Symmetric Potential Well Research Articles

Effects of hydrostatic pressure, aluminium composition, and external magnetic field on intersubband and intrasubband transitions characteristics in symmetric potential quantum wells

Aluminium composition (λ), magnetic field (B), and hydrostatic pressure (P) influences on magneto-optical (MO) properties of intra- and inter-subband transitions in GaAs/AlλGa1−λAs symmetric QWs for four models of confined optic phonons are studied and compared. The findings indicate: the MO characteristics of intra- and inter-subband transitions are significantly influenced by B, λ, and P for phonon absorption and emission. The e–p interaction intensity decreases as P increases, whereas it increases as λ increases. The influences of λ and P on phonon absorption are more significant than those on emission. E–p interaction intensity dependence on λ and P for intra-subband transitions is stronger than that for inter-subband ones. The e–p interaction in QWs is well described by Knipp–Reinecke and Huang–Zhu models. To reduce the influence of P on MO properties, we can increase T and λ or decrease B. At T≈600 K, the influence of P is insignificant and can be neglected.

Effects of vacancies on quantum transport of zigzag graphene nanoribbons

Using the Green’s function method, we investigate the effect of vacancies on conductance and local density of states (LDOS) in zigzag graphene nanoribbons within a symmetric and asymmetric potential wells. The results differ from ideal ribbons in that there are no conductance plateaus and a decline in conductance for varied N (number of atoms in the unit cell) and M (number of unit cells) values following vacancy incorporation. LDOS and conductance in symmetrical and asymmetrical circumstances are examined alongside vacancies and accordingly, they are affecting the respective edge states.

Exact solutions for time-dependent complex symmetric potential well

Using the pseudo-invariant operator method, we investigate the model of a particle with a time-dependent mass in a complex time-dependent symmetric potential well V (x, t) = if (t) |x|. The problem is exactly solvable and the analytic expressions of the Schrödinger wavefunctions are given in terms of the Airy function. Indeed, with an appropriate choice of the time-dependent metric operators and the unitary transformations, for each region, the two corresponding pseudo-Hermitian invariants transform into a well-known time-independent Hermitian invariant which is the Hamiltonian of a particle confined in a symmetric linear potential well. The eigenfunctions of the last invariant are the Airy functions. Then, the phases obtained are real for both regions and the general solution to the problem is deduced.

Asymmetric tristable energy harvester with a compressible and rotatable magnet-spring oscillating system for energy harvesting enhancement

This paper proposes an asymmetric tristable energy harvester with a compressible and rotatable magnet-spring oscillating system to enhance the energy harvesting performance. The linear compressible and rotatable magnet-spring oscillating system is composed of two external magnets, a metal block and two parallel springs. The two external magnets are symmetrically located on the metal block and fixed to the base through the two parallel springs. Due to different compression of the two springs under variable magnetic force, the two external magnets can move along the spring's compression direction and rotate around the center of the metal block, thus to generate tunable and asymmetric potential function. The general governing equations and asymmetric potential function of the proposed harvester are established with Hamilton principle and point dipole method, respectively. The effects of system parameters on the nonlinear dynamic performances of the proposed harvester are numerically investigated. Experiments are carried out to validate the simulations. The results demonstrate that moderate spring stiffness can not only reduce the potential energy barrier and the threshold of snap-through, but also broaden the bandwidth of interwell motion, as well as enhance the energy harvesting efficiency. When the spring stiffness k = 900 N/m, the threshold value of snap-through of the TEH-A is 8 m/s2, the bandwidth of inter-well motion is 2.0∼4.5 Hz, and the maximum harvesting voltage and power reach 6.8 V and 4.6 × 10−5 W, respectively. These values are far better than the tristable energy harvester with symmetric potential wells, whose corresponding values are 3.8∼4.2 Hz, 9.8 m/s2, 0.81 V and 6.6 × 10−7 W, respectively.

THE THEORY OF QUASISTATIONARY ENERGY STATES IN A SPHERICAL POTENTIAL WELL

Quasi-stationary electronic states in a spherically symmetric semiconductor potential well are determined in the semiclassical approximation based on the calculation of matrix elements of the transfer matrix. In this case, three regions of the potential well were taken into account, which differ from each other in geometric dimensions. The calculation was carried out by solving the Schrödenger equation in a spherical coordinate system. KEYWORDS: quasistationary electronic states, spherically symmetric semiconductor potential well, semiclassical approximation, matrix elements of the transfer matrix

Improving the performance of a tri-stable energy harvester with a staircase-shaped potential well

Scavenging energy from ambient vibration by tri-stable energy harvesters (TEHs) has been focused in recent years. Generally, classical TEHs adopt a design of symmetric potential wells. To improve the harvesting efficiency, a TEH with a staircase-shaped potential well (TEH-SSP) is proposed. This staircase-shaped potential well is established by adjusting the distances from the fixed magnets to the axis of symmetry. The distributed model of the proposed energy harvester is established, and the dynamical equations are derived by using the energy approach method. The nonlinear characteristics are explored theoretically and validated by experiments. Compared with the classical TEH with symmetric potential (TEH-SP), the proposed TEH-SSP could execute snap-through from a low frequency, and can obviously enlarge the frequency bandwidth of snap-through. Thus it can create a more dense high output voltages under weak random excitation. Both the theoretical analysis and experimental results prove the effectiveness of the proposed asymmetric design and vindicate that it is very beneficial for harvesting energy from the ambient vibration excitation. This work may serve as a novel insight into the design of more efficient harvesters with a broad bandwidth.

Pair production in asymmetric Sauter potential well

Electron-positron pair production in asymmetric Sauter potential well is studied, where the potential well has been built as the width of the right edge fixed but the left side of the well is changeable at different values. We study the momentum spectrum, the location distribution and the total pair numbers in this case of asymmetric potential well and compare them with the symmetric case. The relationship between created electron energy, the level energy in the bound states and the photon energy in the symmetric potential well is used to the studied problem for the created electrons in the asymmetric potential well and its validity is confirmed by this approximation. By the location distribution of the electrons we have also shown the reason why the momentum spectrum has an optimization in the asymmetric well compared with the symmetric one.

Ratchet universality in the bidirectional escape from a symmetric potential well.

The present work discusses symmetry-breaking-induced bidirectional escape from a symmetric metastable potential well by the application of zero-average periodic forces in the presence of dissipation. We characterized the interplay between heteroclinic instabilities leading to chaotic escape and breaking of a generalized parity symmetry leading to directed ratchet escape to an attractor either at ∞ or at -∞. Optimal enhancement of directed ratchet escape is found to occur when the wave form of the zero-average periodic force acting on the damped driven oscillator matches as closely as possible to a universal wave form, as predicted by the theory of ratchet universality. Specifically, the optimal approximation to the universal force triggers the almost complete destruction of the nonescaping basin for driving amplitudes which are systematically lower than those corresponding to a symmetric periodic force having the same period. We expect that this work could be potentially useful in the control of elementary dynamic processes characterized by multidirectional escape from a potential well, such as forced chaotic scattering and laser-induced dissociation of molecular systems, among others.

Boundary Conditions for Sturm-Liouville Equation with Transition Regions and Barriers or Wells

By means of expansions of rapidly in infinity decreasing functions in delta functions and their derivatives, we derive generalized boundary conditions of the Sturm-Liouville equation for transitions and barriers or wells between two asymptotic potentials for which the solutions are supposed as known. We call such expansions “moment series” because the coefficients are determined by moments of the function. An infinite system of boundary conditions is obtained and it is shown how by truncation it can be reduced to approximations of a different order (explicitly made up to third order). Reflection and refraction problems are considered with such approximations and also discrete bound states possible in nonsymmetric and symmetric potential wells are dealt with. This is applicable for large wavelengths compared with characteristic lengths of potential changes. In Appendices we represent the corresponding foundations of Generalized functions and apply them to barriers and wells and to transition functions. The Sturm-Liouville equation is not only interesting because some important second-order differential equations can be reduced to it but also because it is easier to demonstrates some details of the derivations for this one-dimensional equation than for the full three-dimensional vectorial equations of electrodynamics of media. The article continues a paper that was made long ago.

Dynamics of symmetric and asymmetric potential well-based piezoelectric harvesters: A comprehensive review

Small and micro-scale energy harvesting is an essential and viable option for the powering of portable and maintenance free electronic devices, wireless sensor nodes, and similar applications. In this regard, piezoelectric harvesters have presented promising outcomes. This article provides a sequential, comprehensive, and informative survey of potential well based models and studies related to piezoelectric harvesters (PEH). Piezoelectric materials used for energy harvesting are discussed briefly, following which a non-dimensional generalized model is derived to set the discussion on a common platform. Dynamics of various potential well configurations are presented using the generalized model before discussing specific models and related studies. The survey is classified into symmetric and asymmetric potential well categories. Under the symmetric head, lumped and distributed parameter linear models and tuning methods for improving the broadband response are discussed. Subsequently, studies related to nonlinear mono-stable, bi-stable, and tri-stable potentials showing interwell, multi-periodic and chaotic oscillations with improved broadband response are discussed. The asymmetric section studies the influence of asymmetries on the performance of the mono-stable, bi-stable, and tri-stable configurations. Few other configurations outside the cantilever type PEH were mentioned, realizing the widespread research in this field. Important observations and future challenges for performance improvement are also discussed.

Phase space mixing in an external gravitational central potential

This article is devoted to the study of the dynamical behavior of a collisionless kinetic gas in d = 1, 2, 3 space dimensions which is trapped in a rotationally symmetric potential well. Although at the microscopic level the trajectories of individual gas particles are quasi-periodic and characterized by their d fundamental frequencies, at the macroscopic level the gas relaxes in time to a stationary state, provided the potential satisfies a certain non-degeneracy condition. In this article, we provide a mathematically precise formulation for this relaxation process which is due to phase space mixing. In particular, we prove that a physically relevant class of macroscopic observables computed from the one-particle distribution function, such as particle and energy densities, pressure and stress tensors, converge in time to the corresponding observables associated with an averaged distribution function. The latter can be determined from the initial datum and depends only on integrals of motion. Thus, the final state of the gas is described by an effective distribution function depending only on integrals of motion, which considerably reduces the degrees of freedom of the gas configuration. We discuss some applications to gravitational physics, including the propagation of a collisionless gas in typical potentials arising in stellar dynamics and the modeling of dark matter halos, and we also generalize our results to a relativistic gas whose individual particles follow bound timelike trajectories in the exterior region of a static, spherically symmetric black hole spacetime.

The kinetic inductance of single-walled carbon nanotube metal type

The results of theoretical studies of the kinetic inductance Lk metallic single-walled carbon nanotube type, taking into account the electron-electron interaction in the approximation of a right circular cylinder are presented. The system under consideration is a cylindrically symmetric potential well with a final height of the wall. As part of the response of the theory of two-dimensional electron gas in the external electromagnetic disturbance obtained in the form of kinetic inductance depending on the nanotube diameter, frequency and intensity of the radiation. Established that Lk value increases with increasing frequency, and decreases with increasing radiation intensity.

Intra-well and cross-well chaos in membranes and shells liable to buckling

This work presents the mathematical modeling for the nonlinear vibration analysis of membrane and shell structures of arbitrary shape. These structures are usually the optimal form in many engineering applications. However, they may buckle under specific loading conditions and, in most cases, are sensitive to geometric imperfections. These structures when liable to unstable buckling present for load levels lower than the static critical load a multiwell potential functions, which has an underlying influence on the nonlinear dynamic behavior and stability of the structure in a dynamic environment. The energy barrier of each well is a key factor, and depending on the force control parameters and initial conditions, intra-well and cross-well motions may occur. Also coexisting attractors are the norm, influencing the structure's dynamic integrity. In these structures, escape from a potential well is preceded by global and local bifurcations, usually leading to chaos, which influence the number of coexisting attractor, their period and the topology of the basins of attraction. In the present work, three problems are addressed: an axially loaded cylindrical shell, a pressure-loaded spherical cap and a spherical membrane under internal pressure. These problems illustrate the possible multiwell functions observed in several membrane and shell problems, namely potential functions with one, two and three (symmetric or asymmetric) potential wells. A detailed parametric analysis is conducted through bifurcation diagrams of the Poincare map, time responses and Floquet stability criterion to clarify the influence of the multiwell potential function on the bifurcation scenario, basins of attraction and system safety.

Escape of a harmonically forced classical particle from asymmetric potential well

The paper addresses the problem of escape of harmonically forced classical particle from asymmetric potential well. Two benchmark models of the potential wells are used – truncated parabolic well with small cubic perturbation, and more common essentially nonlinear quadratic-cubic well. Transient escape dynamics in both models is analyzed in the framework of isolated resonance approximation. Despite substantial difference between both models, the observed escape scenarios are qualitatively similar. In each case, as parameters of the system are modified, the amplitude of the slow phase flow can gradually achieve the escape threshold; this simple mechanism is referred to as "maximum" scenario. The other, potentially more dangerous scenario, involves abrupt transition of the system response from relatively small amplitude to the escape. This pattern is related to passage of the slow-flow phase trajectory through dynamical saddle of the resonant manifold, and thus is referred to as "saddle" scenario. Both escape scenarios are easily observed in full-scale numeric simulations, in complete agreement with the theoretical predictions. The aforementioned resonance escape mechanisms are similar to those which were reviled in the case of symmetric potential well. Also, both mechanisms are generic enough to remain unchanged even in presence of small damping. These findings point on an unexpected universality of the resonance escape mechanisms. Description of the motion outside the well is not included in the current study.

三稳态能量收集系统的同宿分岔及混沌动力学分析

Nonlinear dynamic performances such as homoclinic bifurcation and chaos were investigated for tristable vibration energy harvesting systems. The analytical expression of the symmetric and asymmetric homoclinic solution was obtained through the Padé approximation, which was consistent with the numerical solution. According to the Melnikov theory, the qualitative method of studying the homoclinic bifurcation of the energy harvesting system with a triple well was developed, and the necessary condition for the occurrence of homoclinic bifurcation was obtained. Numerical simulations yielded bifurcation diagrams and maximum Lyapunov exponents that demonstrated the inter-well responses predicted with the Melnikov method. Compared with the system with symmetric potential energy, the system with asymmetric potential energy has a lower threshold of homoclinic bifurcation. For a low excitation level, the system with asymmetric potential energy witnesses inter-well chaos, while the response of the system with symmetric potential energy still keeps trapped in a single well. The change of symmetry of the system potential energy function improves the output voltage due to the increase in the probability of generating a large periodical inter-well oscillation response. The research on the homoclinic bifurcation of nonlinear energy harvesting systems with symmetric and asymmetric triple potential wells provides an effective tool for the parametric design of high-performance energy harvesters.

Polynomial time algorithms for estimating spectra of adiabatic Hamiltonians

Much research regarding quantum adiabatic optimization has focused on stoquastic Hamiltonians with Hamming symmetric potentials, such as the well studied "spike" example. Due to the large amount of symmetry in these potentials such problems are readily open to analysis both analytically and computationally. However, more realistic potentials do not have such a high degree of symmetry and may have many local minima. Here we present a somewhat more realistic class of problems consisting of many individually Hamming symmetric potential wells. For two or three such wells we demonstrate that such a problem can be solved exactly in time polynomial in the number of qubits and wells. For greater than three wells, we present a tight binding approach with which to efficiently analyze the performance of such Hamiltonians in an adiabatic computation. We provide several basic examples designed to highlight the usefulness of this toy model and to give insight into using the tight binding approach to examining it, including: (1) adiabatic unstructured search with a transverse field driver and a prior guess to the marked item and (2) a scheme for adiabatically simulating the ground states of small collections of strongly interacting spins, with an explicit demonstration for an Ising model Hamiltonian.

Application of relativistic coupled cluster theory to elastic scattering of electrons from confined Ca atoms

We have developed relativistic coupled-cluster (RCC) theory to study elastic scattering of electrons from free and confined Ca atoms. For this purpose, we first investigated electron correlation effects on the atomic properties of the Ca atom trapped inside an attractive spherically symmetric potential well of an endohedral fullerene C60 cluster and obtained electron correlation energy, ionization potential and dipole polarizability of this atom. Our results are compared with the reported calculations employing multi-configuration Hartree–Fock (MCHF) method. We found that trends in correlation energy with respect to the potential depth are the same, but magnitudes are very large in the relativistic calculations. Finally, using the obtained charge density and phase shift analysis technique we determined the differential and total cross-sections for elastic scattering of electrons from free and confined Ca atoms and demonstrated the role of potential depth in these properties.

Electron states for gapped pseudospin-1 fermions in the field of a charged impurity

The electron states of gapped pseudospin-1 fermions of the $\alpha-{\cal T}_3$ lattice in the Coulomb field of a charged impurity are studied. The free $\alpha-{\cal T}_3$ model has three dispersive bands with two energy gaps between them depending on the parameter $\Theta$ which controls the coupling of atoms of honeycomb lattice with atoms in the center of each hexagon, thus, interpolating between graphene $\Theta=0$ and the dice model $\Theta=\pi/4$. The middle band becomes flat one with zero energy in the dice model. The bound electron states are found in the two cases: the centrally symmetric potential well and a regularized Coulomb potential of the charged impurity. As the charge of impurity increases, bound state energy levels descend from the upper and central continua and dive at certain critical charges into the central and lower continuum, respectively. In the dice model, it is found that the flat band survives in the presence of a potential well, however, is absent in the case of the Coulomb potential. The analytical results are presented for the energy levels near continuum boundaries in the potential well. For the genuine Coulomb potential, we present the recursion relations that determine the coefficients of the series expansion of wave functions of bound states. It is shown that the condition for the termination of the series expansion gives two equations relating energy and charge values. Hence, analytical solutions can exist for a countably infinite set of values of impurity charge at fixed $\Theta$.

Exact Solutions for Distributions of First-Passage, Direct-Transit, and Looping Times in Symmetric Cusp Potential Barriers and Wells.

For a particle diffusing in one dimension, the distribution of its first-passage time from point a to point b is determined by the durations of the particle trajectories that start from point a and are terminated as soon as they touch point b for the first time. Any such trajectory consists of looping and direct-transit segments. The latter is the final part of the trajectory that leaves point a and goes to point b without returning to point a. The rest of the trajectory is the looping segment that makes numerous loops which begin and end at the same point a without touching point b. In this article we discuss general relations between the first-passage time distribution and those for the durations of the two segments. These general relations allow us to find exact solutions for the Laplace transforms of the distributions of the first-passage, direct-transit, and looping times for transitions between two points separated by a symmetric cusp potential barrier or well of arbitrary height and depth, respectively. The obtained Laplace transforms are inverted numerically, leading to nontrivial dependences of the resulting distributions on the barrier height and the well depth.

Analytical description of molecular mechanism of fast relaxation of spherical micelles with the extended Becker–Döring differential equation

To improve and expand kinetic analysis of fast relaxation (via attachments and detachments of surfactant molecules) in an ensemble of spherical micelles in surfactant solutions, a general scheme for reducing the linearized difference Becker–Döring equations to the differential equation of an arbitrary order with respect to the aggregation number is proposed. A perturbation theory is formulated for any model of spherical micelles, where the main approximation corresponds to the kinetic Aniansson equation for the case of a symmetrical potential well of the aggregation work and the perturbation operator is written in the Hermitian form. The latter allows one to use standard perturbation techniques to find the fast relaxation times with the help of extended differential kinetic equation. The calculations were carried out in the second order of the perturbation theory, and the longest fast relaxation times were found as a function of the surfactant concentration for the droplet and the quasi-droplet models of direct spherical micelles and the star model of diblock polymeric spherical micelles. In the case of the droplet model, inclusion of corrections gives the concentration dependence of the longest fast relaxation time that virtually coincides with results of numerical solution of the system of linearized Becker–Döring difference equations. For the quasi-droplet model, the fast relaxation time found in the main approximation considerably deviate from the numerical result (up to 50%). Addition of corrections allows us to reduce these deviations to a considerably smaller value (to 10%). For the star micelle model, a fine agreement between the analytical and numerical solutions is obtained.