Finite Potential Well Research Articles

Multilevel dynamics of matter waves scattering in finite potential wells

We explore the dynamics of matter wave scattering in finite potential wells using analytical solutions of the Schrödinger equation within the framework of a quantum shutter model. We find that the incident wave interferes with the bound states of the quantum well, resulting in time-domain oscillations. These oscillations exhibit Rabi-type frequencies, characterised by the energy differences between the incident wave and the bound states of the quantum well. We show that in systems with double-bound states, the interference pattern is characterised by quantum beats in the time-dependent probability density. The period of these beatings depends on the energy difference between the bound states, which can be tuned by controlling the potential parameters. In the general case where bound, anti-bound, and resonant states coexist in the system spectrum, complex oscillations in the probability density arise from the interactions of the incident wave with different quantum states. We demonstrate that the bound states sector can effectively describe this complex behaviour, providing a simple and reliable analytical expression for the probability density in multilevel systems. This formula highlights the significant role of bound states, whose interaction with the incident wave dominates the transient probability density. This contrasts with conventional systems with potential barriers and wells, where resonances govern the wave dynamics.

Unraveling the effects of non-parabolicity on electron energy levels in InP/InAs/InP heterostructures

In this research, electron energy levels were calculated analytically using Nelson’s formula, the shooting method, and Garrett’s formula for effective mass. These calculations were performed for a rectangular finite deep potential well, focusing on the InP/InAs/InP heterostructure, which is a narrow-bandgap semiconductor system. Our results demonstrate that the nonparabolicity of the dispersion has a more significant effect on higher energy levels compared to lower ones, with deviations of up to 15% for the third energy level. An equation estimating the number of observable energy levels in the potential well is suggested, revealing that considering nonparabolicity leads to a 20% increase in the number of levels compared to the parabolic dispersion case. The relationship between the widths of infinite and finite potential wells for equivalent energy levels follows a linear behaviour, with bonding coefficients ranging from 95,93% to 97,49% and a maximum difference of 1.5% between parabolic and non-parabolic cases. The transcendental equation for the energy levels is linearized, yielding a fourth-order equation that provides results within 98% accuracy compared to the original equation. These findings contribute to the understanding of the energy distribution in InP/InAs/InP heterostructures with a view to their application in optoelectronic devices such as lasers, light-emitting diodes

A Closed-Form Model for 2-DEG Charge and Surface Potential in N-Polar Gallium Nitride Heterostructures

A closed-form model to calculate the 2-D electron gas (2-DEG) charge density and surface potential in an N-polar GaN/AlGaN heterostructure is presented. The proposed model is developed from the solution of Schrödinger’s equation for a finite triangular potential well. Asymptotic expressions of Airy functions are used to compute an initial value of subband energy. A new, unified expression to update the surface potential due to Poisson’s equation is presented. The modified Fang–Howard (M.F.H.) function is then employed to develop a closed-form correction to the subband energy using first-order perturbation theory. No fitting parameters are used. The results from our model are validated with numerical data for a wide range of bias conditions, channel layer thickness, and spacer layer mole fraction. Finally, the gate capacitance of the heterostructure is evaluated using automatic differentiation and validated against experimental data.

Simulation of Time Independent Schrödinger Equation for Finite Potential Well Using the Graphical Solution Method

This paper investigates applying the approximate method; Graphical Solution Method (GSM), to theoretically solve the Time Independent Schrödinger Equation (TISE) in one dimension for a finite square well using MATLAB. With just few lines of MATLAB coding, calculating and plotting accurate eigenvalues (energy), eigenvectors (wave functions) and the bound eigenstates are possible for the finite square well of a negative potential (depth of the well) of -400 eV and a well width of 0.1 nm for an electron confined to this quantum well. These eigenvalues, eigenvectors and eigenstates are obtained and discussed. The found energy eigenvalues and states are discrete and yield physical acceptable solutions. The even and odd solutions of the TISE are also considered. The graphical solutions for the finite potential well are shown. The locations of discrete eigenvalues for even and odd solutions are also presented. These eigenvalues are tested confirming the correct eigenfunctions. The precision of these solutions depend on well width L and on the interval dx used to integrate the equation. Exact analytical solutions for this case are obtained and compared with results from the GSM. The accuracy and the convergence of the numerical results are easily checked. The results showed that the GSM can be considered as a suitable mean for determining the one dimensional solutions for the finite square well.

Magneto-optical absorption in an asymmetric finite quantum well

We examine the influence of internal (confinement frequency, Al-doped concentration, β parameter) and extrinsic (magnetic field, temperature) parameters on the energy differences and magneto-optical absorption properties of a finite semi-parabolic plus semi-inverse squared (FSPSISP) quantum well. Our results demonstrated that the energy difference between the ground and excited states is substantially dependent on the amount of Al-doped in the system, the confinement frequency, and the magnetic field, and slightly dependent on parameter characterizing the inverse squared part of the confinement potential, but not temperature. The magneto-optical absorption coefficient (MOAC) increases with confinement frequency, Al-doped concentration, magnetic field, and temperature, but decreases with β. The full width at half maximum (FWHM) increases with all parameters. By optimizing these parameters, we can enhance the MOAC. Our results provide deeper insight into the optical properties of low-dimensional systems with finite confinement potential.

Effect of the electronic confinement potential on the electron–LO-phonon interaction energy in a quantum well

We investigated the effect of the electronic confinement potential on the electron–LO-phonon (el–LO-ph) interaction energy in a quantum well by using the variational method. In this work, the el–LO-ph interaction energy in the infinite (I) and finite (F) square potential quantum wells (SPQW) is calculated as a function of the well-width for all four different confined-optical-phonon (COP) models. The result shows that the el-LO-ph interaction energy in a quantum well is affected strongly by the electronic confinement potential for all four COP models. Except for the Ridley-Babiker model, the el-LO-ph interaction energy in the ISPQW is always larger than that in the FSPQW for both the lowest phonon modes of different COP models in the narrow well case. The Fuchs-Kliewer and the Knipp-Reineck models have a good agreement in the description of the el-LO-ph interaction for the ISPQW, while that is the Fuchs-Kliewer and the Huang-Zhu models for the FSPQW. Among the above four models, the Fuchs-Kliewer model is the most effective.

Energies of the ground and excited states of confined two-electron atom in finite potential well

For the ground and excited configurations, we have carried out the energy eigenvalues and corresponding wavefunctions of two-electron atom enclosed by finite confining potential by using Quantum Genetic Algorithm (QGA) procedure and Hartree-Fock Roothaan (HFR) method. Besides, we have analysed in detail orbital energies and probabilities of finding electron for the configurations 1S (1s2), 3S (1s2s), 3P (1s2p), 3D (1s3d) and 3F (1s4f). The results reveal that the potential barrier height plays a significant role on average energies, state energies, orbital energies and probabilities of finding electrons in orbitals. As dot size is extremely small and extremely large, the energies for 1snl configurations are getting close to each other, but between these two extreme situations they are separating from each other. While going to higher excited states, singlet and triplet states are almost the same due to the weakening of the exchange energy.

Composite basis set of plane wave and Gaussian function or spline function

By combining plane waves with Gaussian or spline functions, a new composite basis set is constructed in this work. As a non local basis vector, the plane wave basis group needs a large number of plane waves to expand all parts of the physical space, including the intermediate regions that are not important to our problems. Our basis set uses the local characteristics of Gaussian function or spline function at the same time, and controls the energy interval by selecting different plane wave vectors, in order to realize the partition solution of Hamiltonian matrix. Orthogonal normalization of composite basis sets is performed by using Gram-Schmidt’s orthogonalization method or Löwdin’s orthogonalization method. Considering the completeness of plane wave vector, a certain value of positive and negative should be selected at the same time. Here, by changing the absolute value of wave vector, we can select the eigenvalue interval to be solved. The plane wave with a specific wave vector value is equivalent to a trial solution in the region with gentle potential energy. The algorithm automatically combines local Gaussian or spline functions to match the difference in wave vector value between the trial solution and the strict solution. By selecting the absolute value of the wave vector in the plane wave function, the calculation of large Hamiltonian matrices turns into the calculation of multiple small matrices, together with reducing the basis numbers in the region where the electron potential changes smoothly, therefore, we can significantly reduce the computational time. As an example, we apply this basis set to a one-dimensional finite depth potential well. It can be found that our method significantly reduce the number of basis vectors used to expand the wave function while maintaining a suitable degree of computational accuracy. We also study the influence of different parameters on calculation accuracy. Finally, the above calculation method can be directly applied to the density functional theory (DFT) calculation of plasmons in silver nanoplates or other metal nanostructures. Given a reasonable tentative initial state, the ground state electron density distribution of the system can be solved by self-consistent solution through using DFT theory, and then the electromagnetic field distribution and optical properties of the system can be solved by using time-dependent density functional theory.

Students’ difficulties with solving bound and scattering state problems in quantum mechanics

In upper level quantum mechanics courses, the typical process of devoting significant time to the example of a finite square potential well does not provide students with sufficient mastery of the relevant physics concepts and mathematical techniques to solve new problems.

Accurate and fully analytical expressions for quantum energy levels in finite potential wells for nanoelectronic compact modeling

Accurate and fully analytical expressions for quantum energy levels in finite potential wells for nanoelectronic compact modeling

Calculation of the energy levels and wave functions of electrons in nanowires by the shooting method

ABSTRACT In this work, the energy levels and wave functions in rectangular and cylindrical nanowires with a finite potential well are calculated. The Schrodinger equation in Cartesian and cylindrical coordinate systems was solved by the shooting method. The calculations take into account the nonparabolicity of the energy spectrum of electrons. The graphs of the dependence of the energy levels on the sizes of nanowires are obtained. When calculating the energy levels and wave functions, changes in the effective mass of electrons were taken into account. The calculations were performed for the quantum well of the InP/InAs/InP heterostructure.

On quasilinear elliptic problems with finite or infinite potential wells

Abstract We consider quasilinear elliptic problems of the form − div ( ϕ ( ∣ ∇ u ∣ ) ∇ u ) + V ( x ) ϕ ( ∣ u ∣ ) u = f ( u ) , u ∈ W 1 , Φ ( R N ) , -{\rm{div}}\hspace{0.33em}(\phi \left(| \nabla u| )\nabla u)+V\left(x)\phi \left(| u| )u=f\left(u),\hspace{1.0em}u\in {W}^{1,\Phi }\left({{\mathbb{R}}}^{N}), where ϕ \phi and f f satisfy suitable conditions. The positive potential V ∈ C ( R N ) V\in C\left({{\mathbb{R}}}^{N}) exhibits a finite or infinite potential well in the sense that V ( x ) V\left(x) tends to its supremum V ∞ ≤ + ∞ {V}_{\infty }\le +\infty as ∣ x ∣ → ∞ | x| \to \infty . Nontrivial solutions are obtained by variational methods. When V ∞ = + ∞ {V}_{\infty }=+\infty , a compact embedding from a suitable subspace of W 1 , Φ ( R N ) {W}^{1,\Phi }\left({{\mathbb{R}}}^{N}) into L Φ ( R N ) {L}^{\Phi }\left({{\mathbb{R}}}^{N}) is established, which enables us to get infinitely many solutions for the case that f f is odd. For the case that V ( x ) = λ a ( x ) + 1 V\left(x)=\lambda a\left(x)+1 exhibits a steep potential well controlled by a positive parameter λ \lambda , we get nontrivial solutions for large λ \lambda .

A SPICE compatible physics-based intrinsic charge and capacitance model of InAs-OI-Si MOS transistor

In this work, we present a core model of terminal charge and intrinsic capacitances of InAs-on-Si MOS transistors. The eigenenergy and corresponding wavefunctions are deduced by solving time independent Schrödinger wave equation inside the finite potential well. A second-order non-parabolic correction to the sub-band energy, effective mass of electron, and conduction band density of states have been employed and incorporated into the core model. The developed model effectively captures the variation in gate capacitance and reproduce the staircase like capacitance-voltage characteristics. Effect of conduction band nonparabolicity on gate capacitance is studied for different band-nonparabolicity factor. The model is validated with self-consistent Schrödinger-Poisson solver predicted result for different drain bias and channel thicknesses. The model calculated results are in a reasonable agreement with numerical simulation data. The SPICE compatibility of our model is demonstrated through Verilog-AMS implementation of the model and SPICE simulation of two benchmark circuits. • We propose a core model for intrinsic charge and capacitances of InAs-on-insulator MOS Transistor over Silicon substrate. • The model is derived from explicit solution of Schrödinger and Poisson equations without using any empirical parameters. • The model includes conduction band nonparabolicity effect on sub-band energy, terminal charges and capacitances. • Inversion charge present inside buffer region and its effect on capacitance is modeled using a semi analytical approach. • The model is developed using Verilog-AMS language, and circuit simulation is demonstrated using commercial SPICE simulator.

Dirac fermions in zigzag graphene nanoribbon in a finite potential well

The electronic and transport properties of Dirac fermions in zigzag graphene nanoribbon (ZGNR) under the influence of a finite electrostatic potential are investigated. The Green's function method is employed to formulate the problem. It is used to compute the transmission coefficient, charge density distribution, and the local density of states (LDOS) in ZGNR in a finite potential well. It is found that the edge states in ZGNR are strongly dependent on the number of atoms (N), width of the ribbon (M*a) and the strength of the potential well. These edge states have exciting effects on the transmission coefficient through the potential well. Moreover, it is also observed that the LDOS and transmission behaviour depend strongly on the number of configurations in the system under consideration.

Spectral collapse in multiqubit two-photon Rabi model

We have shown that the smallest possible singel-qubit critical coupling strength of the N-qubit two-photon Rabi model is only 1/N times that of the two-photon Rabi model. The spectral collapse can thus occur at a more attainable value of the critical coupling. For both of the two-qubit and three-qubit cases, we have also rigorously demonstrated that at the critical coupling the system not only has a set of discrete eigenenergies but also a continuous energy spectrum. The discrete eigenenergy spectrum can be derived via a simple one-to-one mapping to the bound state problem of a particle of variable effective mass in the presence of a finite potential well and a nonlocal potential. The energy difference of each qubit, which specifies both the depth of the finite potential well and the strength of the nonlocal potential, determines the number of bound states available, implying that the extent of the incomplete spectral collapse can be monitored in a straightforward manner.

Energy Levels in Nanowires and Nanorods with a Finite Potential Well

The energy of electrons and holes in cylindrical quantum wires with a finite potential well was calculated by two methods. An analytical expression is approximately determined that allows one to calculate the energy of electrons and holes at the first discrete level in a cylindrical quantum wire. The electron energy was calculated by two methods for cylindrical layers of different radius. In the calculations, the nonparabolicity of the electron energy spectrum is taken into account. The dependence of the effective masses of electrons and holes on the radius of a quantum wires is determined. An analysis is made of the dependence of the energy of electrons and holes on the internal and external radii, and it is determined that the energy of electrons and holes in cylindrical layers with a constant thickness weakly depends on the internal radius. The results were obtained for the InP/InAs heterostructures.

Manipulating the spectral collapse in two-photon Rabi model

We have investigated the eigenenergy spectrum of the two-photon Rabi model with a full quadratic coupling, particularly the special feature “spectral collapse”. The critical coupling strength is reduced by half from that of the two-photon Rabi model, implying that the spectral collapse can now occur at a more attainable value of the critical coupling. At the critical coupling some discrete eigenenergy levels still survive below the continuous energy spectrum, i.e. an incomplete spectral collapse, and the set of discrete eigenenergies has a one-to-one mapping with that of a particle of variable effective mass in a finite potential well. Since the energy difference between the two atomic levels specifies the depth of the potential well, the number of bound states available (or the extent of the “spectral collapse”) can be straightforwardly monitored. Obviously, this bears a great resemblance to the spectral collapse of the two-photon Rabi model, at least qualitatively. Moreover, since the full quadratic coupling includes an extra term proportional to the photon number operator only, our analysis indicates that one may manipulate the critical coupling of the two-photon Rabi model by incorporating an adjustable proportionality constant to this extra term.

Stability and quantum escape dynamics of spin-orbit-coupled Bose-Einstein condensates in the shallow trap.

The Bose-Einstein condensates in a finite depth potential well provide an ideal platform to study the quantum escape dynamics. In this paper, the ground state, tunneling, and diffusion dynamics of the spin-orbit coupling (SOC) of Bose-Einstein condensates with two pseudospin components in a shallow trap are studied analytically and numerically. The phase transition between the plane-wave phase and zero-momentum phase of the ground state is obtained. Furthermore, the stability of the ground state is discussed, and the stability diagram in the parameter space is provided. The bound state (in which condensates are stably trapped in the potential well), the quasibound state (in which condensates tunnel through the well), and the unstable state (in which diffusion occurs) are revealed. We find that the finite depth potential well has an important effect on the phase transition of the ground state, and, interestingly, SOC can stabilize the system against the diffusion and manipulate the tunneling and diffusion dynamics. In particular, spatial anisotropic tunneling and diffusion dynamics of the two pseudospin components induced by SOC in quasibound and unstable states are observed. We provide an effective model and method to study and control the quantum tunneling and diffusion dynamics.

Multiple approaches to incorporating scattering states in non-degenerate perturbation theory

The second-order perturbative Stark effect on the ground state of hydrogen is a typical example presented in many standard texts on quantum mechanics. Some texts miss the fact that the scattering states are significant contributors to the perturbative energy correction. The inclusion of scattering states has wider applicability than to just the Stark effect. An explicit calculation involving a finite-square well with a perturbation is used to illustrate the importance of including scattering states into the calculation. The second-order correction to the ground-state energy is obtained in three distinct ways. The first involves altering the problem by imposing additional boundary conditions at large distances to make the positive-energy spectrum discrete. The second makes use of the continuum scattering states directly. The third bypasses the use of scattering states by solving a differential equation for the first-order correction to the wave function.

Energy and stability of negatively charged trion in cylindrical quantum dot under temperature effect

The uncorrelated energy of negatively charged trion confined in a semiconductor cylindrical quantum dot (QD) under temperature effect is theoretically investigated using a variational procedure within the effective mass approximation. We have used a wave function satisfying for the strong confinement regime and also presenting well results for the weak confinement regimes. The trion confinement is described by a finite and infinite square potential wells under the temperature effect. Our findings show that the uncorrelated energy of the negatively charged trion is more important for the small size of the QD. We have found a strong dependence of negatively charged trion energy on the confinement potential. Besides, our numerical results have shown that the effect of temperature leads to an increase in uncorrelated energy. The temperature has a significant effect on the stability of the three non-bound charge carriers.