Solution Of Schroedinger Research Articles

Compact drain current modeling of planar InGaAs quantum well MOSFET

In this article, we propose a physics-based compact drain current model of planar InGaAs channel-based quantum well MOS transistor. The effects of essential physical phenomenon such as quantum confinement, multiple sub-band energies, wavefunctions and perturbations in sub-band energies are considered in the model by deriving the time-independent Schroedinger wave equation. The potential and inversion carrier profiles are obtained through direct solution of Schroedinger and Poisson equations inside the device. The proposed model also considers other important physical aspects such as band non-parabolicity, velocity overshoot and threshold voltage roll-off. The model is thus physics-based and does not include any empirical fitting parameter. Professional numerical simulator data for a variety of bias voltages and channel thicknesses have been used to validate the expected outcomes of our model. A reasonable agreement between the transistor characteristics as predicted by our model and that available experimentally is obtained, thus justifying the accuracy of our model.

Algebra of physical space and the geometric spacetime solution of Dirac's equation

AbstractThe algebra of physical space (APS) is a name for the Clifford or geometric algebra, which can be closely associated with the geometry of special relativity and relativistic spacetime. For example, the Dirac Hamiltonian can be presented as the scalar product of the electron's four‐momentum and Dirac's four‐vector of gamma matrices, $({\gamma_0},\vec \gamma)$, the latter of which is a Clifford algebra. We show here that a geometric spacetime or four‐space solution of Dirac's equation conforms to the principles of APS, an early example of which is Schroedinger's solution of Dirac's equation for a free electron, which exhibits Zitterbewegung. In a four‐space solution the spacetime coordinates, $\vec r$ and the scaled time ct, are treated on an equal footing as physical observables to avoid any suggestion of a preferred frame of reference. The geometric spacetime theory is studied here for the Coulomb problem. The positive‐energy spectrum of states is found to be identical within numerical error to that of standard Dirac's theory, but the wave function exhibits Zitterbewegung. It is shown analytically how the geometric spacetime solution can be reduced to the standard solution of Dirac's equation, in which Zitterbewegung is absent. The rigor of APS and of its conforming geometric spacetime solution provide strong support for further investigation into the physical interpretation of the geometric spacetime Dirac's wave function and Zitterbewegung. © 2012 Wiley Periodicals, Inc.

Remarks on the overlapping-sphere method for molecular orbitals

In preference to the overlapping-sphere multiple-scattering method for solving the self-consistent-field problem for a molecule or atomic cluster, the authors recommend a cellular method. Each nucleus is surrounded by a cell, and the cells taken together fill all space and do not overlap. The cells in such a case as a diatomic molecule extend out to infinity. For an approximation one can assume a spherically symmetrical potential in each cell. The general solution of Schroedinger's equation within a cell is then set up as a linear combination of solutions of the spherical problem for a single energy, but for different l and m values. In order to have the functions regular at the nucleus and at infinity, one must join two independent solutions for each l value, one regular at the origin and the other at infinity, over the surface of a sphere of radius r/sub 1/, on which the functions are made continuous, but the normal derivatives are discontinuous. The coefficients of the functions are so chosen that the sum is continuous with continuous derivative over the spheres of radius r/sub 1/, and over the planes separating neighboring atoms. In practice, one uses a finite sum of functions, perhapsmore » for i = 0, 1, ..., 10, and applies the conditions of continuity at enough points to determine the coefficients. By expressing the potential and the radial functions as power series in r, or in r - r/sub 0/, one can get a convenient analytic solution of Schroedinger's equation. The various steps involved in this method have been studied, but they have not yet been combined into a computer program, so that this paper is regarded as a progress report.« less

Bragg quantum wells in weak confinement regime

The optical properties of Bragg quantum wells are studied for exciton confinement under center-of-mass quantization. A variational model of Wannier exciton envelope function, that embodies the correct boundary conditions for center-of-mass, is adopted for calculation. The present non-adiabatic exciton model is compared with adiabatic results and with heuristic “hard sphere” model. The radiative self-energy of a single-quantum well (SQW) and multi-quantum wells (MQWs) are computed in the semiclassical framework, and in effective mass approximation, by self-consistent solution of Schroedinger and Maxwell equations. This microscopic solution is free from “fitting” parameter values, except for the non-radiative broadening, and also the exciton dead-layer and the additional boundary condition are not taken ad hoc, but come coherently from the variational principle and self-consistent Schroedinger-Maxwell solution. Dispersion curves of exciton-polariton propagating in a MQW, under Bragg condition, are studied by selected numerical examples. The case of optical gap in correspondence of higher excited states is studied, and, moreover, the interesting effect of gap enhancement or inhibition, in correspondence of non-resonant Bragg energy, will be addressed.

Determination of band offsets in strainedInxGa1−xAs∕GaAsquantum wells by capacitance-voltage profiling and Schrödinger-Poisson self-consistent simulation

The results of measurements and numerical simulation of charge carrier distribution and energy states in strained quantum wells In_xGa_{1-x}As/GaAs (0.06 < x < 0.29) by C-V-profiling are presented. Precise values of conduction band offsets for these pseudomorphic QWs have been obtained by means of self-consistent solution of Schroedinger and Poisson equations and following fitting to experimental data. For the conduction band offsets in strained In_xGa_{1-x}As/GaAs - QWs the expression DE_C(x) = 0.814x - 0.21x^2 has been obtained.

Study of light emission from GaN/AlGaN quantum wells under power-dependent excitation

We have performed a study of excitation power-dependent spectra of GaN/AlGaN single quantum wells (QWs). First, the experimental “blueshift” of the emission energy, due to screening of internal piezoelectric fields, was compared with the model calculations based on self-consistent solution of Schroedinger and Poisson equations. We found that, even for the highest applied levels of excitation power (2.5 MW/cm2), only 0.5×1012 cm−2 carriers were present in the QW layers. Second, we analyzed the evolution of power-dependent spectra of two single QW having different widths. For the thinner QW (2.1 nm), the peak corresponding to a QW photoluminescence (PL) emission dominates the entire spectrum in the whole range of the used excitation power. In the case of the wider QW (4.4 nm), for sufficiently high excitation power, we observe the effect of PL quenching. Using the rate equation model we show that the observed effect of the PL quenching can be associated with the reduction of exciton binding energy due to the many body interactions in the QW.

Power and initial pulse width dependent soliton velocity in ferrite films

Magnetostatic backward volume wave propagation in yttrium–iron–garnet thin films supports soliton formation and propagation, which is modeled by the nonlinear Schroedinger equation. Previous experimental data have shown that the propagation velocity depends both on the initial pulse power and the duration; however, in the nonlinear Schroedinger solution the soliton velocity, or wave number deviation, is a free parameter that is independent of the initial pulse characteristics. It is shown that these dependences on the initial pulse parameters arise simply from the nonlinear nature of the magnetostatic backward volume wave dispersion relation. Moreover, this same dependence is predicted in the nonlinear Schroedinger model, but the deviation in wave number is determined by the initial pulse characteristics and is no longer a free parameter.

Semi-analytical model for charge control in SiGe quantum well MOS structures

This paper discusses a semi-analytical model to calculate charge distribution in a silicon Metal-Oxide-Semiconductor (MOS) structure with a buried, strained Si 1−xGe x quantum well. This charge distribution model is applicable to a device structure that consists of the following layer sequence: n-type substrate; p + doping spike; Si spacer layer; SiGe quantum well; Si cap layer; insulator and gate. The sheet density of mobile charge in the SiGe well is calculated using a quantum mechanical potential well description with the addition of a first-order perturbation analysis to accommodate the application of a gate bias. The carrier density in the inversion layer at the oxide/Si interface is found in a similar manner using the triangular well approximation. These expressions are coupled with an explicit algorithm to compute the electrostatic potential and depletion charge for various gate biases. Case studies are presented for quantum well MOS structures reported in the literature, and the carrier distributions found utilizing the semi-analytical model are in reasonable agreement with those calculated numerically via self-consistent solution of Schroedinger's and Poisson's equations. This semi-analytical framework allows a means to relate the effects of changing device physical parameters on circuit oriented characteristics. With an appropriate carrier transport description, this one-dimensional treatment can be expanded to quantum well MOS-Field Effect Transistors (MOSFETs) to quantify their static and dynamic characteristics.

On the numerical solution of schroedinger's radial equation

Higher-order difference schemes are considered for the numerical solution of Schroedinger's radial equation. They are a family of difference equations which are extensions of the well-known Numerov difference equation and give highly convergent approximate solutions, the least being O( h 6) compared to O( h 4) in the Numerov equation. An algorithm to find eigenvalues and eigenfunctions using one-directional “shooting” is discussed. The stability and convergence of these schemes are also discussed. An example and numerical results are given, and the order of convergence which is estimated from the results is found to be close to the theoretical value.

Electron effective mass in solids—A generalization of Bardeen's formula

A formula is derived for the effective mass of an electron in a crystal which replaces the sum over excited states in the usual sum rule by an integral over the surface of the unit cell. The integrand of the surface integral involves the wave function(s) at the symmetry point or band extremum k 0 and a second solution of Schroedinger's equation at the same energy but satisfying inhomogeneous boundary conditions on the cell surface. The procedure is applicable regardless whether there is a degeneracy at k 0, and spin-orbit coupling may be taken into account. The result thus represents a generalization of Bardeen's formula for the effective mass of an s -band at k = 0 in the Wigner-Seitz spherical approximation to an arbitrary band at any point k 0, using the polyhedral cell. A related variational principle for the components of the effective mass matrix is also derived.

A numerical solution of Schroedinger's equation in the continuum

A numerical solution of Schroedinger's equation in the continuum