Electronic optics in graphene in the semiclassical approximation
arXiv:1807.02056 · doi:10.1016/j.aop.2018.08.004
Abstract
We study above-barrier scattering of Dirac electrons by a smooth electrostatic potential combined with a coordinate-dependent mass in graphene. We assume that the potential and mass are sufficiently smooth, so that we can define a small dimensionless semiclassical parameter . This electronic optics setup naturally leads to focusing and the formation of caustics, which are singularities in the density of trajectories. We construct a semiclassical approximation for the wavefunction in all points, placing particular emphasis on the region near the caustic, where the maximum of the intensity lies. Because of the matrix character of the Dirac equation, this wavefunction contains a nontrivial semiclassical phase, which is absent for a scalar wave equation and which influences the focusing. We carefully discuss the three steps in our semiclassical approach: the adiabatic reduction of the matrix equation to an effective scalar equation, the construction of the wavefunction using the Maslov canonical operator and the application of the uniform approximation to the integral expression for the wavefunction in the vicinity of a caustic. We consider several numerical examples and show that our semiclassical results are in very good agreement with the results of tight-binding calculations. In particular, we show that the semiclassical phase can have a pronounced effect on the position of the focus and its intensity.
103 pages, 11 figures
References in corpus (46)
- The electronic properties of graphene
- Two-Dimensional Gas of Massless Dirac Fermions in Graphene
- Topological Insulators
- Topological insulators and superconductors
- Experimental Observation of Quantum Hall Effect and Berry's Phase in Graphene
- Berry Phase Effects on Electronic Properties
- Chiral tunneling and the Klein paradox in graphene
- Colloquium: Topological Band Theory
- Electronic States of Graphene Nanoribbons
- Unconventional Integer Quantum Hall effect in graphene
- Kwant: a software package for quantum transport
- Quantum interference and Klein tunneling in graphene heterojunctions
- Veselago Lens for Electrons: Focusing and Caustics in Graphene p-n Junctions
- Size, Shape and Low Energy Electronic Structure of Carbon Nanotubes
- Gauge fields in graphene
- Strong suppression of weak (anti)localization in graphene
- Selective transmission of Dirac electrons and ballistic magnetoresistance of \textit{n-p} junctions in graphene
- Evidence of Klein tunneling in graphene p-n junctions
- Detecting Topological Currents in Graphene Superlattices
- Three-Dimensional Topological Insulators
- Adhesion and electronic structure of graphene on hexagonal boron nitride substrates
- Electron optics with ballistic graphene junctions
- Generic Miniband Structure of Graphene on a Hexagonal Substrate
- Klein Backscattering and Fabry-Perot Interference in Graphene Heterojunctions
- Observation of negative refraction of Dirac fermions in graphene
- Topological Berry phase and semiclassical quantization of cyclotron orbits for two dimensional electrons in coupled band models
- Magneto-transport phenomena related to the chiral anomaly in Weyl semimetals
- Caustics due to Negative Refractive Index in Circular Graphene p-n Junctions
- Chiral tunneling in single and bilayer graphene
- Band gaps in incommensurable graphene on hexagonal boron nitride
- Effect of structural relaxation on the electronic structure of graphene on hexagonal boron nitride
- A Two-dimensional Dirac fermion microscope
- Current flow paths in deformed graphene: from quantum transport to classical trajectories in curved space
- Berry phase in graphene: a semiclassical perspective
- Local Energy Gap in Deformed Carbon Nanotubes
- Helicons in Weyl semimetals
- Modeling Klein tunneling and caustics of electron waves in graphene
- Scattering of two-dimensional massless Dirac electrons by a circular potential barrier
- Semiclassical theory of potential scattering for massless Dirac fermions
- Electronic Properties of Two-Dimensional Carbon
- Graphene quantum dot on boron nitride: Dirac cone replica and Hofstadter butterfly
- Emergence of massless Dirac fermions in graphene's Hofstadter butterfly at switches of the quantum Hall phase connectivity
- Diffraction catastrophes and semiclassical quantum mechanics for Veselago lensing in graphene
- Symmetry breaking and (pseudo)spin polarization in Veselago lenses for massless Dirac fermions
- New formulas for Maslov's canonical operator in a neighborhood of focal points and caustics in 2D semiclassical asymptotics
- Catastrophe optics of caustics in single and bilayer graphene: fine structure of caustics
Cited by in corpus (14)
- Nonlinear transport of ballistic Dirac electrons tunneling through a tunable potential barrier in graphene
- Generalized Dirac Oscillators with position-dependent mass
- Generation of hard twisted photons by charged particles in cholesteric liquid crystals
- Tunneling in the Brillouin Zone: Theory of Backscattering in Valley Hall Edge Channels
- Dirac equation in curved spacetime: the role of local Fermi velocity
- Tunneling in an anisotropic cubic Dirac semi-metal
- Semiclassical theory for plasmons in spatially inhomogeneous media
- Semiclassical theory for plasmons in two-dimensional inhomogeneous media
- Quantum Charged Spinning Massless Particles in 2+1 dimensions
- Short wavelength band structure of photons in cholesteric liquid crystals
- Spin-dependent edge states in two-dimensional Dirac materials with a flat band
- Multichannel scattering for the Schrödinger equation on a line with different thresholds at both infinities
- Exploring spatial dispersion in helical wired media: An effective field theory approach
- Bound states of quasiparticles with quartic dispersion in an external potential: WKB approach