Masses and Majorana fermions in graphene

2020

Two-dimensional (2D) materials, composed of single atomic layers, have attracted vast research interest since the breakthrough discovery of graphene. One major benefit of such systems is the simple ability to tune the chemical potential by back-gating, in-principle enabling to vary the Fermi level through the charge neutrality point, thus tuning between electron and hole doping. For 2D Superconductors, this means that one may potentially achieve the strongly-coupled superconducting regime described by Bose Einstein Condensation physics of small bosonic tightly bound electron pairs. Furthermore, it should be possible to access both electron and hole based superconductivity in a single system. However, in most 2D materials, an insulating gap opens up around the charge neutrality point, thus preventing approach to this regime. Graphene is unique in this sense since it is a true semi-metal in which the un-gapped Dirac point is protected by the symmetries. In this work we show that singl...

Physical Review B

2014

Here we present the theoretical clarification of possibility of eight Majorana-like modes (quasi-particles which are self-conjugate) close to the experimentally inaccessible Dirac points of a two-dimensional monolayer Dirac system. The valley-mixing and the spin-degeneracy lifting are the main requirements. These are possible by wedging in the requisite ingredients in the description, viz. the atomically sharp scatterers and the strong spin-orbit coupling (SOC). The latter can possibly be achieved in graphene folding a sheet; the higher curvature of deformations correspond to stronger values of the coupling. In silicene, the buckled structure of the system generates a staggered sub-lattice potential between silicon atoms at A sites and B sites for an applied electric field E z perpendicular to its plane. The stronger SOC in silicene has its origin also in the buckled structure of the system. Tuning of E z , allows for rich behavior varying from a topological insulator (TI)to a normal insulator (NI) with a valley spin-polarized metal (VSPM) at a critical value in between. The VSPM stage is characterized by the valley-spin locking, i.e. the opposite spin polarization at different valleys. We shall see that in this phase, if the inter-valley scattering process and the real spin-flip process in moderation are allowed to take place, we have the right condition for capturing Majoranas in the proximity of a s-wave superconductor.

Electronic properties of materials are commonly described by quasiparticles that behave as nonrelativistic electrons with a finite mass and obey the Schrödinger equation. Here we report a condensed matter system where electron transport is essentially governed by the Dirac equation and charge carriers mimic relativistic particles with zero mass and an effective "speed of light" c * ≈10 6 m/s. Our studies of graphene -a single atomic layer of carbon -have revealed a variety of unusual phenomena characteristic of two-dimensional (2D) Dirac fermions. In particular, we have observed that a) the integer quantum Hall effect in graphene is anomalous in that it occurs at halfinteger filling factors; b) graphene's conductivity never falls below a minimum value corresponding to the conductance quantum e 2 /h, even when carrier concentrations tend to zero; c) the cyclotron mass m c of massless carriers with energy E in graphene is described by equation E =m c c * 2 ; and d) Shubnikov-de Haas oscillations in graphene exhibit a phase shift of π due to Berry's phase.

2010

The fermion-doubling problem can be an obstacle to getting half-a-qubit in two-dimensional fermionic tight-binding models in the form of Majorana zero modes bound to the core of superconducting vortices. We argue that the number of such Majorana zero modes is determined by a Z 2 × Z 2 topological charge for a family of two-dimensional fermionic tight-binding models ranging from noncentrosymmetric materials to graphene. This charge depends on the dimension of the representation (i.e., the number of species of Dirac fermions -where the doubling problem enters) and the parity of the Chern number induced by breaking time-reversal symmetry. We show that in graphene there are as many as 10 order parameters that can be used in groups of 4 to change the topological number from even to odd.

Journal of Experimental and Theoretical Physics, 2010

Phonon exchange induced superconducting pairing of effectively ultrarelativistic electrons in graphene is investigated. The Eliashberg equation obtained for describing pairing in the Cooper channel with allowance for delayed interaction are matrix equations with indices corresponding to the valence and con duction bands. The equations are solved in the high doping limit, in which pairing is effectively a single band process, and in the vicinity of a critical quantum point of underdoped graphene for a value of the coupling constant for which pairing is an essentially multiband process. For such cases, analytic estimates are obtained for the superconducting transition temperature of the system. It is shown that the inclusion of dynamic effects makes it possible to determine the superconducting transition temperature, as well as the critical coupling constant for underdoped graphene, more accurately than in the static approximation of the BCS type. Esti mates of the constants of electron interaction with the scalar optical phonon mode in graphene indicate that an appreciable superconducting transition temperature can be attained under a high chemical doping level of graphene.

From the tight-binding approximation, we solve for the spectrum of energy bands in monolayer graphene. Near the points K and K' where the energy dispersion is 0, we expand the Hamiltonian and reveal a linear Hamiltonian that resembles the Dirac equation for massless fermions where the velocity is approximately one three-hundreths the speed of light. We observe a property analogous to spin in regular spin one-half systems known as pseudospin. Lastly, we identify the unique Landau levels in graphene that characterize graphene's quantized conductance in the quantum Hall effect. This was a term paper for the MIT quantum physics III undergraduate course,

Physical Review B, 2010

We investigate the emergence of extra Dirac points in the electronic structure of a periodically spaced barrier system, i.e., a superlattice, on single-layer graphene, using a Dirac-type Hamiltonian. Using square barriers allows us to find analytic expressions for the occurrence and location of these new Dirac points in k space and for the renormalization of the electron velocity near them in the low-energy range. In the general case of unequal barrier and well widths the new Dirac points move away from the Fermi level and for given heights of the potential barriers there is a minimum and maximum barrier width outside of which the new Dirac points disappear. The effect of these extra Dirac points on the density of states and on the conductivity is investigated.

physica status solidi (RRL) - Rapid Research Letters, 2009

We study the low energy properties of warped monolayer graphene, where the symmetry of the original honeycomb lattice reveals itself. The zero energy solutions are Majorana fermions, whose wavefunction, originating from the corresponding modified Dirac equation is spatially localized. Experimental consequences are discussed.

Physica B: Condensed Matter, 2021

In this paper, we study the massive Dirac equation with the presence of the Morse potential in polar coordinate. The Dirac Hamiltonian is written as two second-order differential equations in terms of two spinor wavefunctions. Since the motion of electrons in graphene is propagated like relativistic fermionic quasi-particles, then one is considered only with pseudospin symmetry for aligned spin and unaligned spin by arbitrary k. Next, we use the confluent Heun's function for calculating the wavefunctions and the eigenvalues. Then, the corresponding energy spectrum obtains in terms of N and k. Afterward, we plot the graphs of the energy spectrum and the wavefunctions in terms of k and r, respectively. Moreover, we investigate the graphene band structure by a linear dispersion relation which creates an energy gap in the Dirac points called gapped graphene. Finally, we plot the graph of the valence and conduction bands in terms of wavevectors.