Coulomb Impurity Research Articles

Coulomb impurity on a Dice lattice: Atomic collapse and bound states

The modification of the quantum states in a Dice lattice due to a Coulomb impurity are investigated. The energy band structure of a pristine Dice lattice consists of a Dirac cone and a flat band at the Dirac point. We use the tight binding formalism and find that the flat band states transform into a set of discrete bound states whose electron density is localized on a ring around the impurity mainly on two of the three sublattices. The energy is proportional to the strength of the Coulomb impurity. Beyond a critical strength of the Coulomb potential atomic collapse states appear that have some similarity with those found in graphene with the difference that the flat band states contribute with an additional ring-like electron density that is spatially decoupled from the atomic collapse part. At large value of the strength of the Coulomb impurity the flat band bound states anti-cross with the atomic collapse states.

The influences of hydrogen-like impurities and magnetic field on magnetopolaron in asymmetric semi-exponential and parabolic potential quantum wells

In the current research, the impacts of anisotropic parabolic confinement potential (APCP) on strong-coupled impurity bound magnetopolaron (SCIMP) in asymmetrical semi-exponential quantum wells (ASEQWs) containing hydrogen-like impurities (HLIs) at their center under uniform magnetic fields applied along the direction of z axis have been theoretically investigated. Lee-Low-Pine unitary transformation and linear combination operation methods were applied for the derivation of formulations for ground state energy (GSE) and the mean number of LO phonons (MNLOP) of SCIMPs. Also, GSE and MNLOP as functions of magnetic field cyclotron frequency, Coulombic impurity potential (CIP) strength, APCP effective confinement strengths along x and y directions, and two positive parameters of asymmetrical semi-exponential confinement potential (ASECP) were acquired. Numerical results were expressed for a typical CsI. It was found that when considering magnetic field, APCP, CIP, ASECP, GSE and MNLOP were increased.

Coulomb bound states and atomic collapse in tilted Dirac materials

Searching for new states of matter and unusual quasi-particles in Dirac materials attracts a significant interest in condensed matter physics. Here we investigate the bound state formation in tilted Dirac materials in the presence of the one- and two-dimensional Coulomb potentials. Based on the exact solution of the Dirac equation, we obtain the energy spectrum of the bound states and the corresponding wave functions in the one-dimensional case. It is shown that the lower-energy bound states dive into the continuum below the band gap with the increase of the tilted parameter. The atomic collapse appears when the strength of the Coulomb potential exceeds the critical value. We show that the tilt of Dirac cone facilitates lowing the critical Coulomb potential strength where atomic collapse happens. Most notably, it is found that the bound states are absent for an overtilted Dirac dispersion. For the two-dimensional Coulomb potential, we perform the second-order perturbation theory to study the effect of band tilting on the bound state formation with a single point-like Coulomb impurity. The result indicates that the ground state becomes more unstable near the critical point for a stronger tilt parameter.

Influence of charged impurity screening and surface optical phonons on the n=0 Landau level splitting of graphene with polar substrate

The effect of charged Coulombic impurity screening on the n = 0 Landau level splitting of graphene with a polar substrate under a static magnetic field is investigated theoretically considering the electron-phonon Fröhlich interaction from surface optical phonons of the substrate. The screening is improved using a static dielectric function under the random-phase approximation. A method of linear combination operators is adopted to deal with the influence from a magnetic field and a weak electron-phonon coupling theory is used to discuss the effect from the surface optical phonons. It is found that the splitting due to phonons can be broadened by the charged impurity screening and the broadening increases as increasing of the magnetic field. Our results may provide a possible way to enlarge the energy gap of monolayer graphene. • The n = 0 Landau level splitting of graphene with a substrate under a magnetic field affected by Coulombic impurity screening is obtained. • The screening uses a static dielectric function under the random-phase approximation. • The splitting due to phonons can be broadened by the charged impurity screening. • The broadening increases as increasing of the magnetic field.

Optical pulses in carbon nanotubes with strong electron-electron interaction due to impurities

In this paper, we investigate electromagnetic pulse propagation in a medium with impurity carbon nanotubes, which are associated with the acoustic subsystem. The electron spectrum for the carbon system is obtained from the model that takes into account Coulomb impurities. Based on the Maxwell's and the wave equations for the vector of the medium displacements, we obtain the effective equations for the vector potential of the electromagnetic field and the component of the displacement vector of the medium. The dependence of the pulse shape on the various system parameters is analyzed.

Probing Majorana zero modes by measuring transport through an interacting magnetic impurity

Motivated by recent experiments we consider transport across an interacting magnetic impurity coupled to the Majorana zero mode (MZM) observed at the boundary of a topological superconductor (SC). In the presence of a finite tunneling amplitude we observe hybridization of the MZM with the quantum dot, which is manifested by a half-integer zero-bias conductance $G_0=e^2/2h$ measured on the metallic contacts. The low-energy feature in the conductance drops abruptly by crossing the transition line from the topological to the non-topological superconducting regime. Differently from the in-gap Yu-Shiba-Rosinov-like bound states, which are strongly affected by the on-site impurity Coulomb repulsion, we show that the MZM signature in the conductance is robust and persists even at large values of the interaction. Interestingly, the topological regime is characterized by a vanishing Fano factor, $F=0$, induced by the MZM. Combined measurements of the conductance and the shot noise in the experimental set-up presented in the manuscript allow to detect the topological properties of the superconducting wire and to distinguish the low-energy contribution of a MZM from other possible sources of zero-bias anomaly. Despite being interacting the model is exactly solvable, which allows to have an exact characterization of the charge transport properties of the junction.

Disorder effects in topological insulator nanowires

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 $N\ensuremath{\sim}{10}^{19}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$ 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 $\mathrm{\ensuremath{\Gamma}}$ is smaller than the subband gap $\mathrm{\ensuremath{\Delta}}$. Here we calculate $\mathrm{\ensuremath{\Gamma}}$ for recently experimentally studied large-dielectric-constant ${({\mathrm{Bi}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x})}_{2}{\mathrm{Te}}_{3}$ wires in a small-dielectric-constant environment (no superconductor). We show that provided by such a dielectric-constant contrast, the confinement of electric field of impurities within the wire allows more distant impurities to contribute into $\mathrm{\ensuremath{\Gamma}}$, leading to $\mathrm{\ensuremath{\Gamma}}\ensuremath{\sim}3\mathrm{\ensuremath{\Delta}}$. We also calculate a TI wire resistance as a function of the Fermi level and carrier concentration due to scattering on Coulomb and neutral impurities and do not find observable discrete subband-spectrum-related oscillations at $N\ensuremath{\gtrsim}{10}^{18}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$.

Temperature effect of tunable zero Landau Level splitting energy of monolayer graphene on substrates

The temperature effects of the tunable zero-Landau lever (LL) energy are studied using the Lee-Low-Pines (LLP) variational and quantum statistical theory in the presence of a magnetic field B on monolayer graphene of different polar substrates. This zero-LL energy was induced by the surface optical phonon polaron in monolayer graphene on polar substrates. We found that zero-LL splitting energy can be tuned by controlling the polarization strength η of the polar substrate, confinement strength parameter g, magnetic field B , and temperature T . The magnetic field has a negligible impact on the zero-LL splitting energy. And the temperature effect of zero-LL splitting energy of graphene substrates is further enhanced by Coulomb potential . The results obtained in the study provide new information about the fundamental properties of graphene on substrates at different temperatures T . • Temperature effect of tunable zero-LL splitting energy of monolayer graphene(MG) on substrates is studied. • The temperature has significant effect on zero-LL splitting without coulomb impurity for the MG on the substrate. • The magnetic field has a small impact on the zero-LL splitting energy of MG on the substrate. • The coulomb bound potential reinforces the temperature effect of zero-LL splitting energy. • Temperature effect supplies the theoretical interpretations of the experimental measurements.

Modulation Doping: A Strategy for 2D Materials Electronics

It remains a remarkable challenge to develop practical techniques for controllable and nondestructive doping in two-dimensional (2D) materials for their use in electronics and optoelectronics. Here, we propose a modulation doping strategy, wherein the perfect n-/p-type channel layer is achieved by accepting/donating electrons from/to the defects inside an adjacent encapsulation layer. We demonstrate this strategy in the heterostructures of BN/graphene, BN/MoS2, where the previously believed useless deep defects, such as the nitrogen vacancy in BN, can provide free carriers to the graphene and MoS2. The carrier density is further modulated by engineering the surroundings of the encapsulation layer. Moreover, the defects and carriers are naturally separated in space, eliminating the effects of Coulomb impurity scattering and thus allowing high mobility in the 2D limit. This doping strategy provides a highly viable route to tune 2D channel materials without inducing any structural damage, paving the way for high-performance 2D nanoelectronic devices.

Massless Electronic Excitations in Graphene Near Coulomb Impurities

Massless Electronic Excitations in Graphene Near Coulomb Impurities

Atomistic Band-Structure Computation for Investigating Coulomb Dephasing and Impurity Scattering Rates of Electrons in Graphene.

In this paper, by introducing a generalized quantum-kinetic model which is coupled self-consistently with Maxwell and Boltzmann transport equations, we elucidate the significance of using input from first-principles band-structure computations for an accurate description of ultra-fast dephasing and scattering dynamics of electrons in graphene. In particular, we start with the tight-binding model (TBM) for calculating band structures of solid covalent crystals based on localized Wannier orbital functions, where the employed hopping integrals in TBM have been parameterized for various covalent bonds. After that, the general TBM formalism has been applied to graphene to obtain both band structures and wave functions of electrons beyond the regime of effective low-energy theory. As a specific example, these calculated eigenvalues and eigen vectors have been further utilized to compute the Bloch-function form factors and intrinsic Coulomb diagonal-dephasing rates for induced optical coherence of electron-hole pairs in spectral and polarization functions, as well as the energy-relaxation time from extrinsic impurity scattering of electrons for non-equilibrium occupation in band transport.

Disorder effects in topological insulator thin films

Thin films of topological insulators (TI) attract large attention because of expected topological effects from the inter-surface hybridization of Dirac points. However, these effects may be depleted by unexpectedly large energy smearing $\Gamma$ of surface Dirac points by the random potential of abundant Coulomb impurities. We show that in a typical TI film with large dielectric constant $\sim 50$ sandwiched between two low dielectric constant layers, the Rytova-Chaplik-Entin-Keldysh modification of the Coulomb potential of a charge impurity allows a larger number of the film impurities to contribute to $\Gamma$. As a result, $\Gamma$ is large and independent of the TI film thickness $d$ for $d > 5$ nm. In thinner films $\Gamma$ grows with decreasing $d$ due to reduction of screening by the hybridization gap. We study the surface conductivity away from the neutrality point and at the neutrality point. In the latter case, we find the maximum TI film thickness at which the hybridization gap is still able to make a TI film insulating and allow observation of the quantum spin Hall effect, $d_{\max} \sim 7$ nm.

Temperature Dependence of the Ground State of Impurity Bound Magneto-Acoustic Polarons in Monolayer Graphene

The properties of impurity bound magneto-acoustic strong-coupling polarons in monolayer graphene are examined. The energies of the ground state of the impurity bound magneto-acoustic polaron were obtained by using the linear combination operator and the well-known Lee-Low-Pines transformation methods. By employing the quantum statistics theory, the temperature effects on the ground state energy of magneto-acoustic polarons in monolayer graphene were calculated. The results of the numerical model with regards to the temperature, Coulombic impurity potential, Debye cut-off wavenumber and magnetic field strength are discussed, and it was revealed that the tuning of these factors can offer a way of adjusting the ground state energy of the magneto-acoustic polaron.

Effects of an anisotropic parabolic potential and Coulomb's impurity potential on the energy characteristics of asymmetrical semi-exponential CsI quantum wells

Because of its unique optoelectronic properties, people have studied the characteristics of polarons in various quantum well (QW) models. Among them, the asymmetrical semi-exponential QW (ASEQW) is a new model for studying the structure of QWs in recent years. It is of great significance to study the influences of the impurity and anisotropic parabolic confinement potential (APCP) on the crystal’s properties, because some of the impurities, usually regarded as Coulomb’s impurity potential (CIP), will exist in the crystal more or less, and the APCP has flexible adjustment parameters. However, the energy characteristics of the ASEQW under the combined actions of impurities and APCP have not been studied, which is the motivation of this paper. Using the linear combination operation and Lee–Low–Pines unitary transformation methods, we investigate the vibrational frequency and the ground state energy of the strong coupling polaron in an ASEQW with the influences of the CIP at the origin of coordinates and APCP, and make a comparison between our results and previous literature’s. Our numerical results about the energy properties in the ASEQW influenced by the CIP and APCP may have important significances for experimental design and device preparation.

The Coulomb Impurity in 2D Materials with Strong Spin–Orbit Interaction

It is shown that the spin–orbit interaction (SOI) produced by the Coulomb fields of charged impurities provides an efficient mechanism for bound states formation. The mechanism is realized in 2D materials with a sufficiently strong Rashba SOI provided that the impurity locally breaks the structure inversion symmetry in the direction normal to the layer.

Atomic collapse in disordered graphene quantum dots

In this paper, we numerically study a Coulomb impurity problem for interacting Dirac fermions restricted in disordered graphene quantum dots. In the presence of randomly distributed lattice defects and spatial potential fluctuations, the response of the critical coupling constant for atomic collapse is mainly investigated by local density of states calculations within the extended mean-field Hubbard model. We find that both types of disorder cause an amplification of the critical threshold. As a result, up to thirty-four percent increase in the critical coupling constant is reported. This numerical result may explain why the Coulomb impurities remain subcritical in experiments, even if they are supercritical in theory. Our results also point to the possibility that atomic collapse can be observed in defect-rich samples such as Ar$^{+}$ ion bombarded, He$^{+}$ ion irradiated, and hydrogenated graphene.

Effect of disorder on the transverse magnetoresistance of Weyl semimetals

We study the effect of random potential created by different types of impurities on the transverse magnetoresistance of Weyl semimetals. It is shown that the magnetic field and temperature dependence of magnetoresistance is strongly affected by the type of impurity potential. Two limiting cases are analyzed in detail: ($i$) the ultra-quantum limit, when the applied magnetic field is so high that only the zeroth and first Landau levels contribute to the magnetotransport, and ($ii$) the semiclassical situation, for which a large number of Landau levels comes into play. A formal diagrammatic approach allowed us to obtain expressions for the components of the electrical conductivity tensor in both limits. In contrast to the oversimplified case of the $\delta$-correlated disorder, the long-range impurity potential (including that of Coulomb impurities) introduces an additional length scale, which changes the geometry and physics of the problem. It is shown that the magnetoresistance can deviate from the linear behavior as a function of magnetic field for a certain class of impurity potentials.

Influence of temperature on ground state energy of bound polaron in asymmetric Gaussian potential quantum well in presence of electromagnetic field**This work is supported by the National Natural Science Foundation of China under Grant No. 11975011.

By a combination method of Lee–Low–Pines unitary transformation method and Pekar-type variational method, the ground state energy (GSE) of the bound polaron is studied in the asymmetrical Gaussian potential quantum well considering the temperature and electromagnetic field. The impacts of the temperature and asymmetrical Gaussian potential, electromagnetic field and phonon–electron coupling upon the GSE are obtained. The results show that the GSE of the bound polaron not only oscillates as the temperature changes regardless of the electromagnetic field and asymmetrical Gaussian potential and Coulomb impurity potential (CIP) and electron–phonon coupling but also has different rules with the electromagnetic field and asymmetrical Gaussian potential and CIP and electron–phonon coupling at different temperature zones.

Effect of temperature on the first excited state splitting energy of the impurity magneto acoustic polaron in monolayer graphene

The temperature properties of the first excited state (FES) splitting energy (FESSE) of a strongly-coupled impurity magneto acoustic polaron (IMAP) in monolayer graphene is studied by using the linear combination operator method, unitary transformation and quantum statistical theory. The expression of the FESSE versus the temperature, the Coulombic impurity potential strength (CIPS), the Debye cutoff wave number (DCWN) and the magnetitude of the magnetic field strength (MMFS) has been derived. Numerical calculations shown that ① the FES energy curves can be divided into two curves, positive and negative; ② the FESSE is a decreasing function of the CIPS; ③ the FESSE increases with the decrease of temperature, but increases with increasing the MMFS; ④ the smaller the DCWN, the greater the FESSE.

Comparison of bound magneto-polaron in circular, elliptical, and triangular quantum dot qubit

In the present work, we consider an electron which is strongly coupled to the LO-phonon in circular, elliptical, and triangular quantum dots with Coulomb impurity. The eigenenergies and eigenfunctions of the ground and the first-excited states of the electron are obtained under magnetic and electric fields by using the Pekar variational method. We have also obtained the Shannon entropy and the electron probability density. This system can be applied as a two-level qubit. The entropy shows the oscillatory periodic evolution as function of the time due to the form of the confinement. It is seen that entropy has an arrangement periodic behavior for the higher symmetry (circular quantum dot).