished theoretical framework within which to appreciate those heterostructures, in response to the tight binding version, perturbation conception, staff idea and the concept that of the moire superlattice (all of that are elucidated). It explains how graphene heterostructures supply new possibilities for tailoring band constitution, resembling growing extra Dirac issues or beginning band gaps and the way they take place themselves in delivery measurements, optical absorption spectra and the fractal Hofstadter spectra. additionally thought of are the heterostructures of bilayer graphene and resonant tunneling in aligned graphene/insulator/graphene devices.
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Additional resources for Electronic Properties of Graphene Heterostructures with Hexagonal Crystals
The potential is strongest near the substrate, and therefore a greater proportion of the integral I Q,Q ⊗ ,g N comes from the region near the substrate, where the graphene 2 pz orbitals are broad and therefore have both rapidly decaying Fourier components and significant overlap with their neighbours. 6 Microscopic Models 29 Fig. 4 Solid lines show the dimensionless integral I , as a function of the effective Bohr radius of the graphene 2 pz orbitals, for various choices of interlayer separation d.
Taniguchi, P. Moon, M. Koshino, P. C. 1 Introduction The fractal spectrum of electron magnetic bands in crystals subjected to a strong magnetic field [1, 2] is one of the most spectacular results of the quantum theory of solids . For electrons on a two-dimensional lattice, it consists in the fracturing of their band structure into multiple gaps and magnetic bands at each value B p = qp π0 /S of the field providing a rational fraction of magnetic flux quanq tum, π0 , per unit cell area, S, of the crystal, [1, 3].
The matrix elements of the resulting perturbation, taken between sublattice Bloch states i and j (i, j = A or B), acting on the lowenergy Dirac spinors of the graphene K + valley, are given by the long wavelength components of 28 2 Monolayer Graphene on a hBN Underlay πHi j = = −2e2 4ψ∂0 dz RN L 2 φ∓K + i (r, z)φ K + j (r, z) (r − R N )2 + (z − d)2 −2e2 ⊗ ⊗ I|K + +g|,|K + +g⊗ |,|g N | ei(g −g+g N )·r ei(g·τ i −g ·τ j ) . 8) N In Eq. 8) R N are positions of nitrogen sites and L 2 is the total area of the graphene sheet; φ K + ,i (r, z) are Bloch wavefunctions of graphene ψ-electrons exactly at the K + point.