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We study systems of two and three electrons confined to circular rings. The electrons are considered spinless, and we assume that one electron occupies a single ring. We use the framework of such a model to calculate the linear entropy and, thus, the spatial entanglement between the confined electrons. The geometry of the problem for the case of two electrons incorporates situations in which the planes of the two rings form an arbitrary angle with each other. The resulting Schrödinger’s equation is solved numerically with very high accuracy by means of the exact diagonalization method. We compute the ground state energy and entanglement for all configurations under consideration. We also study the case of three electrons confined to identical, parallel and concentric rings which are located in three different equidistant planes. The vertically separated system of rings is allowed to gradually merge into a single ring geometry, which would represent the equivalent system of a ring with three electrons. It is observed that the system of three electrons gives rise to a richer structure, as the three rings merge into a single one. 

Systems composed of several multi-layer compounds have been extremely useful in tailoring different quantum physical properties of nanomaterials. This is very much true when it comes to semiconductor materials and, in particular, to heterostructures and heterojunctions. The formalism of a position-dependent effective mass has proved to be a very efficient tool in those cases where quantum wells emerge either in one or two dimensions. In this work, we use a variety of mathematical theorems, as well as numerical computations, to study different scenarios pertaining to choices of a specific piecewise constant effective mass for a particle that causes its energy eigenvalues to reach an extremum. These results are relevant when it comes to practical technological applications such as modifying the optical energy gap between the first excited state and the ground state energy of the system. At the end of our contribution, we also question the physical validity of some approximations for systems with particles that possess a position-dependent mass especially for those cases in which the mass distribution is divergent. 

We study a two-dimensional system of interacting electrons confined in equidistant planar circular rings. The electrons are considered spinless and each of them is localized in one ring. While confined to such ring orbits, each electron interacts with the remaining ones by means of a standard Coulomb interaction potential. The classical version of this two-dimensional quantum model can be viewed as representing a system of electrons orbiting planar equidistant concentric rings where the kinetic energy may be discarded when one is searching for the lowest possible energy. Within this framework, the lowest possible energy of the system is the one that minimizes the total Coulomb interaction energy. This is the equilibrium energy that is numerically determined with high accuracy by using the simulated annealing method. This process allows us to obtain both the equilibrium energy and position configuration for different system sizes. The adopted semi-classical approach allows us to provide reliable approximations for the quantum ground state energy of the corresponding quantum system. The model considered in this work represents an interesting problem for studies of low-dimensional systems, with echoes that resonate with developments in nanoscience and nanomaterials. 

Abstract It is common in mesoscopic systems to find instances where several charges interact among themselves. These particles are usually confined by an external potential that shapes the symmetry of the equilibrium charge configuration. In the case of classical charges moving on a plane and repelling each other via the Coulomb potential, they possess a ground state à la Thomson or Wigner crystal. As the number of particles N increases, the number of local minima grows exponentially and direct or heuristic optimization methods become prohibitively costly. Therefore the only feasible approximation to the problem is to treat the system in the continuum limit. Since the underlying framework is provided by potential theory, we shall by‐pass the corresponding mathematical formalism and list the most common cases found in the literature. Then we prove a (albeit known) mathematical correspondence that will enable us to re‐discover analytical results in electrostatics. In doing so, we shall provide different methods for finding the equilibrium surface density of charges, analytical and numerical. Additionally, new systems of confined charges in three‐dimensional surfaces will be under scrutiny. Finally, we shall highlight exact results regarding a modified power‐law Coulomb potential in thed‐dimensional ball, thus generalizing the existing literature. 

Abstract It is conjectured that the Pauli exclusion principle alone may be responsible for a particular geometric arrangement of confined systems of identical fermions even when there is no interaction between them. These geometric structures, called Pauli crystals, are predicted for a two‐dimensional (2D) system of free fermions under harmonic confinement. In this work, the possibility of this outcome is pursued and a theoretical model is considered that may capture both qualitatively and quantitatively, the key features of the abovementioned setup. The results for and 6 particles show that the minimum energy configuration corresponds to and is in good quantitative agreement with the reported values of Pauli crystals seen in single‐shot imaging data obtained via the configuration density technique. Numerical results for larger systems of and 30 particles show that the crystalline configurations observed are not the same as the classical Wigner crystal structures that emerge should the confined charged particles interact with a Coulomb potential. An important question floated is whether such crystalline structures do really exist in a quantum system or whether they are artifacts of the methods used to analyze them.