Self-consistent Problem Research Articles

Kinetic Theory of the Wall Sheath for Arbitrary Conditions in a Gas-Discharge Plasma

The self-consistent problem of the structure of a perturbed wall sheath in a dc gas-discharge plasma near a flat surface under negative potential relative to the plasma has been solved for an arbitrary relation between the Debye radius and the ion mean free path. The solution has been obtained without artificial separation of this layer into the quasi-neutral “presheath” and the wall sheath in which quasi-neutrality is violated substantially. The actual ion distribution function in the unperturbed plasma, the dependence of the charge-exchange cross section on the ion energy, and the nonzero electric field in the unperturbed plasma have been considered. It is shown that when the average electron energy is conserved the structure of the perturbed wall sheath weakly depends on the form of the electron distribution function. It has been established that the mean energy of ions in the unperturbed plasma substantially affects the structure of the quasi-neutral presheath as well as the structure of a part of the wall sheath in which quasi-neutrality is not observed even under the assumption that the mean electron energy is much higher than the mean energy of ions. The calculations of ion flow parameters and the structure of the perturbed wall sheath are in conformity with experimental data obtained by other authors, which could not be adequately interpreted earlier.

A DFT-based tight-binding approach to the self-consistent description of molecule metal-nanoparticle interactions

The interaction within a hybrid system consisting of a spherical metal nanoparticle (NP) and a nearby organic dye molecule is formulated in a combined quantum–classical approach. Whereas the NP’s polarization field is treated in classical multipole form, the electronic charge density of the molecule is described quantum mechanically. An efficient solution of the resulting self-consistency problem becomes possible by using the discrete representation of the charge density in terms of atom-centered Mulliken charges within the density functional theory-based tight binding approach. Results for two different dye molecules are presented, which focus on the dependence of the interaction on the NP’s radius, the distance between NP and molecule and their mutual orientation.

Extreme Properties of Oscillations of an Elliptical Float

The extreme amplitude and frequency properties of oscillations of a thin (symmetric with respect to both axes) elongated flat body of elliptical form are studied in the neighborhood of an interface of stably stratified ideal liquids. The numerical and analytical solving procedures were developed for a self-consistent boundary value problem on an asymptotically longtime interval, which leads to their significant radiation decay and evolution in oscillation frequency. Similar characteristics found earlier by the authors for floats of other forms are compared, and the significant quantitative and qualitative differences are established.

Iterative approaches to the self-consistent nuclear energy density functional problem

Large-scale applications of energy density functional (EDF) methods depend on fast and reliable algorithms to solve the associated non-linear self-consistency problem. When dealing with large single-particle variational spaces, existing solvers can become very slow, and their performance dependent on manual fine-tuning of numerical parameters. In addition, convergence can sensitively depend on particularities of the EDF's parametrisation under consideration. Using the widely-used Skyrme EDF as an example, we investigate the impact of the parametrisation of the EDF, both in terms of the operator structures present and the size of coupling constants, on the convergence of numerical solvers. We focus on two aspects of the self-consistency cycle, which are the diagonalisation of a fixed single-particle Hamiltonian on one hand and the evolution of the mean-field densities and potentials on the other. Throughout the article we use a coordinate-space representation, for which the behaviour of algorithms can be straightforwardly analysed. We propose two algorithmic improvements that are easily implementable in existing solvers, heavy-ball dynamics and potential preconditioning. We demonstrate that these methods can be made virtually parameter-free, requiring no manual fine-tuning to achieve near-optimal performance except for isolated cases. The combination of both methods decreases substantially the CPU time required to obtain converged results. The improvements are illustrated for the MOCCa code that solves the self-consistent HFB problem in a 3d coordinate space representation for parametrisations of the standard Skyrme EDF at next-to-leading order in gradients and its extension to next-to-next-to-leading order.

System of ideal quantum dots with Coulomb interaction

The wave functions of single-electron states localised on a system of single-level ideal quantum dots in a semiconductor matrix are constructed. The standard Kane theory, which describes the renormalisation of the effective mass of electrons in bulk III − V semiconductors, is transformed, for the first time as far as we know, for small-size quantum dots. The renormalised electron mass in a quantum dot depends on its ground state energy, which, in turn, depends on this mass. Thus, a self-consistent problem is obtained for calculating the electron binding energy. The radius of the Debye screening of the Coulomb interaction in a system of quantum dots is calculated at room temperature. The Coulomb repulsion of electrons localised on dots limits from above the possible number of filled dots. The condition is formulated for the optimal concentration of quantum dots. A classical distribution function for the probability of quantum dot filling is constructed. It is found the state, where electrons fill exactly half the maximum possible number of filled dots, exhibits the minimal energy (and therefore the most stable), the condition being fairly universal. In particular, it can be used to estimate the limiting efficiency of quantum-dot diode lasers.

Modeling of the structure of ferroelectric polymer systems using the synthesis of discrete and continual approaches

A model of a ferroelectric polymer system is proposed that is a synthesis of various models of polymer systems and low-molecular ferroelectrics. The problem of self-consistency between the orientational order parameter of macromolecules and the appearance of spontaneous polarization is solved. The temperature dependences of the intermolecular distances, the dipole and quadrupole order parameters, the susceptibility at different values of the intermolecular interaction parameters are calculated.

Boundary-Value Problems of the Dynamic Behavior of Two-Dimensional Elastic Systems with Moving Objects

The interdependent dynamic behavior of a two-dimensional elastic system in the form of a one-dimensional mechanical object moving on a band is considered. The Lagrangian density of the two-dimensional system depends on the generalized coordinates and their derivatives up to and including the second order, and the Lagrangian of the moving object as one of the generalized coordinates contains the motion law, which is an unknown function of this problem. The physically and mathematically correct conditions at the moving boundary have been found as a result of formulating the self-consistent boundary-value problem based on the Hamilton variational principle. The problem of the unseparated motion of a rod, which performs bending and torsional vibrations, along a plate with consideration for the rotational inertia of its components is formulated as an example. The differential and integral laws of the change in energy and wave momentum are derived for both the entire complex system and its isolated parts. The relationships true at the moving boundary are established between the components of the energy flux density vector and the wave momentum flux density tensor.

Self-consistency and covariance of light-front quark models: Testing via P , V , and A meson decay constants, and P→P weak transition form factors

In this paper, we test the self-consistencies of the standard and the covariant light-front quark model and study the zero-mode issue via the decay constants of pseudoscalar ($P$), vector ($V$) and axial-vector ($A$) mesons, as well as the $P\to P$ weak transition form factors. With the traditional type-I correspondence between the manifestly covariant and the light-front approach, the resulting $f_{V}$ as well as $f_{^1\!A}$ and $f_{^3\!A}$ obtained with the $\lbd=0$ and $\lbd=\pm$ polarization states are different from each other, which presents a challenge to the self-consistency of the covariant light-front quark model. However, such a self-consistency problem can be "resolved" within the type-II scheme, which requires an additional replacement $M\to M_0$ relative to the type-I case. Moreover, the replacement $M\to M_0$ is also essential for the self-consistency of the standard light-front quark model. In the type-II scheme, the valence contributions to the physical quantities~(${\cal Q}$) considered in this paper are alway the same as that obtained in the standard light-front quark model, $[{\cal Q}]_{\rm val.}=[{\cal Q}]_{\rm SLF}$, and the zero-mode contributions to $f_{V,^1\!A,^3\!A}$ and $f_-(q^2)$ exist only formally but vanish numerically, which implies further that $[{\cal Q}]_{\rm val.}\dot{=} [{\cal Q}]_{\rm full}$. In addition, the manifest covariance of the covariant light-front quark model is violated in the traditional type-I scheme, but can be recovered by taking the type-II scheme.

Theoretical study of the laser annealing process in FinFET structures

In this paper we present a computational tool for the simulation of laser annealing processes in FinFET structures. This is a complex self-consistent problem, where heating is evaluated by means of the time harmonic solution of the Maxwell equations. The main features of our computational code include: A versatile graphical user interface for the structure design; The assignment of materials and the simulation analysis; An interface with the finite element method solver for the automatic generation of the mesh and the runtime control; Parameters for numerous materials (optical/thermal properties and mass transport) as a function of temperature and phases; An efficient coupling with electromagnetic simulations for the self-consistent source estimate (i.e. the power dissipation) in nanostructured topographies; Experimental validation of nanostructured samples; Multiple-dopant models simulating dopant redistribution, including diffusion solubility and segregation; Alloy model, e.g. SiGe (where the melting point depends on the alloy fraction); Multiple phases (e.g. amorphous, liquid, crystal). As a particular application of the tool we present a study of the laser process design by varying the laser fluence, polarization of the electromagnetic field and the pitch of the devices. Results are in excellent agreement with the experiment and could serve as guidelines for the realization of targeted laser annealing processes.

Communication: Can excitation energies be obtained from orbital energies in a correlated orbital theory?

This work shows that vertical excitation energies (characterized as single-electron processes) can be expressed in terms of one-particle solutions from a self-consistent field problem built by means of correlated operators. There are two alternative ways of enforcing this proposal for i → a transitions in a system (M): (1) by using only eigenvalues obtained for the cationic species reached after the removal of an electron from orbital i (M +) or (2) by combining these quantities with the eigenvalue associated with orbital i from the neutral M system. We demonstrate that those eigenvalues derived from the equation-of-motion formalism in terms of the coupled cluster approach including single and double substitutions for ionization potentials and electron affinities show excellent performance in reproducing these electronic transition energies by either path, with mean absolute deviations (MADs) between 0.02 and 0.06 eV. Moreover, the Kohn-Sham Density Functional Theory (KS-DFT) methods from the Quantum Theory Project (QTP) family provide nice results in terms of the second approach (MADs from 0.21 to 0.47 eV). However, DFT is not as successful as long as one takes into account only the eigenvalues of M +, although the respective excitation energies from QTP functionals are still reasonable (MADs between 0.55 and 0.74 eV). Ultimately, these relations can be used as a new consistency condition to develop KS-DFT approximations to the correlated orbital theory.

Formation of the Source Function Generating a Specified Radiation Field

The spectrum of symmetric modes forming a specified distribution of the radiation field is calculated using solution to the self-consistent problem of radiation from the end of a circular dielectric waveguide. The self-consistent problem is reduced to solution of a system of integral equations. In the case of the synthesis of the source function in the space of eigenfunctions of the boundary value problem for a circular dielectric waveguide, these equations are the Fredholm integral equations of the first kind.

Revisiting the positive DC corona discharge theory: Beyond Peek's and Townsend's law

The classical positive Corona Discharge theory in a cylindrical axisymmetric configuration is revisited in order to find analytically the influence of gas properties and thermodynamic conditions on the corona current. The matched asymptotic expansion of Durbin and Turyn [J. Phys. D: Appl. Phys. 20, 1490–1495 (1987)] of a simplified but self-consistent problem is performed and explicit analytical solutions are derived. The mathematical derivation enables us to express a new positive DC corona current-voltage characteristic, choosing either a dimensionless or dimensional formulation. In dimensional variables, the current voltage law and the corona inception voltage explicitly depend on the electrode size and physical gas properties such as ionization and photoionization parameters. The analytical predictions are successfully confronted with experiments and Peek's and Townsend's laws. An analytical expression of the corona inception voltage φon is proposed, which depends on the known values of physical parameters without adjustable parameters. As a proof of consistency, the classical Townsend current-voltage law I=Cφ(φ−φon) is retrieved by linearizing the non-dimensional analytical solution. A brief parametric study showcases the interest in this analytical current model, especially for exploring small corona wires or considering various thermodynamic conditions.

Metal-insulator transition in the hybridized two-orbital Hubbard model revisited

In this work we study the two-orbital Hubbard model on a square lattice in the presence of hybridization between nearest-neighbor orbitals and a crystal-field splitting. We use a highly reliable numerical technique based on the density matrix renormalization group to solve the dynamical mean field theory self-consistent impurity problem. We find that the orbital mixing always leads to a finite local density states at the Fermi energy in both orbitals when at least one band is metallic. When one band is doped, and the chemical potential lies between the Hubbard bands in the other band, the coherent quasiparticle peak in this orbital has an exponential behavior with the Hubbard interaction U.

Green Functions and Self-Consistency: Insights From the Spherium Model

We report an exhaustive study of the performance of different variants of Green function methods for the spherium model in which two electrons are confined to the surface of a sphere and interact via a genuine long-range Coulomb operator. We show that the spherium model provides a unique paradigm to study electronic correlation effects from the weakly correlated regime to the strongly correlated regime, since the mathematics are simple while the physics is rich. We compare perturbative GW, partially self-consistent GW and second-order Green function (GF2) methods for the computation of ionization potentials, electron affinities, energy gaps, correlation energies as well as singlet and triplet neutral excitations by solving the Bethe-Salpeter equation (BSE). We discuss the problem of self-screening in GW and show that it can be partially solved with a second-order screened exchange correction (SOSEX). We find that, in general, self-consistency deteriorates the results with respect to those obtained within perturbative approaches with a Hartree-Fock starting point. Finally, we unveil an important problem of partial self-consistency in GW: in the weakly correlated regime, it can produce artificial discontinuities in the self-energy caused by satellite resonances with large weights.

Approaching the self-consistency challenge of electrocatalysis with theory and computation

This opinion piece centers around challenges involved in developing first-principles electrochemical methods. In recent years, theory and computation have become indispensable tools to navigate the parameter space that controls the activity and stability of electrocatalytic materials and electrochemical devices. Viable methods process as input details on materials structure, composition and reaction conditions. Their output includes metrics for stability and activity, phase diagrams, as well as mechanistic insights on reaction mechanisms and pathways. The core challenge, connecting input to output, is a self-consistency problem that couples the electrode potential to variables for the electronic structure of the solid electrode, solvent properties and ion distributions in the electrolyte as well as specific properties of a boundary region in-between. We will discuss a theoretical framework and computational approaches that strive to accomplish this feat.

Discrete crack dynamics: A planar model of crack propagation and crack-inclusion interactions in brittle materials

The Multipole Method (MPM) is used to simulate the many-body self-consistent problem of interacting elliptical micro-cracks and inclusions in single crystals. A criterion is employed to determine the crack propagation path based on the stress distribution; the evolution of individual micro-cracks and their interactions with existing cracks and inclusions is then predicted using what we coin the Discrete Crack Dynamics (DCD) method. DCD is fast (semi-analytical) and particularly suitable for the simulation of evolving low-speed crack networks in brittle or quasi-brittle materials. The method is validated against finite element analysis predictions and previously published experimental data.

Quark-Parton Model and Relativistic Quantum Mechanics

An attempt to treat the asymptotic freedom and the quark confinement as a self-consistent problem in the framework of relativistic quantum mechanics is realized. It is shown that the confinement of quarks induces a change of their helicities together with a simultaneous alteration of orbital momenta, so that the total angular momentum of each quark is conserved. This observation may cast light on the so-called proton spin puzzle after some additional numerical estimations.

Nonlinear waves in an elastic guide interacting with an elastic-inertial foundation

This paper deals with the study of the transverse wave propagation in a string resting on an elastic-inertial foundation. A self-consistent dynamic problem is herein considered for a system composed of a one-dimensional flexible guide (string) and an elastic-inertial foundation. As a foundation model the Vesnitsky’s model was chosen. The set of equations is reduced to one fourth-order cubic-nonlinear equation relative to transverse displacements of the string. Depending on the string and the elastic-inertial foundation mass ratio, the evolutionary equation has the following three extreme events: the modified Ostrovsky equation, the Riemann equation with a cubic nonlinearity, the equation for an anharmonic oscillator with a cubic nonlinearity.

Oscillations and Waves Associated with a Source

A self-consistent boundary value problem of excitation is considered in which the source function on the right-hand side of the Helmholtz equation is the solution of a homogeneous boundary value problem. It is shown that the source of an associated oscillation can be put into correspondence with a function describing a standing wave and the source of an associated wave, to a function describing a traveling wave. It is established that such sources, in addition to an eigenwave (eigenoscillation), produce an associated wave (oscillation). It is proposed to classify these waves (oscillations) as waves associate with a source, because they exist only in the presence of sources of this type.

Ion energy distribution function in the wall layer at a negative wall potential with respect to the plasma

On the basis of the kinetic approach, the self-consistent problem of the gas discharge ion distribution function in the sheath near a surface at a negative potential with respect to the plasma is solved. For the first time, the solution takes into account the dependence of the ion charge exchange cross section from the atom on the ion energy, as well as the real ion distribution function in the unperturbed plasma. It is shown that the dependence of the charge exchange cross section on the ion energy significantly affects the shape of the ion distribution function. It is found that the mean energy of the ions near the wall depends on the electron mean energy in the unperturbed plasma. It was also found that, at the same electron mean energy, the form of the distribution function has practically no effect on the ion distribution function in the wall sheath. The calculations are in good agreement with the known mass spectrometric measurements of the ion distribution function. The obtained results give an opportunity to develop a self-consistent solution of the wall layer structure problem including the quasi-neutral presheath.