The effect of supercritical charge impurities in graphene is very similar to the supercritical atomic collapses in QED for Z > 137, but with a much lower critical charge. In this sense graphene can be considered as a natural testing ground for the analysis of quantum field theory vacuum instabilities. We analyze the quantum transition from subcritical to supercritical charge regimes in gapped graphene in a common framework that preserves unitarity for any value of charge impurities. In the supercritical regime it is possible to introduce boundary conditions which control the singular behavior at the impurity. We show that for subcritical charges there are also non-trivial boundary conditions which are similar to those that appear in QED for nuclei in the intermediate regime 118<Z<137. We analyze the behavior of the energy levels associated to the different boundary conditions. In particular, we point out the existence of new bound states in the subcritical regime which include a negative energy bound state in the attractive Coulomb regime. A remarkable property is the continuity of the energy spectral flow under variation of the impurity charge even when jumping across the critical charge transition. We also remark that the energy levels of Hydrogenoid bound states at critical values of charge impurities act as focal points of the spectral flow.
We obtain a novel bound state spectrum of the low energy excitations near the Fermi points of graphene in the presence of a charge impurity. The effects of possible short range interactions induced by the impurity are modelled by suitable boundary conditions. The spectrum in the subcritical region of the effective Coulomb coupling is labelled by a parameter which characterizes the boundary conditions and determines the inequivalent quantizations of the system. In the supercritical region we obtain a renormalization group flow for the effective Coulomb coupling.
We study the problem of Coulomb field-induced charging of the ground state in a system of 2D massive Dirac particles (gapped graphene). As in its 3D QED counterpart, the critical Coulomb coupling is renormalized to higher values, compared to the massless case. We find that in gapped graphene a novel supercritical regime is possible, where the screening charge is comparable to the impurity charge, thus leading to suppression of the Coulomb field at nanometer scales. We corroborate this with numerical solution of the tight-binding problem in the honeycomb lattice.
We study the problem of the electron excitation spectrum in the presence of point-like and regularized Coulomb impurities in gapped graphene. To this end, we use the Dirac model and in the point-like case theory of self-adjoint extensions of symmetric operators. In the point-like case, we construct a family of self-adjoint Hamiltonians describing the excitations for any charge of an impurity. Spectra and (generalized) eigenfunctions for all such Hamiltonians are found. Then, we consider the spectral problem in the case of a regularized Coulomb potential of impurities for a special regularizati…
We investigate theoretically indirect exchange interaction between magnetic impurities mediated by one-dimensional gapped helical states. Such states, containing massive Dirac fermions, may be realized on the edge of a two-dimensional topological insulator when time-reversal symmetry is weakly broken. We find that the indirect exchange interaction consists of Heisenberg, Dzyaloshinsky-Moriya, in-plane and out-of-plane Ising terms. These terms decay exponentially when Fermi level lies inside the bandgap whereas the Dzyaloshinsky-Moriya term has smallest amplitude. Outside the energy gap the mas…
We study the combined effect of a conical topological defect and a Coulomb charge impurity on the dynamics of Dirac fermions in gapped graphene. Beyond a certain strength of the Coulomb charge, quantum instability sets in, which demarcates the boundary between sub and supercritical values of the charge. In the subcritical regime, for certain values of the system parameters, the allowed boundary conditions in gapped graphene cone can be classified in terms of a single real parameter. We show that the observables such as local density of states, scattering phase shifts and the bound state spectr…
We study the physics of Dirac fermions in a gapped graphene monolayer containing two Coulomb impurities. For the case of equal impurity charges, we discuss the ground-state energy using the linear combination of atomic orbitals (LCAO) approach. For opposite charges of the Coulomb centers, an electric dipole potential results at large distances. We provide a nonperturbative analysis of the corresponding low-energy scattering problem.
In this paper, we numerically study the bound electron states induced by long range Coulomb impurity in gapped graphene and the quasi-bound states in supercritical region based on the lattice model. We present a detailed comparison between our numerical simulations and the prediction of the continuum model which is described by the Dirac equation in (2+1)-dimensional Quantum Electrodynamics (QED). We also use the Fano's formalism to investigate the quasi-bound state development and design an accessible experiments to test the decay of the supercritical vacuum in the gapped graphene.
We formulate and solve the perhaps simplest two-body bound state problem for interacting Dirac fermions in two spatial dimensions. A two-body bound state is predicted for gapped graphene monolayers in the presence of weakly repulsive electron-electron interactions and a Coulomb impurity with charge $Ze>0$, where the most interesting case corresponds to $Z=1$. We introduce a variational Chandrasekhar-Dirac spinor wave function and show the existence of at least one bound state. This state leaves clear signatures accessible by scanning tunneling microscopy. One may thereby obtain direct informat…
In a recent paper (Phys. Rev. B {\bf 78}, 075316 (2008)), Sajeev and Moiseyev demonstrated that the bound-to-resonant transitions and lifetimes of autoionizing states in spherical quantum dots can be controlled by varying the confinment strength. In the present paper, we report that such control can in some cases be compromised by the presence of Coulomb impurities. It is demonstrated that a screened Coulomb impurity placed in the vicinity of the dot center can lead to bound-to-resonant transitions and to avoided crossings-like behavior when the screening of the impurity charge is varied. It i…
Spatial distribution of local tunneling conductivity was investigated for deep and shallow impurities on semiconductor surfaces. Non-equilibrium Coulomb interaction and interference effects were taken into account and analyzed theoretically with the help of Keldysh formalism. Two models were investigated: mean field self-consistent approach for shallow impurity state and Hubbard-{I} model for deep impurity state. We have found that not only above the impurity but also at the distances comparable to the lattice period both effects interference between direct and resonant tunneling channels and …
The electronic energy spectrum of graphene electron subjected to a homogeneous magnetic field in the presence of a charged Coulomb impurity is studied analytically within two-dimensional Dirac-Weyl picture by using variational approach. The variational scheme we used is just based on utilizing the exact eigenstates of two-dimensional Dirac fermion in the presence of a uniform magnetic field as a basis for determining analytical energy eigenvalues in the presence of an attractive/repulsive charged Coulomb impurity. This approach allows us to determine under which conditions bound state solution…
We analyze the scattering sector of the Hamiltonians for both gapless and gapped graphene in the presence of a charge impurity using the 2D Dirac equation, which is applicable in the long wavelength limit. We show that for certain range of the system parameters, the combined effect of the short range interactions due to the charge impurity can be modelled using a single real parameter appearing in the boundary conditions. The phase shifts and the scattering matrix depend explicitly on this parameter. We argue that this parameter for graphene can be fixed empirically, through measurements of ob…
Recent experimental measurements for near-field radiative heat transfer between two bodies have been able to approach the gap distance within $2 \; \textrm{nm}$, where the contributions of Coulomb fluctuation and electron's tunneling are comparable. Using the nonequilibrium Green's function method in the $G_{0}W_{0}$ approximation, based on a tight-binding model, we obtain for the energy current a Caroli formula from the Meir-Wingreen formula in the local equilibrium approximation. Also, the Caroli formula is consistent with the evanescent part of the heat transfer from the theory of fluctuati…
Application of a perpendicular electric field induces a band gap in bilayer graphene, and it also creates a "Mexican hat" structure in the dispersion relation. This structure has unusual implications for the hydrogen-like bound state of an electron to a Coulomb impurity. We calculate the ground state energy of this hydrogen-like state as a function of the applied interlayer voltage and the effective fine structure constant. Unlike in the normal hydrogen atom, the resulting wavefunction has many nodes even in the ground state. Further, the electron state undergoes "atomic collapse" into the Dir…
We calculate exactly the vacuum polarization charge density in the field of a subcritical Coulomb impurity, $Z|e|/r$, in graphene. Our analysis is based on the exact electron Green's function, obtained by using the operator method, and leads to results that are exact in the parameter $Zα$, where $α$ is the "fine structure constant" of graphene. Taking into account also electron-electron interactions in the Hartree approximation, we solve the problem self-consistently in the subcritical regime, where the impurity has an effective charge $Z_{eff}$, determined by the localized induced charge. We …
Intervalley charged-impurity scattering processes are examined. It is found that the scattering probability is enhanced due to the Coulomb interaction with the impurity by the Sommerfield factor $F_Z\propto ε^{2\sqrt{1-4g^2}-2}$, where $ε$ is the electron energy and $g$ is the dimensionless constant of the Coulomb interaction.
We theoretically describe the spin excitation spectrum of a two dimensional electron gas embedded in a quantum well with localized magnetic impurities. Compared to the previous work, we introduce equations that allow to consider the interplay between the Coulomb interaction of delocalized electrons and the $sd$ exchange coupling between electrons and magnetic impurities. Strong qualitative changes are found : mixed waves propagate below the single particle continuum, an anticrossing gap is open at a specific wavevector and the kinetic damping due to the electron motion strongly influences the …
We demonstrate that electrostatic interactions between helical electrons at the edge of a quantum spin Hall insulator and a dynamical impurity can induce quasi-elastic backscattering. Modelling the impurity as a two-level system, we show that transitions between counterpropagating Kramers-degenerate electronic states can occur without breaking time-reversal symmetry, provided that the impurity also undergoes a transition. The associated electrical resistance has a weak temperature dependence down to a non-universal temperature scale. Our results extend the range of known backscattering mechani…
The temperature dependence of Coulomb blockade peaks of a one dimensional quantum dot is calculated. The Coulomb interaction is treated microscopically using the Luttinger liquid model. The electron interaction is assumed to be non-homogeneous with a maximum strength near the quantum dot. The conductance peaks show non-analytic power law behaviour induced by the interaction. It is shown that there is a crossover in the power law which is related to the inhomogeneity of the interaction.
There exist experiments indicating that at certain conditions, such as an appropriate substrate, a gap of the order of 10 meV can be opened at the Dirac points of a quasiparticle spectrum of graphene. We demonstrate that the opening of such a gap can result in the appearance of a fingerprint bump of the Seebeck signal when the chemical potential approaches the gap edge. The magnitude of the bump can be up to one order higher than the already large value of the thermopower occurring in graphene. Such a giant effect, accompanied by the nonmonotonous dependence on the chemical potential, is relat…
In the current work we have investigated the influence of the Rashba spin-orbit coupling on spin-current of a single layer gapped graphene. It was shown that the Rashba coupling has a considerable role in generation of the spin-current of vertical spins in mono-layer graphene. The behavior of the spin-current is determined by density of impurities. It was also shown that the spin-current of the system could increase by increasing the Rashba coupling strength and band-gap of the graphene and the sign of the spin-current could be controlled by the direction of the current-driving electric field.
In three-dimensional materials with a Dirac spectrum, weak short-ranged disorder is essentially irrelevant near the Dirac point. This is manifestly not the case for Coulomb disorder, where the long-ranged nature of the potential produced by charged impurities implies large fluctuations of the disorder potential even when impurities are sparse, and these fluctuations are screened by the formation of electron/hole puddles. In this paper I present a theory of such nonlinear screening of Coulomb disorder in three-dimensional Dirac systems, and I derive the typical magnitude of the disorder potenti…
We explore theoretically the magnetoresistvity of three-dimensional Weyl and Dirac semimetals in transversal magnetic fields within two alternative models of disorder: (i) short-range impurities and (ii) charged (Coulomb) impurities. Impurity scattering is treated using the self-consistent Born approximation. We find that an unusual broadening of Landau levels leads to a variety of regimes of the resistivity scaling in the temperature-magnetic field plane. In particular, the magnetoresitance is non-monotonous for the white-noise disorder model. For $H\to 0$ the magnetoresistance for short-rang…
We review the physics of charged impurities in the vicinity of graphene. The long-range nature of Coulomb impurities affects both the nature of the ground state density profile as well as graphene's transport properties. We discuss the screening of a single Coulomb impurity and the ensemble averaged density profile of graphene in the presence of many randomly distributed impurities. Finally, we discuss graphene's transport properties due to scattering off charged impurities both at low and high carrier density.
We present a model of two Anderson impurities coupled to and through a superconducting island. The model parametrizes the strength of the coupling between impurity sites, allowing it to represent a variable distance between the impurities. We systematically explore the effect of the model parameters in the subspaces with total even and odd occupancy, identifying unique features of the charge stability diagrams that distinguish the two parities. For total odd electron parity, we identify a device tuning, in which the splitting between the two lowest energy states is highly insensitive to change…
Energy levels of screened charged impurity in gapped graphene under magnetic field
Department of Physics and Computer Science, Medgar Evers College of City University of New York（纽约市立大学梅加·伊弗斯学院物理与计算机科学系）；Bogolyubov Institute for Theoretical Physics（博戈留博夫理论物理研究所）
We use a self-consistent Hartree-Fock approximation with realistic Coulomb interactions for $π$-band electrons to explore the possibility of broken symmetry states in weakly disordered ABC stacked trilayer graphene. The competition between gapped and gapless broken symmetry states, and normal states is studied by comparing total energies. We find that gapped states are favored and that, unlike the bilayer case, gapless nematic broken symmetry states are not metastable. Among the gapped states the layer antiferromagnetic state is favored over anomalous and spin Hall states.
We propose carbon nanotubes (CNTs) with magnetic impurities as a versatile platform to achieve unconventional Kondo physics, where the CNT bath is gapped by the spin-orbit interaction and surface curvature. While the strong-coupling phase is inaccessible for the special case of half-filled impurities in neutral armchair CNTs, the system in general can undergo quantum phase transitions to the Kondo ground state. The resultant position-specific phase diagrams are investigated upon variation of the CNT radius, chirality, and carrier doping, revealing several striking features, e.g., the existence…
It is shown that a ``vacuum polarization'' induced by Coulomb potential in graphene leads to a strong suppression of electric charges even for undoped case (no charge carriers). A standard linear response theory is therefore not applicable to describe the screening of charge impurities in graphene. In particular, it overestimates essentially the contributions of charge impurities into the resistivity of graphene.
The supercritical instability in a system of two identical charged impurities in gapped graphene described in the continuous limit by the two-dimensional Dirac equation has been studied. The case where the charge of each impurity is subcritical, but their sum exceeds the critical value calculated in the version with a single Coulomb center, is considered. Using the developed variational method, the dependence of the critical distance $R_{\rm cr}$ between the impurities on their total charge is calculated. The $R_{\rm cr}$-value is found to grow as the total impurity charge increases and the qu…
We investigate the processes of electron capture by a Coulomb impurity center residing in a hybrid system consisting of spatially separated two-dimensional layers of electron and Bose-condensed dipolar exciton gases coupled via the Coulomb forces. We calculate the probability of the electron capture accompanied by the emission of a single Bogoliubov excitation (bogolon), similar to regular phonon-mediated scattering in solids. Further, we study the electron capture mediated by the emission of a pair of bogolons in a single capture event and show that these processes not only should be treated …
In our previous work (E. Kogan, Phys. Rev. B {\bf 84}, 115119 (2011)) we calculated RKKY interaction between two magnetic impurities in pristine graphene using the Green's functions (GF) in the coordinate -- imaginary time representation. Now we show that the calculations of the GF in this representation can be simplified by using the Feynman's trick, which allows to easily calculate RKKY interaction in gapped graphene. We also present calculations of the RKKY interaction in gapped or doped graphene using the coordinate -- imaginary frequency representation. Both representations, corresponding…
Three-dimensional topological insulator (TI) nanowires with quantized surface subband spectra are studied as a main component of Majorana bound states (MBS) devices. However, such wires are known to have large concentration $N \sim 10^{19}$ cm$^{-3}$ of Coulomb impurities. It is believed that a MBS device can function only if the amplitude of long-range fluctuations of the random Coulomb potential $Γ$ is smaller than the subband gap $Δ$. Here we calculate $Γ$ for recently experimentally studied large-dielectric-constant (Bi$_{1-x}$Sb$_x$)$_2$Te$_{3}$ wires in a small-dielectric-constant enviro…
In graphene, charged defects break the electron-hole symmetry and can even give rise to exotic collapse states when the defect charge exceeds a critical value which is proportional to the Fermi velocity. In this work, we investigate the electronic properties of twisted bilayer graphene (tBLG) with charged defects using tight-binding calculations. Like monolayer graphene, tBLG exhibits linear bands near the Fermi level but with a dramatically reduced Fermi velocity near the magic angle (approximately 1.1°). This suggests that the critical value of the defect charge in magic-angle tBLG should al…
We study the supercritical instability in gapped graphene with two charged impurities separated by distance R using the two-dimensional Dirac equation for electron quasiparticles. Attention is paid to a situation when charges of impurities are subcritical, whereas their total charge exceeds a critical one. The critical distance R_{cr} in the system of two charged centers is defined as that at which the electron bound state with the lowest energy reaches the boundary of the lower continuum. A variational calculation of the critical distance R_{cr} separating the supercritical (R<R_{cr}) and sub…
The instanton approach to the in-gap fluctuation states is applied to the spectrum of biased bilayer graphene. It is shown that the density of states falls off with energy measured from the band-edge as $ν(ε)\propto \exp(-|ε/ε_t|^{3/2})$, where the characteristic tail energy, $ε_t$, scales with the concentration of impurities, $n_i$, as $n_i^{2/3}$. While the bare energy spectrum is characterized by two energies: the bias-induced gap, $V$, and interlayer tunneling, $t_{\perp}$, the tail, $ε_t$, contains a {\it single} combination $V^{1/3}t_{\perp}^{2/3}$. We show that the above expression for …
The interaction between the Fermi sea of conduction electrons and a non-adiabatic attractive impurity potential can lead to a power-law divergence in the tunneling probability of charge through the impurity. The resulting effect, known as the Fermi edge singularity (FES), constitutes one of the most fundamental many-body phenomena in quantum solid state physics. Here we report the first observation of FES for Dirac Fermions in graphene driven by isolated Coulomb impurities in the conduction channel. In high-mobility graphene devices on hexagonal boron nitride substrates, the FES manifests in a…
We show that the changes in the electronic density of states (DOS) in graphene induced by impurity scattering with short-range potentials are completely different from those caused by the long-range Coulomb potential. The spectral weight of the state that eventually disappears into the valence band (as the strength of scattering increases) does not transform into a resonance state. Therefore no unusual screening effects related to a redistribution of the density of states in the valence band are observed. The states induced by the short-range impurities in graphene, therefore, have distinctive…
We study the Dirac equation for quasiparticles in gapped graphene with two oppositely charged impurities by using the technique of linear combination of atomic orbitals and variational Galerkin--Kantorovich method. We show that for sufficiently large charges of impurities 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. This migration of the electron wave function of supercritical electric dipole is a generalization of the fam…
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 un…
We theoretically consider the carrier density dependence of low-temperature electrical conductivity in high-quality and low-disorder two-dimensional (2D) `metallic' electronic systems such as 2D GaAs electron or hole quantum wells or gated graphene. Taking into account resistive scattering by Coulomb disorder arising from quenched random charged impurities in the environment, we show that the 2D conductivity σ(n) varies as σ~ n^{β(n)} as a function of the 2D carrier density n where the exponent β(n) is a smooth, but non-monotonic, function of density with possible nontrivial extrema. In partic…
We have studied properties of quantum charge fluids interacting with random impurities or heavy polarons using a microscopic Hamiltonian with the full many-body Coulomb interaction with the use of variational many-body formalism. At zero temperature and high enough density the bosonic fluid is superconducting, but when density decreases the Coulomb interaction will be strongly over-screened and impurities or polarons begin to trap charge carriers forming bound quasiparticle like clusters, which we call Coulomb bubbles or clumps. These bubbles are embedded inside the superconductor and form nuc…
The ground-state and the transport properties of graphene subject to the potential of in-plane charged impurities are studied. The screening of the impurity potential is shown to be nonlinear, producing a fractal structure of electron and hole puddles. Statistical properties of this density distribution as well as the charge compressibility of the system are calculated in the leading-log approximation. The conductivity depends logarithmically on $α$, the dimensionless strength of the Coulomb interaction. The theory is asymptotically exact when $α$ is small, which is the case for graphene on a …
We study the distribution of vacuum polarization charge induced by a Coulomb impurity in massive graphene. By analytically computing the polarization function, we show that the charge density is distributed in space in a non-trivial fashion, and on a characteristic length-scale set by the effective Compton wavelength. The density crosses over from a logarithmic behavior below this scale, to a power law variation above it. Our results in the continuum limit are confirmed by explicit diagonalization of the corresponding tight-binding model on a finite-size lattice. Electron-electron interaction …
A dilute distribution of magnetic impurities is assumed to be present in doped graphene. We calculate the interaction energy between two magnetic impurities which are coupled via the indirect-exchange or Ruderman-Kittel-Kasuva-Yosida (RKKY) interaction by the doped conduction electrons. The current model is a half-filled $AB$-lattice structure. Our calculations are based on the retarded lattice Green's function formalism in momentum-energy space which is employed in linear response theory to determine the magnetic susceptibility in coordinate space. Analytic results are obtained for gapped gra…
We have derived algebraic, analytic expressions for the chemical potential without any restriction on temperature for all types of doped, or extrinsic, gapped Dirac cone materials including gapped graphene, silicene, germanene and single-layer transition metal dichalcogenides. As an important intermediate step of our derivations, we have established a reliable piecewise-linear model for cal- culating the density-of-states in molybdenum disulfide, showing good agreement with previously obtained numerical results. For the spin- and valley-resolved band structure, we obtain an additional decrease…
Numerical and closed-form analytic expressions for plasmon dispersion relations and rates of dissipation are first obtained at finite-temperatures for free-standing gapped graphene. These closed-system results are generalized to an open system with Coulomb coupling of graphene electrons to an external electron reservoir. New plasmon modes, as well as new plasmon dissipation channels, are found in this open system, including significant modifications arising from the combined effect of thermal excitation of electrons and an energy bandgap in gapped graphene. Moreover, the characteristics of the…
We study effects of Coulomb interactions between localised states in a potential barrier by measuring resonant-tunneling spectra with a small bias applied along the barrier. In the ohmic regime the conductance of 0.2um--gate lateral GaAs microstructures shows distinct peaks associated with individual localised states. However, when an electric field is applied new states start contributing to the current, which becomes a correlated electron flow through two interacting localised states. Several situations of such correlations have been observed, and the conclusions are confirmed by calculation…
We extend the general formalism discussed in the previous paper [A. B. Culver and N. Andrei, Phys. Rev. B 103, 195106 (2021)] to two models with charge fluctuations: the interacting resonant level model and the Anderson impurity model. In the interacting resonant level model, we find the exact time-evolving wavefunction and calculate the steady state impurity occupancy to leading order in the interaction. In the Anderson impurity model, we find the nonequilibrium steady state for small or large Coulomb repulsion $U$, and we find that the steady state current to leading order in $U$ agrees with…
A study of the formation of excitons as a problem of two Dirac particles confined in two-layer graphene sheets separated by a dielectric when gaps are opened and they interact via a Coulomb potential is presented. We propose to observe Bose-Einstein condensation and superfluidity of quasi-two-dimensional dipole excitons in double layer graphene in the presence of band gaps. The energy spectrum of the collective excitations, the sound spectrum, and the effective exciton mass are functions of the energy gaps, density and interlayer separation. The superfluid density ns and temperature of the Kos…
A study of the formation of excitons as a problem of two Dirac particles in a gapped graphene layer and in two gapped graphene layers separated by a dielectric is presented. In the low energy limit the separation of the center-of-mass and relative motions is obtained. Analytical solutions for the wave function and energy dispersion for both cases when electron and hole interact via a Coulomb potential are found. It is shown that the energy spectrum and the effective exciton mass are functions of the energy gaps as well as interlayer separation in case of two layer gapped graphene.
