Finite difference method for transport properties of massless Dirac fermions

Abstract

We adapt a finite difference method of solution of the two-dimensional massless Dirac equation, developed in the context of lattice gauge theory, to the calculation of electrical conduction in a graphene sheet or on the surface of a topological insulator. The discretized Dirac equation retains a single Dirac point (no "fermion doubling"), avoids intervalley scattering as well as trigonal warping, and preserves the single-valley time reversal symmetry (= symplectic symmetry) at all length scales and energies -- at the expense of a nonlocal finite difference approximation of the differential operator. We demonstrate the symplectic symmetry by calculating the scaling of the conductivity with sample size, obtaining the logarithmic increase due to antilocalization. We also calculate the sample-to-sample conductance fluctuations as well as the shot noise power, and compare with analytical predictions.

11 pages, 12 figures

References in corpus (27)

• Topological Insulators in Three Dimensions
• A topological Dirac insulator in a quantum spin Hall phase : Experimental observation of first strong topological insulator
• Topological invariants of time-reversal-invariant band structures
• Anderson Transitions
• Andreev reflection and Klein tunneling in graphene
• A self-consistent theory for graphene transport
• Weak localisation magnetoresistance and valley symmetry in graphene
• Quantum-limited shot noise in graphene
• Zitterbewegung, chirality, and minimal conductivity in graphene
• Effect of Disorder on Transport in Graphene
• Demonstration of one-parameter scaling at the Dirac point in graphene
• Topological delocalization of two-dimensional massless Dirac fermions
• Random Network Models and Quantum Phase Transitions in Two Dimensions
• Quantum criticality and minimal conductivity in graphene with long-range disorder
• Low energy theory of disordered graphene
• Anomalously large conductance fluctuations in weakly disordered graphene
• Boltzmann transport and residual conductivity in bilayer graphene
• Conductivity and Fano factor in disordered graphene
• Z2 topological term, the global anomaly, and the two-dimensional symplectic symmetry class of Anderson localization
• Models for the integer quantum Hall effect: the network model, the Dirac equation, and a tight-binding Hamiltonian
• Mesoscopic conductance fluctuations in graphene samples
• A proof of the Kramers degeneracy of transmission eigenvalues from antisymmetry of the scattering matrix
• Quantum kinetic equation and universal conductance fluctuations in graphene
• Numerical estimation of the $\beta$-function in 2D systems with spin-orbit coupling
• Quantum Hall Effect of Massless Dirac Fermions in a Vanishing Magnetic Field
• Universal scaling of current fluctuations in disordered graphene
• Calculation of the conductance of a graphene sheet using the Chalker-Coddington network model

Cited by in corpus (51)

• Electronic transport in two dimensional graphene
• The recursive Green's function method for graphene
• Disorder and Electronic Transport in Graphene
• Scalable Tight-Binding Model for Graphene
• Charge transport in graphene with resonant scatterers
• Crossover from quantum to Boltzmann transport in graphene
• Coulomb interaction in graphene: Relaxation rates and transport
• Ballistic transport in disordered graphene
• Revivals of quantum wave-packets in graphene
• A dispersion and norm preserving finite difference scheme with transparent boundary conditions for the Dirac equation in (1+1)D
• Effective mass and tricritical point for lattice fermions localized by a random mass
• I-V curve signatures of nonequilibrium-driven band gap collapse in magnetically ordered zigzag graphene nanoribbon two-terminal devices
• Guiding Dirac fermions in graphene with a carbon nanotube
• Staggered grid leap-frog scheme for the (2+1)D Dirac equation
• Finite-difference method for transport of two-dimensional massless Dirac fermions in a ribbon geometry
• High-performance solution of the transport problem in a graphene armchair structure with a generic potential
• Confinement and Fermion Doubling Problem in Dirac-like Hamiltonians
• Tunable Supercurrent at the Charge Neutrality Point via Strained Graphene Junctions
• Nonlinear transport of ballistic Dirac electrons tunneling through a tunable potential barrier in graphene
• Symmetry-dependent transport behavior of graphene double dots
• Disorder induced Dirac-point physics in epitaxial graphene from temperature-dependent magneto-transport measurements
• Interplay between boundary conditions and Wilson's mass in Dirac-like Hamiltonians
• Tangent fermions: Dirac or Majorana fermions on a lattice without fermion doubling
• Origin of the quasi-universality of the graphene minimal conductivity
• Localization landscape for Dirac fermions
• Generalized eigenproblem without fermion doubling for Dirac fermions on a lattice
• Sinc-based method for an efficient solution in the direct space of quantum wave equations with periodic boundary conditions
• Klein Tunneling in the presence of random impurities
• A spectral method for half-integer spin fields based on spin-weighted spherical harmonics
• Finite difference method for Dirac electrons in circular quantum dots
• Transport study of the wormhole effect in three-dimensional topological insulators
• Topologically protected Casimir effect for lattice fermions
• Gap opening of single-layer graphene under the continuum model
• Low-Energy Conductivity of Single- and Double-Layer Graphene from the Uncertainty Principle
• Full counting statistics in disordered graphene at Dirac point: From ballistics to diffusion
• Massless Dirac fermions on a space-time lattice with a topologically protected Dirac cone
• Bulk effects on topological conduction on the surface of 3-D topological insulators
• Trigonal warping, pseudodiffusive transport, and finite-system version of the Lifshitz transition in magnetoconductance of bilayer-graphene Corbino disks
• Helical Luttinger liquid on a space-time lattice
• Reflectionless Klein tunneling of Dirac fermions: Comparison of split-operator and staggered-lattice discretization of the Dirac equation
• Anderson localization in a two-dimensional random gap model
• Geometry-independent tight-binding method for massless Dirac fermions in two dimensions
• Engineering Dirac electrons emergent on the surface of a topological insulator
• Four-band effective square lattice model for Bernal-stacked bilayer graphene
• Mapping universal conductance fluctuations
• Spin-polarized scanning tunneling microscopy measurement scheme for determining the quantum geometric tensor
• Local Optical Conductivity of Bilayer Graphene with Kink Potential
• Finite-size scaling analysis of the conductivity of Dirac electrons on a surface of disordered topological insulators
• Scaling behavior of disordered lattice fermions in two dimensions
• Coherent-scatterer enhancement and Klein-tunneling suppression by potential barriers in gapped graphene with chirality-time-reversal symmetry
• Effect of Uncorrelated On-site Scalar Potential and Mass Disorder on Transport of Two-Dimensional Dirac Fermions