Two-dimensional Dirac Systems Research Articles

Dynamical current–current correlation in two-dimensional parabolic Dirac systems

We theoretically investigate the current–current correlation of the two-dimensional (2D) parabolic Dirac system in hexogonal lattice. The analytical expressions of the random phase approximation (RPA) susceptibility, Ruderman–Kittel–Kasuya–Yosida (RKKY) Hamiltonian, and the diamagnetic orbital susceptibility in noninteracting case base on the density–density or current–current correlation function are derived and quantitatively analyzed. In noninteracting case, the dynamical polarization within RPA, and spin transverse susceptibility as well as the RKKY interaction (when close to the half-filling) are related to the current–current response in the 2D parabolic Dirac systems. Both the cases of anisotropic dispersion and isotropic dispersion are discussed.

Microscopic Linear Response Theory of Spin Relaxation and Relativistic Transport Phenomena in Graphene

We present a unified theoretical framework for the study of spin dynamics and relativistic transport phenomena in disordered two-dimensional Dirac systems with pseudospin-spin coupling. The formalism is applied to the paradigmatic case of graphene with uniform Bychkov-Rashba interaction and shown to capture spin relaxation processes and associated charge-to-spin interconversion phenomena in response to generic external perturbations, including spin density fluctuations and electric fields. A controlled diagrammatic evaluation of the generalized spin susceptibility in the diffusive regime of weak spin-orbit interaction allows us to show that the spin and momentum lifetimes satisfy the standard Dyakonov-Perel relation for both weak (Gaussian) and resonant (unitary) nonmagnetic disorder. Finally, we demonstrate that the spin relaxation rate can be derived in the zero-frequency limit by exploiting the SU(2) covariant conservation laws for the spin observables. Our results set the stage for a fully quantum-mechanical description of spin relaxation in both pristine graphene samples with weak spin-orbit fields and in graphene heterostructures with enhanced spin-orbital effects currently attracting much attention.

Electron states in the field of charged impurities in two-dimensional Dirac systems (Review Article)

We review the theoretical and experimental results connected with the electron states in two-dimensional Dirac systems paying a special attention to the atomic collapse in graphene. Two-electron bound states of a Coulomb impurity are considered too. A rather subtle role of a magnetic field in the supercritical charge problem in graphene is discussed. The electron states in the field of two equally charged impurities are studied and the conditions for supercritical instability to occur are determined. It is shown that the supercriticality of novel type is realized in gapped graphene with two unlikely charged impurities. For sufficiently large charges of impurities, it is found that the wave function of the occupied electron bound state of the highest energy changes its localization from the negatively charged impurity to the positively charged one as the distance between the impurities increases. The specifics of the atomic collapse in bilayer graphene is considered and it is shown that the atomic collapse in this material is not related to the phenomenon of the fall-to-center.

Modifications of the Lifshitz-Kosevich formula in two-dimensional Dirac systems

Starting from the Luttinger-Ward functional we derive an expression for the oscillatory part of the grand potential of a two dimensional Dirac system in a magnetic field. We perform the computation for the clean and the disordered system, and we study the effect of electron-electron interactions on the oscillations. Unlike in the two dimensional electron gas (2DEG), a finite temperature and impurity scattering also affects the oscillation frequency. Furthermore, we find that in graphene, compared to the 2DEG, additional interaction induced damping effects occur: to two-loop order electron-electron interactions do lead to an additional damping factor in the amplitude of the Lifshitz-Kosevich-formula.

Robustness of persistent currents in two-dimensional Dirac systems with disorder

We consider two-dimensional (2D) Dirac quantum ring systems formed by the infinite mass constraint. When an Aharonov-Bohm magnetic flux is present, e.g., through the center of the ring domain, persistent currents, i.e., permanent currents without dissipation, can arise. In real materials, impurities and defects are inevitable, raising the issue of robustness of the persistent currents. Using localized random potential to simulate the disorders, we investigate how the ensemble averaged current magnitude varies with the disorder density. For comparison, we study the nonrelativistic quantum counterpart by analyzing the solutions of the Schr\"{o}dinger equation under the same geometrical and disorder settings. We find that, for the Dirac ring system, as the disorder density is systematically increased, the average current decreases slowly initially and then plateaus at a finite nonzero value, indicating remarkable robustness of the persistent currents. The physical mechanism responsible for the robustness is the emergence of a class of boundary states - whispering gallery modes. In contrast, in the Schr\"{o}dinger ring system, such boundary states cannot form and the currents diminish rapidly to zero with increase in the disorder density. We develop a physical theory based on a quasi one-dimensional approximation to understand the striking contrast in the behaviors of the persistent currents in the Dirac and Schr\"{o}dinger rings. Our 2D Dirac ring systems with disorders can be experimentally realized, e.g., on the surface of a topological insulator with natural or deliberately added impurities from the fabrication process.

Detection of topological states in two-dimensional Dirac systems by the dynamic spin susceptibility

We discuss dynamic spin susceptibility (DSS) in two-dimensional (2D) Dirac electrons with spin-orbit interactions to characterize topological insulators. The imaginary part of the DSS appears as an absorption rate in response to a transverse ac magnetic field, just as in an electron spin resonance experiment for localized spin systems. We found that when the system is in a static magnetic field, the topological state can be identified by an anomalous resonant peak of the imaginary part of the DSS as a function of the frequency of the transverse magnetic field $\omega$. In the absence of a static magnetic field, the imaginary part of the DSS becomes a continuous function of $\omega$ with a threshold frequency $\omega_{\rm c}$. In this case, the topological and the trivial phases can also be distinguished by the values of $\omega_{\rm c}$ and by the line shapes. Thus the DSS is an experimentally observable physical quantity to characterize a topological insulator directly from bulk properties, without observing a topological transition.

Scanning tunneling spectroscopy study of the Dirac spectrum of germanene

The temperature dependence of the density of states of germanene, synthesized on Ge/Pt crystals, has been investigated with scanning tunneling spectroscopy. After correction for thermal broadening, a virtually perfect V-shaped density of states, which is a hallmark of a two-dimensional Dirac system, has been found. In an attempt to directly measure the energy dispersion relation via quasiparticle interference we have recorded spatial maps of the differential conductivity near the edges and defects of germanene. Unfortunately, we did not find any sign of Friedel oscillations. The absence of these Friedel oscillations hints to the occurrence of Klein tunneling.

Dirac cones beyond the honeycomb lattice: A symmetry-based approach

Recently, several new materials exhibiting massless Dirac fermions have been proposed. However, many of these do not have the typical graphene honeycomb lattice, which is often associated with Dirac cones. Here, we present a classification of these different two-dimensional Dirac systems based on the space groups, and discuss our findings within the context of a minimal two-band model. In particular, we show that the emergence of massless Dirac fermions can be attributed to the mirror symmetries of the materials. Moreover, we uncover several novel Dirac systems that have up to twelve inequivalent Dirac cones, and show that these can be realized in (twisted) bilayers. Hereby, we obtain systems with an emergent SU(2N) valley symmetry with N=1,2,4,6,8,12. Our results pave the way to engineer different Dirac systems, besides providing a simple and unified description of materials ranging from square- and $\beta$-graphynes, to Pmmn-Boron, TiB$_2$, phosphorene, and anisotropic graphene.

Dynamical class of a two-dimensional plasmonic Dirac system.

A current goal in plasmonic science and technology is to figure out how to manage the relaxational dynamics of surface plasmons in graphene since its damping constitutes a hinder for the realization of graphene-based plasmonic devices. In this sense we believe it might be of interest to enlarge the knowledge on the dynamical class of two-dimensional plasmonic Dirac systems. According to the recurrence relations method, different systems are said to be dynamically equivalent if they have identical relaxation functions at all times, and such commonality may lead to deep connections between seemingly unrelated physical systems. We employ the recurrence relations approach to obtain relaxation and memory functions of density fluctuations and show that a two-dimensional plasmonic Dirac system at long wavelength and zero temperature belongs to the same dynamical class of standard two-dimensional electron gas and classical harmonic oscillator chain with an impurity mass.

Two-dimensional Dirac signature of germanene

The structural and electronic properties of germanene coated Ge2Pt clusters have been determined by scanning tunneling microscopy and spectroscopy at room temperature. The interior of the germanene sheet exhibits a buckled honeycomb structure with a lattice constant of 4.3 Å and a buckling of 0.2 Å. The zigzag edges of germanene are reconstructed and display a 4× periodicity. The differential conductivity of the interior of the germanene sheet has a V-shape, which is reminiscent of the density of states of a two-dimensional Dirac system. The minimum of the differential conductivity is located close to the Fermi level and has a non-zero value, which we ascribe to the metallic character of the underlying Ge2Pt substrate. Near the reconstructed germanene zigzag edges the shape of the differential conductivity changes from a V-shape to a more parabolic-like shape, revealing that the reconstructed germanene zigzag edges do not exhibit a pronounced metallic edge state.

Semiclassical approach to bound states of a pointlike impurity in a two-dimensional Dirac system

The goal of this paper is to provide an intuitive and useful tool for analyzing the impurity-bound-state problem. We develop a semiclassical approach and apply it to an impurity in two-dimensional systems with parabolic or Dirac-like bands. Our method consists of reducing a higher-dimensional problem into a sum of one-dimensional ones using the two-dimensional Green's functions as a guide. We then analyze the one-dimensional effective systems in the spirit of the wave-function-matching method as in the standard one-dimensional quantum model. We demonstrate our method on two-dimensional models with parabolic and Dirac-like dispersion, with the later specifically relevant to topological insulators.

NONCOMMUTATIVE GRAPHENE

We consider a noncommutative description of graphene. This description consists of a Dirac equation for massless Dirac fermions plus noncommutative corrections, which are treated in the presence of an external magnetic field. We argue that, being a two-dimensional Dirac system, graphene is particularly interesting to test noncommutativity. We find that momentum noncommutativity affects the energy levels of graphene and we obtain a bound for the momentum noncommutative parameter.

Spin-orbit scattering in quantum diffusion of massive Dirac fermions

We theoretically study the effects of spin-orbit scattering on weak (anti)localization in two-dimensional massive Dirac systems. We clarify that weak antilocalization and localization of a single massive Dirac cone come from the diffusion of a singlet Cooperon in the massless limit and one of triplet Cooperons in the large-mass limit, respectively. Spin-orbit scattering behaves like random magnetic scattering to the triplet Cooperon, and suppresses the weak localization in the large-mass regime, different from in conventional systems where spin-orbit scattering leads to a crossover from weak localization to antilocalization. This behavior suggests an experiment to detect the weak localization of bulk subbands in topological insulator thin films, in which an enhancement of ``weak antilocalization'' is expected after doping heavy-element impurities. Finally, we compare the conventional electron and Dirac fermion systems in the quantum diffusion transport under ordinary, spin-orbit, and magnetic scattering.

Conserved charges of order-parameter textures in Dirac systems

A simple expression for the induced fermion current in the presence of a texture in mass order parameters in two-dimensional condensed-matter Dirac systems is derived using the representation theory of Clifford algebras. In particular, it is shown that every texture in three mutually anticommuting order parameters, in graphene for example, implies an induced density of a properly defined conserved charge. The sufficient condition for the general charge to be the familiar electrical charge is that the remaining two anticommuting order parameters allowed by the particle-hole symmetry are the two phase components of some superconducting order. This allows eight different types of electrically charged texture in graphene or in the $\ensuremath{\pi}$-flux Hamiltonian on the square lattice. The generalized charge of mass textures on the surfaces of thin films of topological insulators or in spinless Dirac fermions hopping on the honeycomb lattice is also discussed.

Imaging Universal Conductance Fluctuations in Graphene

We study conductance fluctuations (CF) and the sensitivity of the conductance to the motion of a single scatterer in two-dimensional massless Dirac systems. Our extensive numerical study finds limits to the predicted universal value of CF. We find that CF are suppressed for ballistic systems near the Dirac point and approach the universal value at sufficiently strong disorder. The conductance of massless Dirac fermions is sensitive to the motion of a single scatterer. CF of order e(2)/h result from the motion of a single impurity by a distance comparable to the Fermi wavelength. This result applies to graphene systems with a broad range of impurity strength and concentration while the dependence on the Fermi wavelength can be explored via gate voltages. Our prediction can be tested by comparing graphene samples with varying amounts of disorder and can be used to understand interference effects in mesoscopic graphene devices.

Asymptotic behaviour of the solution of the two-dimensional Dirac system with rapidly oscillating coefficients

An asymptotic expansion is constructed and substantiated for the solution of the boundary-value problem for the two-dimensional elliptic system of Dirac equations with rapidly oscillating coefficients, which holds uniformly with respect to the complex variable and the two real variables.

Basic functions associated with a two-dimensional dirac system

Basic functions associated with a two-dimensional dirac system