Electron Probability Density Research Articles

Electron-related properties in a GaAs/GaAlAs Ultra-thin Core/Shell Film through external field direction for energy and photonic devices

In this paper, we investigate electron-related properties in GaAs/GaAlAs Ultra-thin Core/Shell Film Quantum dot (UTFQD), examining the effects of external electric fields, impurity position, and aluminum fraction. Using the finite element method (FEM) within the effective mass approximation, we numerically solve the Schrödinger equation to study electron probability density, polarizability, and diamagnetic susceptibility. Our results reveal that electric fields can shift electron probability density due to the Stark effect. Without an electric field, the density centers near the shell's core. However, with electric fields, the density shifts opposite to the field's direction. Moreover, polarizability and diamagnetic susceptibility are influenced by impurity position and aluminum fraction. Polarizability increases with electric field intensity, indicating a transition from geometrical confinement to the Stark effect. Diamagnetic susceptibility rises with more aluminum, indicating stronger electron confinement. These findings suggest that adjusting electric fields, impurity position, and aluminum fraction in GaAs/GaAlAs core/shell QDs can control electron behavior, providing flexibility for photonic devices. This study offers a foundation for optimizing quantum dot-based components in photonics.

Hopf-link GaAs-AlGaAs quantum ring under geometric and external field settings

Within the framework of effective mass approximation and with the use of finite element method, we calculate the energy states of an electron in a double toroidal quantum ring with a Hopf link structure. The study includes the influence of externally applied static electric and magnetic fields, and considers different geometric combinations of radii and link positions. We have found that geometric manipulations have significant impact on the energy values and the distribution of electronic probability densities in either one or both linked rings. Similar effects can be achieved with the application of the electromagnetic probes.

Inversion Symmetry Breaking in the Probability Density by Surface‐Bulk Hybridization in Topological Insulators

AbstractThe probability density distribution is analyzed in a topological insulator (TI) slab of finite thickness, where the bulk and surface states are allowed to hybridize. By using an effective continuum Hamiltonian approach as a theoretical framework, the wave functions are analytically obtained for each state near the ‐point. The results reveal that, under particular combinations of the hybridized bulk and surface states, the spatial symmetry of the electronic probability density with respect to the center of the slab can be spontaneously broken. This symmetry breaking arises as a combination of the parity of the solutions, their spin projection, and the material constants.

The Shape Complexity of a Particle in the Circular Quantum Well with Homogeneous Perpendicular Magnetic Field

The shape complexity of a particle confined in the circular quantum well with an external homogeneous perpendicular magnetic field is investigated. First, the Shannon entropy and the averaging electron probability density, as well as the shape complexities in the position and momentum spaces for several quantum states, are computed and discussed. The quantum confinement effect on the shape complexity of this system is extensively explored. Then, the influence of the magnetic field on the shape complexities of the particle confined in the circular quantum well is discussed. It is demonstrated that the shape complexity exhibits scaled invariance under the influence of the magnetic field. However, when the radius of the quantum well is fixed, the shape complexity of the given quantum state changes with the magnetic field strength. Results suggest that the shape complexity of a particle confined in the quantum well can be changed by varying both the strength of the magnetic field and the size of the well. This study has some practical applications in quantum information measurement of the confined particle and can guide future researches for the disorder characteristics of confined particle in semiconductor quantum dots.

A particular behavior of levels energy evolution and theoretical study of the quantum probability distribution in GaAs/AlGaAs asymmetric stepped semiconductor quantum wells for terahertz optoelectronics

The study of the evolution of the energy levels as a function of the step layer’s width Lw2 in an asymmetric stepped quantum well AlxGa1-xAs/GaAs/AlyGa1-yAs/AlxGa1-xAsx=24%,y=14%, revealed a particular behavior, marked by the presence of plateaus within certain deep well’s width, Lw2, ranges. We initially computed the eigenenergies and eigenfunctions by numerically solving the Schrödinger equation. This was done within the effective mass and envelope function approximation, taking into account the influence of non-parabolicity. To explain the origin of the plateaus and their impact on intersubband transition energies, we determined the distribution of the probability density of the presence of the electron in each region of the stepped quantum well. The results of these calculations demonstrated that the electron's probability density is higher in the step well than in the deep one, particularly for electrons with energy levels above 113.47 meV. This elevation in probability density exhibits maximum values that remain nearly constant within specific Lw2 domains, providing an explanation for the observed energy plateaus. Such plateaus allow more energy tunability and thus open up additional possibilities for applications in the THz domain either in the linear or nonlinear optics.

Nonlinear optical properties of coupled quantum dots in peanut configuration

ABSTRACT This paper presents a theoretical investigation of nonlinear properties of coupled quantum dots in the peanut configuration. A complete Hamiltonian is formulated using the adiabatic method and the confinement energy is represented in a cylindrical coordinate system. The effective energy for the slow subsystem is analysed and graphed as a function of a fixed value of axial coordinate for both the asymmetric and symmetric peanut QD cases. The electron motion in the presence of an electric field in the vertical direction is considered, and the eigenfunctions and energy spectrum of electron motion are determined for the same direction utilising the finite element method. The dependence of the first three energy levels on the electric field is shown, and the electron probability density for the ground state and the first excited state is plotted. In addition, calculations for the nonlinear optical properties are presented, particularly optical rectification, second harmonic generation and third harmonic generation. The results demonstrate that these properties can be effectively controlled by varying the external electric field. The findings suggest that coupled peanut QDs hold significant potential for applications in high-performance optoelectronic devices.

Investigation of band parameters and electrochemical analysis of multi core-shell CdSe/CdS/ZnS quantum dots

Highly emissive core−multi-shell (CdSe/CdS/ZnS) quantum dots (QDs) within size range of ∼ 2 nm–5 nm were prepared via hot injection method. Capping agent (polymers) were used to increase the lifetime of QDs, without a protective atmosphere. Red shift was observed in an absorption spectrum of QDs peaks at 362 nm, 365 nm, 558 nm, and 556 nm for CdSe/CdS and CdSe/CdS/ZnS with respect to CdSe core. Voltammetry measurements (Eg ∼ 2. 09 eV) for band gaps were in good agreement with the UV–visible spectroscopic data (Eg ∼ 2.65 eV). QDs exhibited three distinguished peaks comparing to (111), (220), and (311) planes of cubic structures of CdSe, CdSe/CdS and CdSe/CdS/ZnS within crystallite size of 20–27 nm as observed by XRD spectra. A larger active surface area 0.06 cm−2, computed by CV increased the catalytic activity of sensors by exposing the electroactive sites. Linear, nonlinear optical susceptibility and absorption coefficients were computed by an effective mass model. Quantum confinement was confirmed by taking electron probability density as function of radial distance of quantum dots.

5p1/2nℓj autoionizing states of strontium for 0≤ℓ≤5

We report a comprehensive experimental and theoretical study of the autoionization of core-excited $5{p}_{1/2}n{\ensuremath{\ell}}_{j}$ strontium Rydberg states for intermediate values of ${\ensuremath{\ell}}_{j}$ in the range $0\ensuremath{\lesssim}{\ensuremath{\ell}}_{j}\ensuremath{\lesssim}5$. For a given value of ${\ensuremath{\ell}}_{j}$ autoionization, which results from the energy exchange between the two excited valence electrons, is found to scale as $\ensuremath{\sim}1/{n}^{3}$, mirroring the decrease in the Rydberg electron probability density in the vicinity of the inner $5{p}_{1/2}$ electron cloud. In contrast, the ${\ensuremath{\ell}}_{j}$ dependence of the autoionization rate is more complex. For the larger values of ${\ensuremath{\ell}}_{j}$ $(\ensuremath{\ge}4)$, the autoionization rate decreases as ${\ensuremath{\ell}}_{j}^{\ensuremath{-}5}$ due to the centrifugal barrier, which prevents the Rydberg electron from penetrating the core and inner $5{p}_{1/2}$ electron cloud. For the low-${\ensuremath{\ell}}_{j}$ core-penetrating Rydberg states, the scattering phase shift induced by the effective core potential becomes important in determining both the autoionization rates, which no longer show a monotonic variation with ${\ensuremath{\ell}}_{j}$, and the quantum defects.

Impact of nanoscale fluctuations and cap-layer thickness in buried InGaN single quantum wells probed by tip-enhanced Raman scattering

Ternary semiconductors such as InGaN thin films, quantum wells, and superlattices commonly exhibit alloy fluctuations that become increasingly pronounced with higher In-content. The thickness fluctuations of quantum wells and their thin cap-layers further introduce nanoscale inhomogeneities that alter the potential landscape. In this work, we present a combined theoretical and experimental study of InGaN single quantum wells with thin GaN cap-layers to unravel the influence of cap-layer thickness, compositional inhomogeneity, and thickness fluctuations on their electronic and optical properties. A pronounced spectral shift of quantum well emission for thin cap-layers between 1 and 10 nm is observed by micro-photoluminescence spectroscopy. The origin of this shift is explained by calculations of electronic band profiles and probability density overlap of carriers in the quantum well. The impact of alloy fluctuations and homogeneity for different cap-layer thicknesses is studied on both the microscale and nanoscale using UV micro-Raman scattering and tip-enhanced Raman spectroscopy (TERS). On the microscale, the alloy composition as determined by micro-Raman mapping appears very homogeneous except for the thinnest 1 nm cap-layer where small fluctuations are visible. On the nanoscale, TERS reveals local fluctuations on a 20–30 nm length scale. The influence of the cap-layer thickness on the TERS spectra is discussed regarding both the nanoscale homogeneity and the depth resolution of the near-field Raman scattering technique. Our results demonstrate the capabilities of TERS to resolve nanoscale thickness fluctuations and compositional inhomogeneities in ultra-thin semiconductor layers, even when they are buried by thin cap-layers with thicknesses below 10 nm.

Calculations of Rényi entropy, Tsallis entropy and Onicescu information energy for helium, lithium and beryllium atoms using an analytic correlated wave function

A comparative study using an analytical correlated wave function for two K-shell active electrons and two electron uncorrelated wave function for two K-shell bare electrons, on information theoretic quantities such as, Rényi entropy, Tsallis entropy and Onicescu information energy are presented for helium, lithium and beryllium. Obtained differences can be interpreted as a measure of electron correlation and quantified for these quantities. Correlated one electron probability density is delocalized with respect to uncorrelated one-electron probability which increases/decreases the correlated values of these quantities. Results for helium atom are in good agreement with the available data in the literature.

Inequality-induced similarity measures for radial electronic probability densities in hydrogenic atoms

Inequality-induced similarity measures for radial electronic probability densities in hydrogenic atoms

Hydrogen states described by solutions of the Dirac equation: Role of spinor invariants

The solution of the Dirac equation with the Coulomb potential is used to analyze bound electron states in a hydrogen atom. The analysis is based on the fact that such states are characterized by a set of quantum numbers which describe definite values of the complete set of physical quantities that can be determined simultaneously. This set includes the energy, square of the total angular momentum, one of its component and a spinor invariant. The latter, spinor invariant gives two-valued quantum number which determines the sign of its eigenvalue. In addition to the known Dirac and Johnson–Lippman invariant, there exists a new one. Operators of these three spinor invariants do not commute between themselves which results in the degeneracy of the energy levels with respect to the two-valued quantum number. Three different systems of the eigenbispinor corresponding to the three spinor invariants are obtained and the generalized solution with free parameters is calculated. Variation of the free parameters transforms one particular solution into any other. It is shown that the electron probability densities and spin polarizations in an electron cloud depend essentially on the invariant set, demonstrating physical difference of the states corresponding to different spinor invariants.

Investigation of the high-order harmonic generation for a solid with a defect by Bohmian trajectories

We theoretically investigate the high-order harmonic generation (HHG) for a solid by solving the one-dimensional time-dependent Schrödinger equation. The numerical results with the potential well depth of the defect and are demonstrated. The intensity of the second plateau of the harmonic spectra can be enhanced with the small potential well depth of the defect. We investigate the emission time of harmonics in terms of the time-frequency analysis, which shows that with the potential well depth of the defect , one emission peak is weakened and the other is enhanced. By analyzing the time evolution of the electronic probability density distribution and the Bohmian trajectories, the electrons ionized in the negative-x direction will pass directly through the defect to the positive-x direction, which is different from the potential well depth of the defect In addition, we show a velocity curve to illustrate further the movement process of the ionized electrons.

Characteristics of the molecular above-threshold ionization by a bichromatic elliptically polarized field with co-rotating components

We address ionization of a diatomic molecule by a bichromatic elliptically polarized field with co-rotating components. Using the strong-field approximation we investigate symmetry properties of the photoelectron momentum distribution and explore the minima which appear in the photoelectron spectra. We distinguish two types of minima: (i) two-center interference minima which appear due to the destructive interference of the contributions of two electron wave packets emitted from the two centers of the diatomic molecule and (ii) the one-center minima which are caused by the interference of the parts of the wave packet emitted from the same atomic center at different times. The position of the two-center interference minima depends on the molecular orientation. When a molecular orbital is modelled using the atomic orbitals of a specific parity, the position of the two-center interference minima does not depend on the ellipticity of our driving field. However, when a molecular orbital consists of both odd and even atomic orbitals the interference of their contributions and the position of the minima depend on the ellipticity. The position of the interference minima in the photoelectron momentum plane is confirmed using the saddle-point method. The position and the number of the one-center minima do not depend on the molecular orientation, but they strongly depend on the ellipticity of the field components. Finally, comparing the photoelectron spectra of the CO molecule with the spectra of homonuclear molecules and the NO molecule we show that the electron probability density distribution plays a significant role for the high-energy rescattered electrons.

Linear and nonlinear optical properties of a superlattice with periodically increased well width under electric and magnetic fields

In this paper, we have studied the electronic and optical properties of GaAs/AlxGa1-xAs superlattice with periodically increased well width. Under effective mass approximation, the finite element method is used to obtain wavefunctions and corresponding energy eigenvalues for several electric field (F) and magnetic field (B) values. We have shown that the increasing well width has a major effect on the localization of the first two energy states. The direction of the applied electric field shifts the localization position of the probability density of electrons to the left and right. For B = 0 (according to the parameters used), F = 5 kV/cm (especially for the difference between the first two energy levels (E12)) is a critical value. While the E12 value decreases in the range from −30 kV/cm to 5 kV/cm, it increases for the 5 < F < 30 kV/cm range. This behavior causes a red or blue shift in the optical spectrum. Also, F = −30 kV/cm causes more change in the structure than F = 30 kV/cm. In addition, the localization of the electrons is observed in the center of the superlattice under applied magnetic fields. The optical absorption coefficients and the refractive index changes are affected by applied F and B intensities. We can say that the electro-optical features of the superlattice have changed significantly with the combined effect of F and B values. This is desired for semiconductor optical device applications to have stable performances.

On electron propagation in triangular graphene quantum dots

Tight-binding models of electron propagation in single-layer triangular graphene quantum dots with armchair and zigzag edges are developed. The electron hoppings to the nearest and next-to-nearest neighbours on the honeycomb lattice as well as interactions with the confining Dirichlet and Neumann walls are incorporated into the resulting tight-binding Hamiltonians. Associated to the irreducible crystallographic root system A 2, the armchair and zigzag honeycomb Weyl orbit functions together with the related discrete Fourier–Weyl transforms provide explicit exact forms of the electron wave functions and energy spectra. The electronic probability densities corresponding to the armchair and zigzag dots are evaluated and their contrasting behaviour exemplified.

Spin relevant invariants and the general solution of the Dirac equation for the Coulomb fields

The Dirac equation with the Coulomb potential is studied. A new spin relevant invariant is introduced, which does not commute with the Dirac and Johnson–Lippmann integrals of motion. The solutions of the Dirac equation compatible with each of the above three invariants are found and shown to be different. The explicit expressions for the Dirac bispinors are found for each of the three spin related invariants. The generalized invariant is constructed and corresponding to it general solution of the Dirac equation with the Coulomb potential is obtained. The general solution contains free parameters, whose particular values reduce it to any of the three particular solution. It is shown that the distributions of the electron probability density and expectation values of the electron spin direction in the hydrogen-like energy spectrum depend essentially on the spin relevant invariant. This demonstrates physical difference of the states corresponding to different invariants, which can be experimentally manifested.

Machine learning techniques in the examination of the electron-positron pair creation process

We employ two machine learning techniques, i.e., neural networks and genetic-programming-based symbolic regression, to examine the dynamics of the electron-positron pair creation process with full space–time resolution inside the interaction zone of a supercritical electric field pulse. Both algorithms receive multiple sequences of partially dressed electronic and positronic spatial probability densities as training data and exploit their features as a function of the dressing strength in order to predict each particle’s spatial distribution inside the electric field. A linear combination of both predicted densities is then compared with the unambiguous total charge density, which also contains contributions associated with the independent vacuum polarization process. After its subtraction, the good match confirms the validity of the machine learning approach and lends some credibility to the validity of the predicted single-particle densities.

Sensitivity of correlation effects and Shannon entropy in Be@C60 to the nature of confinement potentials

The multi-configuration Dirac-Fock (MCDF) method is used to study the correlation effects in Be@C60 by investigating the localization properties of the electronic probability density distribution. The confinement environment is simulated by two types of model potentials: (A) the commonly employed compact annular square well (ASW) and (B) the diffusive Gaussian annular square well (GASW). The present work aims to investigate the sensitivity of correlation effects and localization properties to the contrasting nature of the confinement potentials. To accomplish this, the variation of the correlation energy and Shannon’s correlation information entropy with the depth of both confinement potentials are presented to give an idea of the sensitivity of these parameters to the nature of the confining potential. The present work concludes that the correlation energy and the associated electronic structure attributes are extremely susceptible to the nature of the confining potential.

Exploration of the high-order harmonic generation from periodic potentials by Bohmian trajectories

The recombination processes of the electrons in solid are illustrated by solving the time-dependent Schrödinger equation. The results show that the Bohmian trajectories and the time evolution of the electronic probability density agrees very well, which demonstrates that we can use the Bohmian trajectories to investigate the recombination processes of the electrons in solid. We select the region where the probability density of the electron reached the strongest and the weakest as the initial position of the calculated Bohmian trajectories, one can see that the Bohmian trajectories have similar structures. In addition, our results show that the emission time of the solid high-order harmonic generation (HHG) spectra from the time-frequency distribution agrees well with the time that the Bohmian trajectories change direction. By regulating the phase of the electric field, the electrons can move farther, which will result in the broad cutoff of the HHG. We have also demonstrated that the similar structure of the Bohmian trajectories of the solid with a defect for different initial positions disappears due to the broken periodic structure, which further illustrates that the motion processes of the electrons in solid depend on the structure of the solid.