Finite Potential Well Research Articles

Approximate approaches to the one-dimensional finite potential well

The one-dimensional finite well is a textbook problem. We propose approximate approaches to obtain the energy levels of the well. The finite well is also encountered in semiconductor heterostructures where the carrier mass inside the well (mi) is taken to be distinct from mass outside (mo). A relevant parameter is the mass discontinuity ratio β = mi/mo. To correctly account for the mass discontinuity, we apply the BenDaniel–Duke boundary condition. We obtain approximate solutions for two cases: when the well is shallow and when the well is deep. We compare the approximate results with the exact results and find that higher-order approximations are quite robust. For the shallow case, the approximate solution can be expressed in terms of a dimensionless parameter σl = 2moV0L2/ℏ2 (or σ = β2σl for the deep case). We show that the lowest-order results are related by a duality transform. We also discuss how the energy upscales with L (E∝1/Lγ) and obtain the exponent γ. Exponent γ → 2 when the well is sufficiently deep and β → 1. The ratio of the masses dictates the physics. Our presentation is pedagogical and should be useful to students on a first course on elementary quantum mechanics or low-dimensional semiconductors.

The static electric polarizability of a particle bound by a finite potential well

We derive an expression for the static electric polarizability of a charged particle bound by a finite potential well by employing the elegant Dalgarno-Lewis perturbative technique. The derivation does not make explicit use of the continuum states, is much simpler than the usual perturbation derivation, and allows us to separate the contributions to the polarizability from the classically forbidden and classically allowed regions of the finite potential well.

S-wave scattering and the zero-range limit of the finite square well in arbitrary dimensions

We examine the zero-range limit of the finite square well in arbitrary dimensions through a systematic analysis of the reduced, s-wave, two-body, time-independent Schrödinger equation. A natural consequence of our investigation is the requirement of a delta function multiplied by a regularization operator to model the zero-range limit of the finite-square well when the dimensionality is greater than one. The case of two dimensions turns out to be surprisingly subtle, and needs to be treated separately from all other dimensions.

Exciton in a strained (001)‐oriented zinc‐blende GaN/AlxGa1‐xN ellipsoidal finite‐potential quantum dot under hydrostatic pressure

AbstractA variational method is used to discuss the binding energies of excitons in a strained zinc‐blende ellipsoidal quantum dot by considering the hydrostatic pressure effect. The analytic expressions for the Hamiltonian of the exciton have been obtained by using the Hylleraas coordinate system. The results indicate that the binding energies of excitons reach a peak value as the quantum dot radius increases and then diminish for the finite potential well. We show that the binding energies of excitons decrease with increasing ellipticity constant, but increase with pressure. In particular, it is found that the binding energies are more sensitive to the variations of the ellipticity constant for the small size of QD (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Linear and nonlinear intersubband optical absorption in symmetric double semi-parabolic quantum wells

The linear and the nonlinear intersubband optical absorption in the symmetric double semi-parabolic quantum wells are investigated for typical GaAs / Al x Ga 1 − x As . Energy eigenvalues and eigenfunctions of an electron confined in finite potential double quantum wells are calculated by numerical methods from Schrödinger equation. Optical properties are obtained using the compact density matrix approach. In this work, the effects of the barrier width, the well width and the incident optical intensity on the optical properties of the symmetric double semi-parabolic quantum wells are investigated. Our results show that not only optical incident intensity but also structure parameters such as the barrier and the well width really affect the optical characteristics of these structures.

Bound polaron in a wurtzite GaN/Al xGa 1 − xN ellipsoidal finite-potential quantum dot

A variational method is used to study the ground state of a bound polaron in a weakly oblate wurtzite GaN/Al x Ga 1 − x N ellipsoidal quantum dot. The binding energy of the bound polaron is calculated by taking the electron couples with both branches of LO-like and TO-like phonons due to the anisotropic effect into account. The interaction between impurity and phonons has also been considered to obtain the binding energy of a bound polaron. The results show that the binding energy of bound polaron reaches a peak value as the quantum dot radius increases and then diminishes for the finite potential well. We found that the binding energy of bound polaron is reduced by the phonons effect on the impurity states, the contribution of LO-like phonon to the binding energy is dominant, the anisotropic angle and ellipticity influence on the binding energy are small.

The phonon effects on the impurity states in cylindrical quantum wire with a finite confining potential

Abstract The variational method and the effective mass approximation are applied to calculate the binding energies of the hydrogenic impurity states in a cylindrical quantum wire with finite deep potential well. The phonon effects on the impurity states are considered by taking both the couplings of the electron–phonon and the impurity ion–phonon into account. The numerical results for the GaAs cylindrical quantum wire are given and discussed. It is found that the ion–phonon interaction reduces the impurity binding energy and supplies key contribution to the energy shift, but the electron–phonon coupling enhances the binding energy less. Longitudinal optical (LO) phonons play more important role than interface optical (IO) phonons in the impurity potential screening. The polaron effect caused by LO phonons is more important when the wire is thinner, otherwise the LO phonons are dominant for the thicker wires.

PHYSICS OF GALACTIC COLLIDERS: HIGH-SPEED SATELLITES IN ΛCDM VERSUS MONDIAN COSMOLOGY

The statistics of high-speed satellite galaxies, as reported in the recent literature, can be a powerful diagnosis of the depth of the potential well of the host halo, and hence discriminate between competing gravitational theories. Naively one expects that high-speed satellites are more common in modified Newtonian dynamics (MOND) than in cold dark matter (CDM) since an isolated MONDian system has an infinite potential well, while CDM halos have finite potential wells. In this Letter, we report on an initial test of this hypothesis in the context of the first generation of cosmological simulations utilizing a rigorous MONDian Poisson solver. We find that such high-speed encounters are approximately a factor of 4 more common in MOND than in the concordance ΛCDM model of cosmic structure formation.

The asymptotic shift for the principal eigenvalue for second order elliptic operators in the presence of small obstacles

Let L be a uniformly elliptic second order differential operator with nice coefficients, defined on a smooth, bounded domain in ℝd, d ≥ 2, with either the Dirichlet or an oblique-derivative boundary condition. In this work we study the asymptotics for the principal eigenvalue of L under hard and soft obstacle perturbations. The hard obstacle perturbation of L is obtained by making a finite number of holes with the Dirichlet boundary condition on their boundaries. The main result gives the asymptotic shift of the principal eigenvalue as the holes shrink to points. The rates are expressed in terms of the Newtonian capacity of the holes and the principal eigenfunctions for the unperturbed operator and its formal adjoint. The soft obstacle corresponds to a finite number of compactly supported finite potential wells. Here we only consider the oblique-derivative Laplacian. The main difference from the hard obstacle problem is that phase transitions occur, due to the various scaling possibilities. Our results generalize known results on similar perturbations for selfadjoint operators. Our approach is probabilistic.

Effect of a terahertz laser field on the electron-DOS in a GaAs/AlGaAs cylindrical quantum wire: finite well model

The influence of a linearly polarized terahertz laser field on the density of states (DOS) for carriers confined in a cylindrical semiconductor quantum wire is investigated here within a nonperturbative scheme, based on the Green's function approach introduced by Xu (1997 Semicond. Sci. Technol. 12 1559). In our finite potential well model, the functional dependence of the DOS on energy is significantly modified by a monochromatic, nonresonant, intense laser radiation polarized along the wire axis. Apart from a uniform blueshift with respect to the laser-free DOS, which is proportional to the laser intensity and to the inverse of the square of the frequency, a remarkable reduction of the DOS, followed by pronounced Franz–Keldysh-like oscillations above the bottom of each 1D sub-band, is obtained. The period of these oscillations is seen to depend more on the laser frequency than on the intensity. Interestingly, with an increase of the laser intensity and/or a reduction of the laser frequency, the DOS profile changes from the usual shape found in 1D electronic systems to a series of discrete peaks similar to that found in quasi-0D systems, such as quantum dots.

Binding energy and stability of charged excitons in a semiconductor cylindrical quantum dot

The binding energy of full three-dimensional (3-D) charged excitons confined in a semiconductor cylindrical quantum dot (QD) is theoretically investigated using a variational procedure within the effective mass approximation. We predicted a trial wave function to not only satisfy the strong confinement regime but also yield the correct results in the weak confinement regime. The trions confinement is described by a finite square potential well. We show that the negatively charged exciton has higher binding energy than the positively charged exciton, when the QD half height is less than the effective Bohr radius (strong confinement regime). At large QD, the negatively charged exciton binding energy crosses down the positively charged exciton binding energy and becomes an unstable system. We have clarified the paradoxes existing in the previous literatures concerning the trions binding energy.

Applications of continuity and discontinuity of a fractional derivative of the wave functions to fractional quantum mechanics

The space fractional Schrödinger equation with a finite square potential, periodic potential, and delta-function potential is studied in this paper. We find that the continuity or discontinuity condition of a fractional derivative of the wave functions should be considered to solve the fractional Schrödinger equation in fractional quantum mechanics. More parity states than those given by standard quantum mechanics for the finite square potential well are obtained. The corresponding energy equations are derived and then solved by graphical methods. We show the validity of Bloch’s theorem and reveal the energy band structure for the periodic potential. The jump (discontinuity) condition for the fractional derivative of the wave function of the delta-function potential is given. With the help of the jump condition, we study some delta-function potential fields. For the delta-function potential well, an alternate expression of the wave function (the H function form of it was given by Dong and Xu [J. Math. Phys. 48, 072105 (2007)]) is obtained. The problems of a particle penetrating through a delta-function potential barrier and the fractional probability current density of the particle are also discussed. We study the Dirac comb and show the energy band structure at the end of the paper.

CALCULATION OF ELECTRONIC STRUCTURE OF A SPHERICAL QUANTUM DOT USING A COMBINATION OF QUANTUM GENETIC ALGORITHM AND HARTREE–FOCK–ROOTHAAN METHOD

The electronic structure of Quantum Dot (QD), GaAs/Al x Ga 1-x As , has been investigated by using a combination of Quantum Genetic Algorithm (QGA) and Hartree–Fock–Roothaan (HFR) method. One-electron system with an on-center impurity is considered by assuming a spherically symmetric confining potential of finite depth. The ground and excited state energies of one-electron QD were calculated depending on the dot radius and stoichiometric ratio. Expectation values of energy were determined by using the HFR method along with Slater-Type Orbitals (STOs) and QGA was used for the wavefunctions optimization. In addition, the effect of the size of the basis set on the energy of QD was investigated. We also calculated the binding energy for a dot with finite confining potential. We found that the impurity binding energy increases for the finite potential well when the dot radius decreases. For the finite potential well, the binding energy reaches a peak value and then diminishes to a limiting value corresponding to the radius for which there are no bound states in the well. Whereas in previous study, in Ref. 40, for the infinite potential well, we found that the impurity binding energy increases as the dot radius decreases.

Selective-area growth of hexagonal nanopillars with single InGaAs/GaAs quantum wells onGaAs(111)B substrate and their temperature-dependent photoluminescence

We report on the fabrication of highly uniform hexagonal nanopillars with singleInGaAs/GaAs quantum wells (QW) on GaAs(111)B substrate by selective-areametal–organic vapour phase epitaxy. The standard size deviation of the fabricatednanopillars with single InGaAs/GaAs QW is about 2% and the standard deviation in theirheight about 5%. With a decrease in temperature, the peak position shifts to shorterwavelength, the peak intensity increases and the peak width decreases. The calculated wellwidth based on the finite square potential well model taking account of the strain effectand the piezoelectric effect is smaller than the value determined from the growthrate and the growth time, which is mainly due to indium segregation and itsincorporation into the subsequent GaAs layer grown at the higher temperature.

Study of shallow donor level binding energies confined in aGaAs–Ga1−xAlxAs spherical quantum dot

We study the level structure of a Coulombic donor impurity that bindsan electron that is confined within a spherical quantum dot in theGaAs–Ga1−xAlxAs system. Initially, the electron is considered to be moving in a spherically symmetricrectangular potential well and then the effects of a hydrogenic donor with effective chargeZD taken at the centre of the well are considered as a strong perturbation within theeffective mass approximation. The perturbation calculations are carried out usingthe Jiang approach (Jiang 1987 Phys. Rev. B 35 9287). We calculate the bindingenergies of the ground and a few higher excited states for both infinite and finitepotential wells as a function of well width and barrier height. The results arecompared with those obtained by other techniques such as the variational and directnumerical methods. It is found that the agreement among all these methods isexcellent, and this would open the way for more investigations using our direct andrelatively easy perturbation treatment, especially in more complicated cases in similarsystems.

Thermal effect on bound exciton in CdTe/Cd1−xZnxTe cylindrical quantum dots

Abstract The temperature dependence of the exciton ground state, lowest and binding energies in cylindrical quantum dot (QD) are theoretically investigated using a variational procedure within the effective mass approximation. The interaction between the charge carriers (electron and hole) and longitudinal optical (LO) phonon modes is taken into account. The excitonic confinement is described by a finite depth potential well. Specific applications of these results are given for CdTe QDs embedded in a CdTe / Cd 1 - x Zn x Te matrix. The total and lowest energies depend strongly on the temperature, up to a critical value of temperature T (around 40 °K), increasing with it. In the small dot sizes region, both energies decrease rapidly with increasing dot radius and approach the quantum well limit energies for large radius. In addition, an enhancement of the Zn fraction in the barrier material increases both the total and binding energies of the exciton. It is found that the exciton binding energy is significantly reduced with increasing temperature and its effect is more important on the ground state energy than on the binding energy. Moreover the exciton is stable even at room temperature. The LO-phonon contribution to the exciton binding energy is important and depends on the dot size, the Zn fraction and the temperature.

A model for scattering due to interface roughness in finite quantum wells

A model for scattering due to interface roughness in finite quantum wells (QWs) is developed within the framework of the Boltzmann transport equation and a simple and explicit expression between mobility limited by interface roughness scattering and barrier height is obtained. The main advantage of our model is that it does not involve complicated wavefunction calculations, and thus it is convenient for predicting the mobility in thin finite QWs. It is found that the mobility limited by interface roughness is one order of amplitude higher than the results derived by assuming an infinite barrier, for finite barrier height QWs where x = 0.3. The mobility first decreases and then flattens out as the barrier confinement increases. The experimental results may be explained with monolayers of asperity height 1–2, and a correlation length of about 33 Å. The calculation results are in excellent agreement with the experimental data from AlxGa1−xAs/GaAs QWs.

Quantum size effects of CO reactivity on metallic quantum dots

We study the reactivity of a metallic quantum dot when exposed to a gas phase CO molecule. First, we perform a Newns–Anderson model calculation in which the valence electrons of the quantum dot are confined by a finite potential well and the molecule is characterized by its lowest unoccupied molecular orbital in the gas phase. A pronounced quantum size effect regarding the charge transfer between the quantum dot and molecule is observed. We then perform a first-principles calculation for a selected size interval. The quantum dot is described within the jellium model and the molecule by pseudopotentials. Our results show that the charge transfer between the quantum dot and the molecule depends critically on the size of the quantum dot, and that this dependence is intimately connected with the electronic structure. The key factor for charge transfer is the presence of states with the symmetry of the chemically active molecular orbital at the Fermi level.

Macroscopic quantum tunnelling of Bose–Einstein condensates in a finite potential well

Bose–Einstein condensates are studied in a potential of finite depth which supports both bound and quasi-bound states. This potential, which is harmonic for small radii and decays as a Gaussian for large radii, models experimentally relevant optical traps. The nonlinearity, which is proportional to both the number of atoms and the interaction strength, can transform bound states into quasi-bound ones. The latter have a finite lifetime due to tunnelling through the barriers at the borders of the well. We predict the lifetime and stability properties for repulsive and attractive condensates in one, two and three dimensions, for both the ground state and excited soliton and vortex states. We show, via a combination of the variational and WKB approximations, that macroscopic quantum tunnelling in such systems can be observed on time scales of 10 ms to 10 s.

Quasiclassical approach to Bose condensation in a finite potential well

We treat the problem of self-consistently interacting bosons in the presence of a finite (but macroscopic) potential well within a quasiclassical approximation for the normal component and the order parameter. We solve the equilibrium problem and show that condensation actually occurs in two steps: one already at low densities with Bose condensation only in the well, and another one corresponding to the usual condensation in bulk. The peak and width of the distribution of trapped particles in the well display a distinct signature of the local condensation. A possible connection to recent experiments with excitons is discussed.