Jellium Approximation Research Articles

Interaction potential between coplanar uniformly charged disk and ring

We consider a system made up of a uniformly charged disk and a uniformly charged ring that are coplanar with each other. The disk and ring have arbitrary radii and contain arbitrary amounts of net charge that is spread uniformly over area and length, respectively. This study is seen as the first necessary step towards a classical model treatment of a two-dimensional system of electrons in a weak perpendicular magnetic field. In this scenario, the uniformly charged disk represents the neutralizing background in a jellium approximation while a uniformly charged ring is viewed as a classical approximation to the cyclotron motion of the electron in presence of a uniform perpendicular magnetic field. In this work, we obtain an exact integral expression for the interaction potential energy of the system as a function of the separation distance between the centers of the two bodies and discuss various emerging features of this model.

Theory of magnetic 3d transition metal dopants in gallium nitride

Using first-principles density functional theory (DFT) methods and size-converged supercell models, we analyze the electronic and atomic structure of magnetic $3d$ transition metal dopants in cubic gallium nitride (c-GaN). All stable defect charge states for Fermi levels across the full experimental gap are computed using a method that correctly resolves the boundary condition problem (without a jellium approximation) and eliminates finite-size errors. The resulting computed defect levels are not impacted by the DFT band-gap problem, they span a width consistent with the experimental gap rather than being limited to the single-particle DFT gap. All defects with electronically degenerate (half-metal) ${T}_{d}$ ground states are found to have significant distortions, relaxing to ${D}_{2d}$ structures driven by the Jahn-Teller instability. This leads to insulating ground states for all substitutional $3d$ dopants, refuting claims in the literature that $+U$ or hybrid functional methods are required to avoid artificial half-metal results. Interpreting the ${d}^{n}$ atomic occupations within a crystal-field model and exchange splittings, we identify a systematic trend across the $3d$ transition metal series. Approaches to estimate excited-state energies as observed in photoluminescence from defect centers are assessed, ranging from a Koopmans-type single-particle energy interpretation to relaxed total energy differences in fully self-consistent DFT. The single-particle interpretations are found to be qualitatively predictive and the calculations are consistent with the limited available experimental data across the $3d$ dopant series. These results provide a baseline understanding to guide future studies and a conceptual framework within which to interpret new results.

Morphology of Ionic Micelles as Studied by Numerical Solution of the Poisson Equation.

The existing concepts of the ionic micelle structure were specified. It was noted that the composition of dispersed phase particles in a liquid dispersion medium should necessarily include adsorbed counterions rigidly bound to these particles. By numerical solution of the Poisson equation for the two most often used approximations, the Poisson–Boltzmann (PB) model and the Jellium-approximation (JA), the electric potential decay from the Stern potential of dispersed phase particles was defined. A new methodological approach to analyze the reaction of micelle potential decay based on small variability of the CMC value was proposed. It made possible to determine the dimension parameter, which in the presence of weak thermal effects approximately corresponds to the micelle hydrodynamic radius, and to calculate the electrokinetic potential of micelles. The results of theoretical calculations were compared with our previous experimental data on the thickness of the SDS micelle hydrophilic layer obtained by SAXS. A good agreement between the calculated and measured values was obtained, and it was noted that for low concentrations the experimental values are more correctly described by the PB model, but for concentrations greater than 100 mM the JA model is more preferable. It was found that the slipping plane is located near the outer Stern plane and is separated from it only by a few molecular layers of water. The influence stronger than the thermal one can shift the slipping plane closer to the micelle core. Accordingly, the smallest hydrodynamic micelle size is determined by the outer Stern plane. The results of our work allowed us to conclude that the micelle is not something soft and watery, but according to its specified structure, it is a more solid-like particle than was previously assumed. The proposed approach can be extended to investigate other effects of a physicochemical nature, in particular, those observed with the addition of an external electrolyte or nanoparticles.

Theoretical study of intrinsic defects in cubic silicon carbide 3C -SiC

Using the local moment counter charge (LMCC) method to accurately represent the asymptotic electrostatic boundary conditions within density functional theory supercell calculations, we present a comprehensive analysis of the atomic structure and energy levels of point defects in cubic silicon carbide ($3C$-SiC). Finding that the classical long-range dielectric screening outside the supercell induced by a charged defect is a significant contributor to the total energy. we describe and validate a modified Jost screening model to evaluate this polarization energy. This leads to bulk-converged defect levels in finite size supercells. With the LMCC boundary conditions and a standard Perdew-Burke-Ernzerhof (PBE) exchange correlation functional, the computed defect level spectrum exhibits no band gap problem: the range of defect levels spans $\ensuremath{\sim}2.4\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$, an effective defect band gap that agrees with the experimental band gap. Comparing with previous literature, our LMCC-PBE defect results are in consistent agreement with the hybrid-exchange functional results of Oda et al. [J. Chem. Phys. 139, 124707 (2013)] rather than their PBE results. The difference with their PBE results is attributed to their use of a conventional jellium approximation rather than the more rigorous LMCC approach for handling charged supercell boundary conditions. The difference between standard dft and hybrid functional results for defect levels lies not in a band gap problem but rather in solving a boundary condition problem. The LMCC-PBE entirely mitigates the effect of the band gap problem on defect levels. The more computationally economical PBE enables a systematic exploration of $3C$-SiC defects, where, most notably, we find that the silicon vacancy undergoes Jahn-Teller-induced distortions from the previously assumed ${T}_{d}$ symmetry, and that the divacancy, like the silicon vacancy, exhibits a site-shift bistability in $p$-type conditions.

Deformation of the Fermi surface of a spinless two-dimensional electron gas in presence of an anisotropic Coulomb interaction potential

We consider the stability of the circular Fermi surface of a two-dimensional electron gas system against an elliptical deformation induced by an anisotropic Coulomb interaction potential. We use the jellium approximation for the neutralizing background and treat the electrons as fully spin-polarized (spinless) particles with a constant isotropic (effective) mass. The anisotropic Coulomb interaction potential considered in this work is inspired from studies of two-dimensional electron gas systems in the quantum Hall regime. We use a Hartree–Fock procedure to obtain analytical results for two special Fermi liquid quantum electronic phases. The first one corresponds to a system with circular Fermi surface while the second one corresponds to a liquid anisotropic phase with a specific elliptical deformation of the Fermi surface that gives rise to the lowest possible potential energy of the system. The results obtained suggest that, for the most general situations, neither of these two Fermi liquid phases represent the lowest energy state of the system within the framework of the family of states considered in this work. The lowest energy phase is one with an optimal elliptical deformation whose specific value is determined by a complex interplay of many factors including the density of the system.

Correlations between conduction electrons in dense plasmas.

Most treatments of electron-electron correlations in dense plasmas either ignore them entirely (random phase approximation) or neglect the role of ions (jellium approximation). In this work, we go beyond both these approximations to derive a formula for the electron-electron static structure factor which properly accounts for the contributions of both ionic structure and quantum-mechanical dynamic response in the electrons. The result can be viewed as a natural extension of the quantum Ornstein-Zernike theory of ionic and electronic correlations, and it is suitable for dense plasmas in which the ions are classical and the conduction electrons are quantum-mechanical. The corresponding electron-electron pair distribution functions are compared with the results of path integral Monte Carlo simulations, showing good agreement whenever no strong electron resonance states are present. We construct approximate potentials of mean force which describe the effective screened interaction between electrons. Significant deviations from Debye-Hückel screening are present at temperatures and densities relevant to high-energy density experiments involving warm and hot dense plasmas. The presence of correlations between conduction electrons is likely to influence the electron-electron contribution to the electrical and thermal conductivity. It is expected that excitation processes involving the conduction electrons (e.g., free-free absorption) will also be affected.

Core-polarization-corrected random-phase approximation with exact exchange for dipole surface plasmons in silver clusters

The surface plasmon in silver clusters is red shifted with respect to standard jellium random-phase approximation (RPA) predictions that work well for simple metal clusters. The reason for the deviation arises primarily from the non-negligible polarization interaction between the valence electrons and ionic cores. In order to quantify this effect in the jellium approximation we introduce a modified RPAE (RPA with exact exchange). The jellium background of Ag cores is treated as a polarizable sphere. This model predicts a dipole surface resonance in excellent agreement with published experimental data. Moreover it yields the blue shift (red shift) with decreasing sizes for cationic (anionic) Ag clusters as observed experimentally.

On the response function of simple metal clusters

We analyse the response function of simple metal clusters considering several theoretical approaches and exploring a large class of momenta and energies. The study is performed on a wide range of cluster sizes to explore the relative weight of the various dominant contributions to the response. We work with Sodium clusters in jellium approximation. The comparison between the various theoretical approaches shows altogether a remarkable consistency. The simple Boson approximation to the response function is furthermore shown to perform remarkably well in spite of its simplicity.

Real-space method for highly parallelizable electronic transport calculations

We present a real-space method for first-principles nano-scale electronic transport calculations. We use the non-equilibrium Green's function method with density functional theory and implement absorbing boundary conditions (ABCs, also known as complex absorbing potentials, or CAPs) to represent the effects of the semi-infinite leads. In real space, the Kohn-Sham Hamiltonian matrix is highly sparse. As a result, the transport problem parallelizes naturally and can scale favorably with system size, enabling the computation of conductance in relatively large molecular junction models. Our use of ABCs circumvents the demanding task of explicitly calculating the leads' self-energies from surface Green's functions, and is expected to be more accurate than the use of the jellium approximation. In addition, we take advantage of the sparsity in real space to solve efficiently for the Green's function over the entire energy range relevant to low-bias transport. We illustrate the advantages of our method with calculations on several challenging test systems and find good agreement with reference calculation results.

Effective mass theory of monolayerδdoping in the high-density limit

Monolayer $\ensuremath{\delta}$-doped structures in silicon have attracted renewed interest with their recent incorporation into atomic-scale device fabrication strategies as source and drain electrodes and in-plane gates. Modeling the physics of $\ensuremath{\delta}$ doping at this scale proves challenging, however, due to the large computational overhead associated with ab initio and atomistic methods. Here, we develop an analytical theory based on an effective mass approximation. We specifically consider the Si:P materials system and the limit of high donor density, which has been the subject of recent experiments. In this case, metallic behavior including screening tends to smooth out the local disorder potential associated with random dopant placement. While smooth potentials may be difficult to incorporate into microscopic, single-electron analyses, the problem is easily treated in the effective mass theory by means of a jellium approximation for the ionic charge. We then go beyond the analytic model, incorporating exchange and correlation effects within a simple numerical model. We argue that such an approach is appropriate for describing realistic, high-density, highly disordered devices, providing results comparable to density functional theory, but with greater intuitive appeal and lower computational effort. We investigate valley coupling in these structures, finding that valley splitting in the low-lying $\ensuremath{\Gamma}$ band grows much more quickly than the $\ensuremath{\Gamma}$-$\ensuremath{\Delta}$ band splitting at high densities. We also find that many-body exchange and correlation corrections affect the valley splitting more strongly than they affect the band splitting.

Semiclassical approach to plasmon–electron coupling and Landau damping of surface plasmons

A semiclassical model is developed to describe plasmon-electron coupling and electronic damping of surface plasmons. It is compared with the ab initio linear response calculations for metallic thin films in the jellium approximation and for a realistic crystalline Mg(0001) surface. The semiclassical model is able to reproduce the quantum oscillations of plasmon linewidth, which was obtained in the previous ab initio calculations. In addition, state-resolved analysis reveals the origin of these oscillations, which result from superposition of the short-period oscillations of individual electron-hole pair transitions. The semiclassical model is further applied to a crystalline Mg(0001) surface, where linewidth dispersion of the surface plasmon is calculated and shows good agreement with earlier ab initio calculation and experiment. Our results suggest that this semiclassical approach is quite promising for the quantitative description of plasmon-electron coupling and associated processes such as surface-enhanced Raman scattering, light emission, and fluorescence.

The metastable states of foam films containing electrically charged micelles or particles: Experiment and quantitative interpretation

The stepwise thinning (stratification) of liquid films containing electrically charged colloidal particles (in our case — surfactant micelles) is investigated. Most of the results are applicable also to films from nanoparticle suspensions. The aim is to achieve agreement between theory and experiment, and to better understand the physical reasons for this phenomenon. To test different theoretical approaches, we obtained experimental data for free foam films from micellar solutions of three ionic surfactants. The theoretical problem is reduced to the interpretation of the experimental concentration dependencies of the step height and of the final film thickness. The surface charges of films and micelles are calculated by means of the charge-regulation model, with a counterion-binding (Stern) constant determined from the fit of surface tension isotherms. The applicability of three models was tested: the Poisson–Boltzmann (PB) model; the jellium-approximation (JA), and the cell model (CM). The best agreement theory/experiment was obtained with the JA model without using any adjustable parameters. Two theoretical approaches are considered. First, in the energy approach the step height is identified with the effective diameter of the charged micelles, which represents an integral of the electrostatic-repulsion energy calculated by the JA model. Second, in the osmotic approach the step height is equal to the inverse cubic root of micelle number density in the bulk of solution. Both approaches are in good agreement with the experiment if the suspension of charged particles (micelles) represents a jellium, i.e. if the particle concentration is uniform despite the field of the electric double layers. The results lead to a convenient method for determining the aggregation number of ionic surfactant micelles from the experimental heights of the steps.

Note: Renormalized jellium model for charged colloids revisited

Recently, it has been demonstrated that electrostatic interactions between charged colloids in contact with a symmetric electrolyte can accurately be described by using the so-called renormalized jellium (RJ) model proposed by Trizac and Levin.1 This mean-field approximation, based on the Poisson–Boltzmann equation (PBE), requires that the charge of the background around a tagged colloid be renormalized self-consistently to coincide with the colloid effective charge; such a requirement leads to precise thermodynamic properties even in highly charged colloidal suspensions.2–5 In this note, we revisit the renormalized jellium approximation and, in particular, provide a simple explicit recipe that facilitates the renormalization procedure being easy of implementing in different situations. We here apply such recipe in the case of charged spheres and discuss its extension to the case of charged rods. The system under consideration is a charged colloidal suspension of volume fraction η composed of, obviously, charged colloids of radius a, and counter ions in contact with a symmetric (1:1) salt reservoir of concentration 2cs , with cs the density of positive or negative salt ions; the solvent is included through the dielectric constant e. The RJ model assumes that the charge of Nc − 1 colloidal particles around a tagged macroion is smeared out in the whole suspension to form a homogeneous background with charge Zbacke, e being the elementary charge. This background charge is Zback = Zbare, with Zbare the colloidal bare charge, and not known a priori, but it is determined self-consistently to be equal to the effective charge, Zeff, of the tagged macroion.1 An extraordinary advantage of the RJ model is that Zeff is directly associated to the system osmotic pressure, the screening parameter, and the effective pair interaction between colloids (when it is explicitly considered at the level of the Yukawa approximation).1, 2 Nevertheless, since it is an explicit function of the system state, i.e., Zeff = Zeff(η, Zbare, cs), its evaluation depends on the specific conditions of the system and, currently, the entire charge renormalization procedure is carried out through an iterative process.6 This iterative procedure demands the construction of a function of the form Zeff = Zeff(Zback, Zbare). The selfconsistently condition is reached in the intersection of the function with the straight line Zeff = Zback. However, it is still possible to reformulate this scheme and avoid the full iterative procedure to gain clarity in the way in which the RJ can be applied and extended to study the physical properties of charge-stabilized colloidal suspensions. The complete original procedure can then be reformulated as follows. We start with the requirement of selfconsistency from the beginning. This means that the condition Zback = Zeff must be explicitly incorporated into the PBE. Our main assumption is that there exists a unique Zeff for a given Zbare; this avoids completely the inclusion of Zback and facilitates drastically the renormalization scheme. Additionally, one should also rephrase the typical boundary conditions (BCs) to solve the PBE with simple BCs at one point [see Eqs. (3) and (4)]. To achieve this, we use the fact that the farfield solution of the PBE has a Yukawa-like form that depends on Zeff. Therefore, in the case of charged spherical colloids, the PBE now reads d2φ dr2 + 2 r dφ dr = −3η ZeffλB a − ρ+(∞)e−φ + ρ−(∞)e, (1)

Second random phase approximation studies in metallic clusters

We study collective states in metallic clusters within the second random phase approximation (RPA) including all kinds of couplings among all one particle--one hole and two particle--two hole configurations. The calculations are done in the jellium approximation. The coupling of one particle--one hole configurations with the two particle--two hole ones strongly pushes down the multipole strength distribution, especially for the dipole case, with respect to RPA. The inclusion of the coupling of the two particle--two hole configurations among themselves enhances this behavior also for the collective states whose description within RPA is widely accepted as satisfactory. Some possible origins of these unpleasant results are discussed. In particular, the need for a better treatment of short-range correlations is underlined.

Time‐dependent density functional calculation of the energy loss of antiprotons colliding with metallic nanoshells

AbstractTime‐dependent density functional theory is used to study the interaction between antiprotons and metallic nanoshells. The ground state electronic properties of the nanoshell are obtained in the jellium approximation. The energy lost by the antiproton during the collision is calculated and compared to that suffered by antiprotons traveling in metal clusters. The resulting energy loss per unit path length of material in thin nanoshells is larger than the corresponding quantity for clusters. It is shown that the collision process can be interpreted as the antiproton crossing of two nearly bi‐dimensional independent metallic systems. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Energy loss and deflection of fast ions under glancing incidence upon a graphene sheet

We investigate the stopping power and the image force on fast ions moving parallel to a graphene sheet caused by the dynamic polarization of carbon valence electrons. We compare the results based on a two-dimensional hydrodynamic model of graphene with those based on a three-dimensional hydrodynamic model which uses in the ground-state the electron density obtained from the Thomas–Fermi class of models for graphene in jellium approximation. It is shown that, at least for high ion speeds, the two-dimensional model underestimates the magnitudes of both the stopping power and the image force. Our results are relevant for consistent modelling of the various interaction forces involved in ion channeling through graphite and through carbon nanotubes.

Genuine converging solution of self-consistent field equations for extended many-electron systems

Calculations of the ground state of inhomogeneous many-electron systems involve a solving of the Poisson equation for Coulomb potential and the Schrödinger equation for singleparticle orbitals. Due to nonlinearity and complexity this set of equations, one believes in the iterative method for the solution that should consist in consecutive improvement of the potential and the electron density until the self-consistency is attained. Though this approach exists for a long time there are two grave problems accompanying its implementation to infinitely extended systems.The first of them is related with the Poisson equation and lies in possible incompatibility of the boundary conditions for the potential with the electron density distribution renewed by means of the Schrödinger equation. Rigorously speaking, it means the iteration process cannot be continued. The resolution of this difficulty is proposed for both infinite conducting systems in jellium approximation and periodic structures. It provides the existence of self-consistent solution for the potential at every iteration step due to realization of a screening effect.The second problem results from the existence of continuous spectrum of Hamiltonian eigenvalues for unbounded systems. It needs to have a definition of Hilbert space basis with eigenfunctions of continuous spectrum as elements, which would be convenient in numerical applications. It is proposed to insert a limiting transition into the definition of scalar product specifying the Hilbert space. It provides self-adjointness of Hamiltonian and, respectively, the orthogonality of eigenfunctions corresponding to the different eigenvalues. In addition, it allows to normalize them effectively to delta-function and to prove the orthogonality of the 'right' and 'left' eigenfunctions belonging to twofold degenerate eigenvalues.

Probing the jellium model with colloidal dispersions of charged liposomes

The structure factor, S( q), is measured for a set of dispersions that consist of liposome vesicles with differing numbers of ionizable surface groups. These systems allow us to probe the dependence of the local structure on the particle bare charge systematically. The height of the nearest neighbor peak in S( q), which is a direct measure of the spatial order of the vesicles, does not increase monotonically with the number of ionizable groups for a given volume fraction. On the contrary, it rather decreases. This feature cannot be explained by means of the classical renormalization scheme based on the cell model. We analyze our experimental data using a renormalization model based on the jellium approximation, which predicts the weakening of the spatial order for moderate and large particle charges and we find a qualitative agreement.

Theory of multiple harmonic generation in reflection from a metal surface

A microscopic theory is presented for multiple harmonic generation (MHG) from a metal surface in the jellium approximation and for laser intensities below the critical value for plasma formation. Calculations are presented for the second and up to the sixth harmonic, in the case of a gold surface and for laser wavelengths 800 and 1064 nm, with femtosecond and picosecond laser pulses. The peaks of the harmonic pulses are delayed relative to the peak of the incident laser pulse by about 20 to 30 fs due to the finite electron-relaxation times. The calculated relative peak intensities of the various harmonics are most sensitive to the values of the relaxation times, and MHG provides a way of determining these times.

Renormalized jellium model for charge-stabilized colloidal suspensions.

We introduce a renormalized jellium model to calculate the equation of state for charged colloidal suspensions. An almost perfect agreement with Monte Carlo simulations is found. Our self-consistent approach naturally allows to define the effective charge of particles at finite colloidal density. Although this quantity may differ significantly from its counterpart obtained from the standard Poisson-Boltzmann cell approach, the osmotic pressures for both models are in good agreement. We argue that by construction, the effective charge obtained using the jellium approximation is more appropriate to the study of colloidal interactions. We also discuss a possibility of a fluid-fluid critical point and show how our equation of state can be used to shed light on the surprising results found in recent sedimentation experiments.