Dynamic conductivity modified by impurity resonant states in doping three-dimensional Dirac semimetals

The impurity effect is studied in three-dimensional Dirac semimetals in the framework of a T-matrix method to consider the multiple scattering events of Dirac electrons off impurities. It has been found that a strong impurity potential can significantly restructure the energy dispersion and the density of states of Dirac electrons. An impurity-induced resonant state emerges and significantly modifies the pristine optical response. It is shown that the impurity state disturbs the common longitudinal optical conductivity by creating either an optical conductivity peak or double absorption jumps, depending on the relative position of the impurity band and the Fermi level. More importantly, these conductivity features appear in the forbidden region between the Drude and interband transition, completely or partially filling the Pauli block region of optical response. The underlying physics is that the appearance of resonance states as well as the broadening of the bands leads to a more complicated selection rule for the optical transitions, making it possible to excite new electron-hole pairs in the forbidden region. These features in optical conductivity provide valuable information to understand the impurity behaviors in 3D Dirac materials.     

Effect of atomic impurities on the helical surface states of the topological insulator Bi2Te3

The effect of atomic impurities including N, O, Na, Ti and Co on the surface states of the topological insulator (TI) Bi(2)Te(3) is studied using pseudopotential first principles methods. The robustness of the TI surface states is particularly investigated against magnetic and non-magnetic atomic adsorption by calculating the electronic band structure, charge transfer, and magnetic moments. Interestingly, it is found that a non-magnetic nitrogen atom has produced a residual magnetic moment and opens a gap in the surface states whereas Na and O atoms preserve the Dirac-like dispersion. The charge transfer from the adatoms produces an electric dipole field that causes Rashba splitting in the surface bands. For atomic impurities with 3d orbitals (Ti and Co), the TI surface states are destroyed and two spin-resolved resonance peaks are developed near the Fermi level in the DOS.     

A computational investigation of topological insulator Bi2Se3 film

Topological insulators have a bulk band gap like an ordinary insulator and conducting states on their edge or surface which are formed by spin–orbit coupling and protected by time-reversal symmetry. We report theoretical analyses of the electronic properties of three-dimensional topological insulator Bi₂Se₃ film on different energies. We choose five different energies (–123, –75, 0, 180, 350 meV) around the Dirac cone (–113 meV). When energy is close to the Dirac cone, the properties of wave function match the topological insulator’s hallmark perfectly. When energy is far way from the Dirac cone, the hallmark of topological insulator is broken and the helical states disappear. The electronic properties of helical states are dug out from the calculation results. The spin-momentum locking of the helical states are confirmed. A 3-fold symmetry of the helical states in Brillouin zone is also revealed. The penetration depth of the helical states is two quintuple layers which can be identified from layer projection. The charge contribution on each quintuple layer depends on the energy, and has completely different behavior along K and M direction in Brillouin zone. From orbital projection, we can find that the maximum charge contribution of the helical states is p_z orbit and the charge contribution on p_yand p_x orbits have 2-fold symmetry.     

Capacitance spectroscopy of a system of gapless Dirac fermions in a HgTe quantum well

Information on the density of states of two-dimensional Dirac fermions in a 6.6-nm-thick HgTe quantum well that corresponds to a transition from the direct to inverted spectrum is obtained for the first time by means of capacitance measurements. It is found that the density of states of Dirac electrons is a linear function of the Fermi energy at E _F > 30 meV with the corresponding velocity vDF = 8.2 × 10⁵ m/s. At lower energies, this dependence deviates from the linear law, indicating a strong effect of disorder, which is associated with fluctuations of a built-in charge, on the density of states of the studied system near the Dirac point. At negative energies, a sharp increase in the density of states is observed, which is associated with the tail of the density of states of valleys of heavy holes. The described behavior is in agreement with the proposed model, which includes both the features of the real spectrum of Dirac fermions and the effect of the fluctuation potential.     

Chirality and correlations in graphene

Graphene is described at low energy by a massless Dirac equation whose eigenstates have definite chirality. We show that the tendency of Coulomb interactions in lightly doped graphene to favor states with larger net chirality leads to suppressed spin and charge susceptibilities. Our conclusions are based on an evaluation of graphene's exchange and random-phase-approximation correlation energies. The suppression is a consequence of the quasiparticle chirality switch which enhances quasiparticle velocities near the Dirac point.     

Transmission resonance for a Dirac particle in a one-dimensional Hulthén potential

We have solved exactly the two-component Dirac equation in the presence of a spatially one-dimensional Hulthén potential, and presented the Dirac spinors of scattering states in terms of hypergeometric functions. We have derived the reflection and transmission coefficients using the matching condition on the wavefunctions, and investigated the condition for the existence of transmission resonance. Furthermore, we have demonstrated how the transmission resonance depends on the shape of the potential.      

Bound electron states in the monolayer gapped graphene with the short-range impurities

Bound electron states in impure graphene with the massive Dirac spectrum are considered. Short-range perturbations for defect and impurities of the types “local chemical potential” and “local gap” are taken into account.     

The effect of bulk impurities on localized surface states

The effect of the occurrence of bulk impurities in a crystal on the existence and localization of surface states is examined by investigating a simple one-dimensional model. It is found that the bulk impurity has a small effect on the existence conditions for localized states associated with the surface, but the degree of localization of the corresponding charge density can be altered appreciably by the presence of such impurities. The penetration depth associated with localized surface levels can be much greater that than estimated for a pure crystal. The charge density associated with the electrons which occupy localized surface states may vary in a real crystal over a broader region near the surface than would be deduced from models of pure crystals. This will affect physical quantities, such as surface impedance, which are dependent on this charge distribution.     

Diagnostics of many-particle electronic states: Non-stationary currents and residual charge dynamics

We propose the method for identifying many particle electronic states in the system of coupled quantum dots (impurities) with Coulomb correlations. We demonstrate that different electronic states can be distinguished by the complex analysis of localized charge dynamics and non-stationary characteristics. We show that localized charge time evolution strongly depends on the properties of initial state and analyze different time scales in charge kinetics for initially prepared singlet and triplet states. We reveal the conditions for existence of charge trapping effects governed by the selection rules for electron transitions between the states with different occupation numbers.     

Electronic transport in graphene nanoribbons with disorder look at the pseudo-spin polarization: Dirac versus tight-binding model

We have compared results of electronic transport using two different approaches: Dirac vs tight-binding (TB) Hamiltonians to assesses disorder-induced effects in graphene nanoribbons. We apply the proposed Hamiltonians to calculate the density of states, the transmission along the ribbon, and the pseudo-spin polarization (P(E)) in metallic armchair graphene nanoribbons. We clearly show differences between these two approaches in the interference processes, especially in the low-lying energy limit, when the systems are found in the presence of random impurities (disorder). This allows us to find fingerprints associated with each model used. As the disorder increases, more robust electronic transmission (through polarized states in a given sublattice) arises when one is dealing with the Dirac model only. We also find with this model unexpected peaks in the P(E) far from the Dirac point for wider nanoribbons. In the other hand, the model TB show the Dirac limit with disturbances of the hyperboloid subbands for certain potentials of the impurities. In general, our study is indicating that a P(E) spectroscopy (analyzing the line width and intensity) can be used to detect fingerprints of the increase of asymmetry in the scattering processes and the transport limits where hyperboloid subbands are important.     

Effects of bulk charged impurities on the bulk and surface transport in three-dimensional topological insulators

In the three-dimensional topological insulator (TI), the physics of doped semiconductors exists literally side-by-side with the physics of ultrarelativistic Dirac fermions. This unusual pairing creates a novel playground for studying the interplay between disorder and electronic transport. In this mini-review, we focus on the disorder caused by the three-dimensionally distributed charged impurities that are ubiquitous in TIs, and we outline the effects it has on both the bulk and surface transport in TIs. We present self-consistent theories for Coulomb screening both in the bulk and at the surface, discuss the magnitude of the disorder potential in each case, and present results for the conductivity. In the bulk, where the band gap leads to thermally activated transport, we show how disorder leads to a smaller-than-expected activation energy that gives way to variable-range hopping at low temperatures. We confirm this enhanced conductivity with numerical simulations that also allow us to explore different degrees of impurity compensation. For the surface, where the TI has gapless Dirac modes, we present a theory of disorder and screening of deep impurities, and we calculate the corresponding zero-temperature conductivity. We also comment on the growth of the disorder potential in passing from the surface of the TI into the bulk. Finally, we discuss how the presence of a gap at the Dirac point, introduced by some source of time-reversal symmetry breaking, affects the disorder potential at the surface and the mid-gap density of states.     

Highly anisotropic Dirac cones in epitaxial graphene modulated by an island superlattice

We present a new method to engineer the charge carrier mobility and its directional asymmetry in epitaxial graphene by using metal cluster superlattices self-assembled onto the moiré pattern formed by graphene on Ir(111). Angle-resolved photoemission spectroscopy reveals threefold symmetry in the band structure associated with strong renormalization of the electron group velocity close to the Dirac point giving rise to highly anisotropic Dirac cones. We further find that the cluster superlattice also affects the spectral-weight distribution of the carbon bands as well as the electronic gaps between graphene states.     

Reversal of Klein reflection by magnetic barriers in bilayer graphene

Massless Dirac fermions in monolayer graphene exhibit total transmission when normally incident on a scalar potential barrier, a consequence of the Klein paradox originally predicted by O Klein for relativistic electrons obeying the 3 + 1 dimensional Dirac equation. For bilayer graphene, charge carriers are massive Dirac fermions and, due to different chiralities, electron and hole states are not coupled to each other. Therefore, the wavefunction of an incident particle decays inside a barrier as for the non-relativistic Schr?dinger equation. This leads to exponentially small transmission upon normal incidence. We show that, in the presence of magnetic barriers, such massive Dirac fermions can have transmission even at normal incidence. The general consequences of this behavior for multilayer graphene consisting of massless and massive modes are mentioned. We also briefly discuss the effect of a bias voltage on such magnetotransport.     

Relativistic quantum mechanics of fermions

We present a modification of the Dirac equation that allows us to formulate a relativistic quantum mechanics for spin-1/2 fermions in an external electromagnetic field, with a probabilistic interpretation similar to that in nonrelativistic quantum mechanics and based on an indefinite charge density. We find that stationary states cannot be interpreted in this manner, and we replace them by quasistationary states. We also include a general discussion of the difficulties and possible generalizations of this approach.     

Theory of magnetic impurities in superconductors. III

The structure of the bound states in superconductors with magnetic impurities is investigated in detail following earlier work. It is shown that there is exactly one bound state per magnetic impurity with a wave function which decays exponentially on a characteristic scale. However, there is no accumulation of charge around the impurity as the bound state charge distribution is compensated by the contribution from band states apart from Ruderman-Kittel oscillations.     

Shot noise in ballistic graphene

We have investigated shot noise in graphene field effect devices in the temperature range of 4.2-30 K at low frequency (f=600-850 MHz). We find that for our graphene samples with a large width over length ratio W/L, the Fano factor F reaches a maximum F ~ 1/3 at the Dirac point and that it decreases strongly with increasing charge density. For smaller W/L, the Fano factor at Dirac point is significantly lower. Our results are in good agreement with the theory describing that transport at the Dirac point in clean graphene arises from evanescent electronic states.     

Relativistic bound-state equations for fermions with instantaneous interactions

In thispaper three types of relativistic bound-state equations for a fermion pair with instantaneous interaction are studied, viz., the instantaneous Bethe-Salpeter equation, the quasi-potential equation, and the two-particle Dirac equation. General forms for the equations describing bound states with arbitrary spin, parity, and charge parity are derived. For the special case of spinless states bound by interactions with a Coulomb-type potential the properties of the ground-state solutions of the three equations are investigated both analytically and numerically. The coupling-constant spectrum turns out to depend strongly on the spinor structure of the fermion interaction. If the latter is chosen such that the nonrelativistic limits of the equations coincide, an analogous spectrum is found for the instantaneous Bethe-Salpeter and the quasi-potential equations, whereas the two-particle Dirac equation yields qualitatively different results.     

The general relativistic hydrogen atom

The general relativistic Dirac equation is formulated in an arbitrary curved space-time using differential forms. These equations are applied to spherically symmetric systems with arbitrary charge and mass. For the case of a black hole (with event horizon) it is shown that the Dirac Hamiltonian is self-adjoint, has essential spectrum the whole real line and no bound states. Although rigorous results are obtained only for a spherically symmetric system, it is argued that, in the presence of any event horizon there will be no bound states. The case of a naked singularity is investigated with the results that the Dirac Hamiltonian is not self-adjoint. The self-adjoint extensions preserving angular momentum are studied and their spectrum is found to consist of an essential spectrum corresponding to that of a free electron plus eigenvalues in the gap (–mc ², +mc ²). It is shown that, for certain boundary conditions, neutrino bound states exist.Supported in part by the National Science Foundation     

Zitterbewegung in Quantum Mechanics

The possibility that zitterbewegung opens a window to particle substructure in quantum mechanics is explored by constructing a particle model with structural features inherent in the Dirac equation. This paper develops a self-contained dynamical model of the electron as a lightlike particle with helical zitterbewegung and electromagnetic interactions. The model admits periodic solutions with quantized energy, and the correct magnetic moment is generated by charge circulation. It attributes to the electron an electric dipole moment rotating with ultrahigh frequency, and the possibility of observing this directly as a resonance in electron channeling is analyzed in detail. Correspondence with the Dirac equation is discussed. A modification of the Dirac equation is suggested to incorporate the rotating dipole moment.