Screening of an Electric Field and the Quasi-Static Capacitance of an Induced Charge in Semiconductors with Hopping Conductivity

Expressions for the screening length of an external electrostatic field in a crystalline semiconductor are derived in the Debye–Hückel and Mott–Schottky approximations taking into account electron (hole) hopping via hydrogen-like donors (acceptors). The feasibility of determining the Debye–Hückel screening length from measurements of a quasi-static capacitance with low and high degrees of basic dopant compensation has been demonstrated even in a strong field, that is, in the Mott–Schottky approximation. To measure the capacitance, the electric signal frequency must be much less than the average frequency of electron hopping via donors.     

Two-Dimensional Coulomb Systems on a Surface of Constant Negative Curvature

We study the equilibrium statistical mechanics of classical two-dimensional Coulomb systems living on a pseudosphere (an infinite surface of constant negative curvature). The Coulomb potential created by one point charge exists and goes to zero at infinity. The pressure can be expanded as a series in integer powers of the density (the virial expansion). The correlation functions have a thermodynamic limit, and remarkably that limit is the same one for the Coulomb interaction and some other interaction law. However, special care is needed for defining a thermodynamic limit of the free energy density. There are sum rules expressing the property of perfect screening. These generic properties can be checked on the Debye–Hückel approximation, and on two exactly solvable models, the one-component plasma and the two-component plasma, at some special temperature.     

Debye-Hückel limit of quantum coulomb systems

In this paper, we consider charge symmetric quantum Coulomb systems with Boltzmann statistics. We prove that the theory of screening of Debye and Hückel is a combined classical and mean field limit of these quantum Coulomb systems.Supported by the Swiss National Foundation for Scientific Research     

Saturation of Electrostatic Potential: Exactly Solvable 2D Coulomb Models

We test the concepts of renormalized charge and potential saturation, introduced within the framework of highly asymmetric Coulomb mixtures, on exactly solvable Coulomb models. The object of study is the average electrostatic potential induced by a unique “guest” charge immersed in a classical electrolyte, the whole system being in thermal equilibrium at some inverse temperature β. The guest charge is considered to be either an infinite hard wall carrying a uniform surface charge or a charged colloidal particle. The systems are treated as two-dimensional; the electrolyte is modelled by a symmetric two-component plasma (TCP) of point-like ±e charges with logarithmic Coulomb interactions. Two cases are solved exactly: the Debye–Hückel limit β e²→ 0 and the Thirring free-fermion point β e²=2. The results at the free-fermion point can be summarized as follows: (i) The induced electrostatic potential exhibits the asymptotic behavior, at large distances from the guest charge, whose form is different from that obtained in the Debye–Hückel (linear Poisson–Boltzmann) theory. This means that the concept of renormalized charge, developed within the nonlinear Poisson–Boltzmann (PB) theory to describe the screening effect of the electrolyte cloud, fails at the free-fermion point. (ii) In the limit of an infinite bare charge, the induced electrostatic potential saturates at a finite value in every point of the electrolyte region. This fact confirms the previously proposed hypothesis of potential saturation.     

Density Profiles in a Classical Coulomb Fluid Near a Dielectric Wall. II. Weak-Coupling Systematic Expansions

In the framework of the grand-canonical ensemble of statistical mechanics, we give an exact diagrammatic representation of the density profiles in a classical multicomponent plasma near a dielectric wall. By a reorganization of Mayer diagrams for the fugacity expansions of the densities, we exhibit how the long-range of both the self-energy and pair interaction are exponentially screened at large distances from the wall. However, the self-energy due to Coulomb interaction with images still diverges in the vicinity of the dielectric wall and the variation of the density is drastically different at short or large distances from the wall. This variation is involved in the inhomogeneous Debye–Hückel equation obeyed by the screened pair potential. Then the main difficulty lies in the determination of the latter potential at every distance. We solve this problem by devising a systematic expansion with respect to the ratio of the fundamental length scales involved in the two coulombic effects at stake. (The application of this method to a plasma confined between two ideally conducting plates and to a quantum plasma will be presented elsewhere). As a result we derive the exact analytical perturbative expressions for the density profiles up to first order in the coupling between charges. The mean-field approach displayed in Paper I is then justified.     

Charge Correlations in a Coulomb System Along a Plane Wall: A Relation Between Asymptotic Behavior and Dipole Moment

Classical Coulomb systems at equilibrium, bounded by a plane dielectric wall, are studied. A general two-point charge correlation function is considered. Valid for any fixed position of one of the points, a new relation is found between the algebraic tail of the correlation function along the wall and the dipole moment of that function. The relation is tested first in the weak-coupling (Debye–Hückel) limit, and afterwards, for the special case of a plain hard wall, on the exactly solvable two-dimensional two-component plasma at coupling =2, and on the two-dimensional one-component plasma at an arbitrary even integer .     

Classical coulomb systems near a plane wall. I

The equilibrium structure of classical Coulomb systems bounded by a plane wall is studied near that wall. Several models are considered: the two-dimensional one-component plasma at a special value of the coupling constant (which makes the model exactly soluble), the two-dimensional and three-dimensional one-component and two-component plasmas in the weak-coupling limit (a Debye-Hückel type of approach is then used). Along a wall, the pair correlation functions decay only as an inverse power of the distancer, namely, asr ^–v for av-dimensional system (v=2,3). The one-body densities are also studied; the first BGY equation is used.     

Part II. Algebraic tails in three-dimensional quantum plasmas

We review various exact results concerning the presence of algebraic tails in three-dimensional quantum plasmas. First, we present a solvable model of two quantum charges immersed in a classical plasma. The effective potential between the quantum charges is shown to decay as 1/r ⁶ at large distances r. Then, we mention semiclassical expansions of the particle correlations for charged systems with Maxwell-Boltzmann statistics and short-ranged regularization of the Coulomb potential. The quantum corrections to the classical quantities, from orderh ⁴ on, also decay as 1/r ⁶. We also give the result of an analysis of the charge correlation for the one-component plasma in the framework of the usual many-body perturbation theory; some Feynman graphs beyond the random phase approximation display algebraic tails. Finally, we sketch a diagrammatic study of the correlations for the full many-body problem with quantum statistics and pure 1/r interactions. The particle correlations are found to decay as 1/r ⁶, while the charge correlation decays faster, as 1/r ¹⁰. The coefficients of these tails can be exactly computed in the low-density limit. The absence of exponential screening arises from the quantum fluctuations of partially screened dipolar interactions.     

Certain properties of an electrically insulated Coulomb plasma. Numerical simulation of microfields and thermodynamic properties. the problem of ball lightning

Many-body dynamics is used to study the (quasi-)steady state of a classical Coulomb plasma. The shortest relaxation time in such a plasma, for both the Debye screening and the thermodynamic properties, is the electron transit time over the average distance between ions. The steady-state energy of the Coulomb interaction of the particles and the steady-state potential near a fixed charge can be described well by the Debye-Hückel theory, even if there is less than a single particle in a Debye sphere, on the average. Distributions of instantaneous values of the microfields in the plasma are derived. The results calculated for the electron turning distance are compared with the results of the quasibinary theory. An attempt is made to link the anomalously long lifetime of the plasma of ball lightning to a retardation of the recombination of a classical Coulomb plasma in the absence of a stochastic external agent.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 10–23, February, 1992.     

Debye–Hückel Theory for Charged Aligned Needles and for Polyelectrolyte Solutions

We consider a simple extension of the familiar Debye–Hückel theory of electrolyte solutions (in which the ions are represented by spheres with embedded point charges) to study the thermodynamic properties and the phase diagram of ionic solutions in which the ions of at least one of the species are deformed into parallel and rigid needle-like ellipsoidal objects that have a continuous line of charge distribution along their axis of revolution. We examine two specific cases: (a) solutions comprising both cationic and anionic needles that are identical in every respect except for the charge sign, and (b) solutions in which only one ionic species is made up of parallel rigid needles while the other species is made up of point ions. The first system is the analog, for ionic needles, of the familiar restricted primitive model of electrolytes, while the second one is a very simple model for a polyelectrolyte solution. For both systems we investigate how the phase diagram is affected by the extent of deformation of the ions, as measured by the spatial spread of their charge distribution.     

Consequences of an attractive force on collective modes and dust structures in a strongly coupled dusty plasma

We present investigations of the combined effects of Debye–Hückel repulsive and overlapping Debye spheres attractive interaction potentials around charged dust particles on collective modes, phase separation and ordered structures in a strongly coupled dusty plasma. We obtain static and dynamical information via Molecular Dynamics simulations in the liquid and crystallized phases and identify the onset of an instability in the transverse mode, by using lattice summation method. The results are useful for understanding the origin of coagulation/agglomeration of charged dust particles and the formation of ordered dust structures in low-temperature laboratory and space plasmas.     

A simple treatment of color screening in a $q\bar{q}$ plasma

The formation of Hückel-Debye screening in a plasma is studied following as close as possible the procedures used in the usual electric case. The equivalent of the Debye length is found, but the screening as a result is found to be less efficient than for the Coulomb interaction, since the correlation functions decay as the exponential of the distance raised at a power less than one.Received: 9 February 2004, Revised: 1 April 2004, Published online: 2 July 2004     

Pressure and Stress Tensor in a Yukawa Fluid

Systems of particles interacting through a screened Coulomb potential of the Debye–Yukawa form are considered. The pressure is obtained from the stress tensor of the field corresponding to the Yukawa interaction, by a suitable statistical average. This approach is especially appropriate for systems living in a curved space. In a curved space, a self contribution to the pressure appears, and it is essential to take it into account for retrieving a correct pressure when the Yukawa interaction tends to the Coulomb interaction.     

Classification of the quantum two-dimensional superintegrable systems with quadratic integrals and the Stäckel transforms

The two-dimensional quantum superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion. There are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones. Analytic formulas for the involved integrals are calculated in all the cases. All the known quantum superintegrable systems with quadratic integrals are classified as special cases of these six general classes. The coefficients of the quadratic associative algebra of integrals are calculated and they are compared to the coefficients of the corresponding coefficients of the Poisson quadratic algebra of the classical systems. The quantum coefficients are similar to the classical ones multiplied by a quantum coefficient -?² plus a quantum deformation of order ?⁴ and ?⁶. The systems inside the classes are transformed using Stäckel transforms in the quantum case as in the classical case. The general form of the Stäckel transform between superintegrable systems is discussed.     

Charge Fluctuations in Finite Coulomb Systems

When described in a grand canonical ensemble, a finite Coulomb system exhibits charge fluctuations. These fluctuations are studied in the case of a classical (i.e., non-quantum) system with no macroscopic average charge. Assuming the validity of macroscopic electrostatics gives, on a three-dimensional finite large conductor of volume V, a mean square charge Q ² which goes as V ^1/3. More generally, in a short-circuited capacitor of capacitance C, made of two conductors, the mean square charge on one conductor is Q ²=TC, where T is the temperature and C the capacitance of the capacitor. The case of only one conductor in a grand canonical ensemble is obtained by removing the other conductor to infinity. The general formula is checked in the weak-coupling (Debye–Hückel) limit for a spherical capacitor. For two-dimensional Coulomb systems (with logarithmic interactions), there are exactly solvable models which reveal that, in some cases, macroscopic electrostatics is not applicable even for large conductors. This is when the charge fluctuations involve only a small number of particles. The mean square charge on one two-dimensional system alone, in the grand canonical ensemble, is expected to be, at most, one squared elementary charge.     

Large-Distance Behavior of Particle Correlations in the Two-Dimensional Two-Component Plasma

The model under consideration is a two-dimensional two-component plasma, i.e., a continuous system of two species of pointlike particles of opposite charges ±1, interacting through the logarithmic Coulomb interaction. Using the exact results for the form-factors of an equivalent Euclidean sine-Gordon theory, we derive the large-distance behavior of the pair correlation functions between charged particles. This asymptotic behavior is checked on a few lower orders of its -expansion ( is the inverse temperature) around the Debye–Hückel limit 0, and at the free-fermion point =2 at which the collapse of positive-negative pairs of charges occurs.     

Semiclassical mechanism for the quantum decay in open chaotic systems

We address the decay in open chaotic quantum systems and calculate semiclassical corrections to the classical exponential decay. We confirm random matrix predictions and, going beyond, calculate Ehrenfest time effects. To support our results we perform extensive numerical simulations. Within our approach we show that certain (previously unnoticed) pairs of interfering, correlated classical trajectories are of vital importance. They also provide the dynamical mechanism for related phenomena such as photoionization and photodissociation, for which we compute cross-section correlations. Moreover, these orbits allow us to establish a semiclassical version of the continuity equation.     

Janco’s disciples in Coulombland

As an introduction to the following two papers, we first give an explanation of the above general title. Both authors met Bernard Jancovici as a professor, and he was such an enthusiastic teacher that we felt like doing a thesis with him. This proved to be a very good idea! Indeed, Jancovici treated each of us both as a student who had much to learn and as a true collaborator from the start. Thus, we can say that we were born to scientific research thanks to him. Moreover, “Janco” also taught us skiing, hiking, wine tasting, etc. As he provided us with such a complete education, he deserves the title of our “spiritual father,” a title which he himself recognizes. In these papers, we would like to give an idea of the scientific approach which Janco taught us, and which is based on the first principles of statistical mechanics. We have chosen to exemplify this point of view through two nice problems of classical and quantum Coulomb systems which we studied after Janco addressed them alone or with us. “Coulombland” refers to systems of particles with Coulomb interactions at large distances. The Coulomb potential is defined as the solution of the Poisson equation inD dimensions. In three dimensions, it is the usual 1/r interaction, while in two dimensions, it takes a logarithmic form. The long range and the harmonicity of the Coulomb potential are responsible for a basic phenomenon called screening. A charged particle in a plasma is surrounded by a polarization cloud, whose total charge exactly compensates the charge of the particle it surrounds. Subsequently, the total effective potential created by a charge and its cloud at large distances is no longer the bare Coulomb potential, and the correlations are expected to decay faster. We review exact analytical results for the large-distance behavior of the correlations in two different situations, namely in the Kosterlitz-Thouless phase of the 2D classical Coulomb gas (Part I), and in the 3D quantum plasmas (Part II). Solvable models and systematic expansions starting from first principles exemplify Janco’s rigorous approach.     

Tomita representation of the Arnol'd cat map

The Tomita Hilbert-space representation of the Arnol'd cat map model of Benattiet al. is described and the operators representing physical quantities are defined for the classical and quantum cases. It is seen that the exponential decay of correlations is preserved upon quantization.