Print version ISSN 0103-9733
Braz. J. Phys. vol.34 no.2a São Paulo June 2004
Molecular dynamics simulations of two-dimensional clusters of charges
Simara S. de MoraesI, II; Débora CoimbraI; J. Ricardo de SousaI; Hidemberg Ordozgoith da FrotaI
IDepartamento de Física, Universidade Federal do Amazonas, 69077000, Manaus, AM, Brazil
IICentro Universitário Luterano de Manaus, Universidade Luterana do Brasil, 69077730, Manaus, AM, Brazil
We study structural and spectral properties of finite classical systems of N two-dimensional charged particles,confined by a parabolic potential µ, and interacting via inverse power-law potentials µ 1/rn' . Molecular dynamics simulations are performed for different cluster sizes (N = 30 to 230) and n, n' values 1, 2, 3 and 10. We also analyze the phase transition from a ring-like configuration to a Wigner structure as a function of parameter n' and anisotropy. We compare our results with Monte Carlo simulations of Bedanov and Peeters, obtaining good agreement. In addition, we determine the Voronoi structure of the cluster. Our work complements that of Cândido, Rino and Studart, who analyzed confinement in a screened parabolic potential.
The two-dimensional classical atom is a model system in which N equally charged classical particles move in a plane, and are kept together by a parabolic potential. For each value of N we have at least one stable configuration. Examples of 2D Coulomb clusters are electrons on the surface of liquid helium  and electrons in quantum dots . After the discovery of Wigner crystallization  of electrons on the surface of liquid helium  (a two-dimensional system), there has been considerable theoretical and experimental progress in the study of the mesoscopic system consisting of a finite number of charged particles, which are laterally confined by a parabolic potential and interact each other through a potential. This system is the classical analog of the well known quantum dot problem, if the interaction potential is the Coulomb one. In a finite system there is a competition between the bulk triangular lattice and the circular confinement potential that tries to force the particles into a ringlike configuration. These quantum dots are atomlike structures that have interesting optical properties and may be of interest for single-electron devices . These systems and their configurations have been observed experimentally , and are important in solid-state physics , plasma physics  as well as in atomic physics . Bolton and Rössler  implemented Monte Carlo simulation of isotropic parabolic confinement, to investigate the minimum energies and spatial distributions for small numbers (1-40) of particles.
Ringlike configurations in two-dimensional clusters were systematically investigated by Bedanov and Peeters  using Monte Carlo method and a Mendeleev-type of table for these classical atomlike structures was constructed, for up to 230 particles. The spectral properties of the ground-state configurations were studied by Schweigert and Peeters  and generalized to screened Coulomb interactions by Cândido et al.  and to logarithmic interactions [14, 15, 16]. In particular, Partoens and Peeters  compared the values obtained using Monte Carlo simulation with the Thomson model for a two-dimensional system constituted for N classical particles confined by potentials µ rn.
Recently, Kong et al.  pointed out the importance and treatment of metastable states in numerical simulations. They also investigated the stability of the ground-state configurations against the functional form of the confinement potential and the exact form of the interparticle interaction potential.
In this work, we study structural and spectral properties of finite classical systems of N charged two-dimensional particles, confined by a parabolic potential interacting by inverse power potentials. The molecular dynamics simulation was performed to simulate clusters containing about 230 particles.
This paper is organized as follows. In Section 2, we described the model and introduced the dimensionless formulation. In Section 3, we presented our results, calculating the energy and the space arrangement of particles. Finally, our conclusions were performed out in last section.
2 Model and Method
We study a 2D system consisting of a finite number N identical classical particles, with dielectric constant e, confined by means of a potential of the form
where m is the particle mass, (xi, yi) are the coordinates of the particles, whose vectorial position is ri, w is the frequency of the confinement potential, a is the anisotropic parameter in the direction of the axis y. The particles interact through the potential
where q is the charge of each particle and e is dielectric constant of the medium.
The total potential energy is
Eq. (4) can be written in a dimensionless form , expressing the coordinates, the energy, a temperature and the force in terms of the units r0, E0, T0 e F0, respectively, given by
where g = mw2. In terms of these dimensionless units, the Hamiltonian of Eq. (4) becomes
This potential has also been studied for Partoens and Peeters , using Monte Carlo methods, as previously mentioned.
Thus, the energy involves the following parameters: the power n of the confinement potential, the power n¢ of the interaction potential, number N of particles and a. The Hamiltonian of Eq. (9) was studied by Partoens and Peeters , who generalized the Thomson model to investigate the effect of different interaction potential between particles on the ground-state configurations. They compared their results with numerical Monte Carlo simulations. For n = 2 and n¢ = 1, the N particles system is under a Coulomb potential and are confined by a parabolic potential. This case was studied by Bedanov and Peeters  using the Monte Carlo method, with a = 1. In agreement with Ref. , the typical values given by the Eqs. (5) - (8) are
1. For electrons above liquid helium (q = e, e = 1, m = me) with confinament potential such that w = 0.1 meV:r0 = 2800 Å , E0 = 5.1 meV, T0 = 60 K e F0 = 29.4 × 1016 N.
2. For electrons in a GaAs heterostructure, with a typical energy of quantum dot confinement w = 1 meV:r0 = 630 Å , E0 = 1.7 meV, T0 = 20 K e F0 = 6.32 × 1016 N.
The Molecular Dynamics Simulation (MD) is a method to solve classical Newton's equations of movement of a particle system. Early on, Alder and Wainwright  applied it to study the hard sphere fluid. Afterwards, many authors analyzed Lennard-Jones fluids using this method [18, 19]. In this work, we use the predictor corrector algorithm to solve the equations of motion for the Hamiltonian of Eq. (9). For references on basic MD implementation see .
In Table I and II, we presented the minimum state configuration and energy E/N for parabolic confinement and a logarithmic interaction between particles, as well as an interaction proportional to |ri rj|1, |ri rj|2 and |ri rj|3. The last represents an original contribution of this work. For simplicity, we adopt the notation r = |ri rj| . For the logarithmic potential, for N of 2 to 6 the particle arrangement is a ring; for N = 7 to N = 16 the particles form two concentric rings; for N = 17 to N = 33, three concentric rings. For N between 34 and 51 we obtain four rings. Our results are in a good agreement with those obtained by Partoens and Peeters . Despite this, for more than one ring, our results for the maximum number of particles that are distributed in a certain number of rings in general differ from those of . Perhaps this discrepancy is a consequence of the approach used in the Thomson model. For N > 6, the system is unstable if we impose only one ring, because the excess charges are placed in the center of the ring. Considering the confinement potential with n = 2 and a = 1, and the interaction potential with n¢ = 1, 2 and 3, in Eq. (9), we found for a repulsive Coulomb interaction (n¢ = 1) a single ring for N varying from 2 to 5; two rings for N varying from 6 to 15; three rings for N = 16 to N = 31; and four rings for N = 32 to N = 51.
We compared our results with those obtained with the Thomson model  shown in Table III, observing a good agreement. We also compared our results with obtained with the Monte Carlo method (MC) for Bedanov e Peerters . The same number of rings are found with DM and MC, because both methods have a good precision. The number of charges per ring is also generally in agreement. Only for N = 21, 29 and 51 the rings do not have the same number of charges. For an interaction proportional to the square of the distance and N varying of 2 to 5 we obtained only one ring; for N varying of 6 to 15 two rings; for N = 16 the N = 29 three rings; for N = 30 to 48 four rings and for N = 49 to 51 five rings. We also observed good agreement with the results of Partoens and Peeters  for systems with less than 18 particles. The discrepancies for larger numbers of particles occur because to stabilize the system in Thomson model, the internal particles are considered located in the external orbit center. Koulakov and Shklovskii  showed for N large, considering Coulomb interaction, the energy is proportional to the particle number as E ~ N5/3. This behavior is plotted in Fig. 1 for all potentials previously mentioned.
Fixing the kind of interaction (Coulomb), we present in Table IV the distribution of charges and respective energy of the ground-state to several confinement potential. In Fig. 2 we show the configurations, with N = 13 and n assuming the values of n = 1, 2, 3, 7, 10 and 11 in confinement potential. As n increases the particles shift to external rings, but the diameter decreases.
In Fig. 3, we presented the Voronoi structure of the particles for the parabolic confinement potential and the Coulomb interaction among the particles with n = 2 (a) and n = 5 (b). We observed that the internal rings form a Wigner structure with hexagonal symmetry for n = 2. However, for n = 5, the particles tend to migrate to the external layers, breaking the hexagonal symmetry in the central area of the distribution. We also analyzed the anisotropy in the y direction, as presented in Fig. 4. As a increases, the ring configurations tend to elliptical ones.
4 Final Remarks
We present results of numerical simulation (MD) of confined interacting particles, in clusters of 30 to 230 particles. The energies and configurations obtained are in agreement with literature values. The importance of these studies lies in their reduced complexity in two-dimensional systems, in relation to three-dimensional ones.
We find that the particle arrangement is ring-like in the ground state. These configuration vary with anisotropy, becoming increasingly elliptical. The energy could be fitted as N5/3. This study highlights the possibility to analyze the energy variation with particle number, extensivity, phase transitions and critical points, taking to account metastable states and field interactions.
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Received on 6 January, 2004