C. A. Downing

Zero-energy states in graphene quantum dots and rings

School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom(Dated: May 5, 2011)We present exact analytical zero-energy solutions for a class of smooth decaying potentials, show-ing that the full confinement of charge carriers in electrostatic potentials in graphene quantum dotsand rings is indeed possible without recourse to magnetic fields. These exact solutions allow usto draw conclusions on the general requirements for the potential to support fully confined states,including a critical value of the potential strength and spatial extent.

Searching for confined modes in graphene channels: The variable phase method

Using the variable phase method, we reformulate the Dirac equation governing the charge carriers in graphene into a nonlinear first-order differential equation from which we can treat both confined-state problems in electron waveguides and above-barrier scattering problems for arbitrary-shaped potential barriers and wells, decaying at large distances. We show that this method agrees with a known analytic result for a hyperbolic secant potential and go on to investigate the nature of more experimentally realizable electron waveguides, showing that, when the Fermi energy is set at the Dirac point, truly confined states are supported in pristine graphene. In contrast to exponentially-decaying potentials, we discover that the threshold potential strength at which the first confined state appears is vanishingly small for potentials decaying at large distances as a power law, but nonetheless further confined states are formed when the strength and spread of the potential reach a certain threshold.

One-dimensional Coulomb problem in Dirac materials

We investigate the one-dimensional Coulomb potential with application to a class of quasirelativistic systems, so-called Dirac-Weyl materials, described by matrix Hamiltonians. We obtain the exact solution of the shifted and truncated Coulomb problems, with the wave functions expressed in terms of special functions (namely, Whittaker functions), while the energy spectrum must be determined via solutions to transcendental equations. Most notably, there are critical band gaps below which certain low-lying quantum states are missing in a manifestation of atomic collapse.

Localization of massless Dirac particles via spatial modulations of the Fermi velocity

The electrons found in Dirac materials are notorious for being difficult to manipulate due to the Klein phenomenon and absence of backscattering. Here we investigate how spatial modulations of the Fermi velocity in two-dimensional Dirac materials can give rise to localization effects, with either full (zero-dimensional) confinement or partial (one-dimensional) confinement possible depending on the geometry of the velocity modulation. We present several exactly solvable models illustrating the nature of the bound states which arise, revealing how the gradient of the Fermi velocity is crucial for determining fundamental properties of the bound states such as the zero-point energy. We discuss the implications for guiding electronic waves in few-mode waveguides formed by Fermi velocity modulation.

Magnetic quantum dots and rings in two dimensions

We consider the motion of electrons confined to a two dimensional plane with an externally applied perpendicular inhomogeneous magnetic field, both with and without a Coulomb potential. We find that as long as the magnetic field is slowly-decaying, bound states in magnetic quantum dots are indeed possible. Several example cases of such magnetic quantum dots are considered in which one can find the eigenvalues and eigenfunctions in closed form, including two hitherto unknown quasi-exactly solvable models treated with confluent and biconfluent Heun polynomials. It is shown how a modulation of the strength of the magnetic field can exclude magnetic vortex-like states, rotating with a certain angular momenta and possessing a definite spin orientation, from forming. This indicates one may induce localization-delocalization transitions and suggests a mechanism for spin-separation.

Massless Dirac fermions in two dimensions: Confinement in nonuniform magnetic fields

We acknowledge financial support from the CNRS through the PICS program (Contract No. 6384 APAG) and from the ANR under Grant No. ANR-14-CE26-0005 Q-MetaMat, as well as support from the EU H2020 RISE project CoExAN (Grant No. H2020-644076), EU FP7 ITN NOTEDEV (Grant No. FP7-607521), and the FP7 IRSES projects CANTOR (Grant No. FP7-612285), QOCaN (Grant No. FP7-316432), and InterNoM (Grant No. FP7-612624).

Optimal traps in graphene

We transform the two-dimensional Dirac-Weyl equation, which governs the charge carriers in graphene, into a nonlinear first-order differential equation for scattering phase shift, using the so-called variable-phase method. This allows us to utilize the Levinson theorem, relating scattering phase shifts of a slow particle to its bound states, to find zero-energy bound states created electrostatically in realistic structures. These confined states are formed at critical potential strengths, which leads us to posit the use of “optimal traps” to combat the chiral tunneling found in graphene: this could be explored experimentally with an artificial network of point charges held above the graphene layer. We also discuss scattering on these states and find that the s states create a dominant peak in the scattering cross section as the energy tends towards the Dirac point energy, suggesting a dominant contribution to the resistivity. DOI: 10.1103/PhysRevB.92.165401

Bielectron vortices in two-dimensional Dirac semimetals

Searching for new states of matter and unusual quasi-particles in emerging materials and especially low-dimensional systems is one of the major trends in contemporary condensed matter physics. Dirac materials, which host quasi-particles which are described by ultrarelativistic Dirac-like equations, are of a significant current interest from both a fundamental and applied physics perspective. Here we show that a pair of two-dimensional massless Dirac–Weyl fermions can form a bound state independently of the sign of the inter-particle interaction potential, as long as this potential decays at large distances faster than Kepler’s inverse distance law. This leads to the emergence of a new type of energetically favorable quasiparticle: bielectron vortices, which are double-charged and reside at zero-energy. Their bosonic nature allows for condensation and may give rise to Majorana physics without invoking a superconductor. These novel quasi-particles arguably explain a range of poorly understood experiments in gated graphene structures at low doping.Two-dimensional Dirac semimetals are known to host fermionic excitations which can mimic physics usually found in ultrarelativistic quantum mechanics. Here, the authors unveil the existence of another type of quasiparticle, bielectron vortices, which are bosonic and may give rise to new types of condensates.

A class of exactly solvable models for the Schrödinger equation

We present a class of confining potentials which allow one to reduce the one-dimensional Schrödinger equation to a named equation of mathematical physics, namely either Bessel’s or Whittaker’s differential equation. In all cases, we provide closed form expressions for both the symmetric and antisymmetric wavefunction solutions, each along with an associated transcendental equation for allowed eigenvalues. The class of potentials considered contains an example of both cusp-like single wells and a double-well.

Nanohelices as superlattices: Bloch oscillations and electric dipole transitions

Subjecting a nanohelix to a transverse electric field gives rise to superlattice behavior with tunable electronic properties. We theoretically investigate such a system and find Bloch oscillations and negative differential conductance when a longitudinal electric field (along the nanohelix axis) is also applied. Furthermore, we study dipole transitions across the transverse-electric-field-induced energy gap, which can be tuned to the eulogized terahertz frequency range by experimentally attainable external fields. We also reveal a photogalvanic effect by shining circularly polarized light onto our helical quantum wire.