Electronic Eigenfunctions Research Articles

The guiding center role in defining the eigenstates of an electron in the fractional quantum Hall effect regime: ladder operators approach

The quantum problem of a two-dimensional electron in a uniform perpendicular magnetic field is considered. Using the formalism of ladder operators, the electron eigenfunctions are derived for all quantum numbers n and m, where n denotes the Landau level and m the eigenvalue of the angular momentum L z , with m taking all possible eigenvalues. We note that existing one-electron orbitals, in most of the known literature on the fractional quantum Hall effect (FQHE), correspond to a restricted range of possible eigenvalues m, some are missing. Similarly detailed calculations using ladder operator techniques show that for a state ∣n, m〉, the quantum number (n − m) represents a precise physical quantity, that is the distance from the origin to the center of the electron orbit. This finding allowed us to obtain, for this known quantum problem, a new set of basis states for which both quantum numbers have a physical meaning namely n and (n − m).

Chemistry without the Born-Oppenheimer approximation.

Chemistry without the Born-Oppenheimer approximation.

Importance of charge self-consistency in first-principles description of strongly correlated systems

First-principles approaches have been successful in solving many-body Hamiltonians for real materials to an extent when correlations are weak or moderate. As the electronic correlations become stronger often embedding methods based on first-principles approaches are used to better treat the correlations by solving a suitably chosen many-body Hamiltonian with a higher level theory. The success of such embedding theories, often referred to as second-principles, is commonly measured by the quality of self-energy Σ which is either a function of energy or momentum or both. However, Σ should, in principle, also modify the electronic eigenfunctions and thus change the real space charge distribution. While such practices are not prevalent, some works that use embedding techniques do take into account these effects. In such cases, choice of partitioning, of the parameters defining the correlated Hamiltonian, of double-counting corrections, and the adequacy of low-level Hamiltonian hosting the correlated subspace hinder a systematic and unambiguous understanding of such effects. Further, for a large variety of correlated systems, strong correlations are largely confined to the charge sector. Then an adequate nonlocal low-order theory is important, and the high-order local correlations embedding contributes become redundant. Here we study the impact of charge self-consistency within two example cases, TiSe2 and CrBr3, and show how real space charge re-distribution due to correlation effects taken into account within a first-principles Green’s function-based many-body perturbative approach is key in driving qualitative changes to the final electronic structure of these materials.

Intensity-dependent nonlinear optical properties in an asymmetric Gaussian potential quantum well-modulated by external fields

In this paper, the effects of external electric, magnetic and non-resonant intense laser fields on the nonlinear optical rectification (NOR), second-harmonic (SH), and third harmonic (TH) generation in a GaAs quantum well with asymmetrical Gaussian potential are theoretically investigated. Firstly, the energy eigenvalues and eigenfunctions of a single electron confined in the structure are obtained by using the diagonalization method within the framework of the effective-mass and parabolic band approaches. Then, using these energy eigenvalues and eigenfunctions, expressions derived within the compact density matrix approximation have been employed to calculate the coefficients of the nonlinear optical response in the structure. The obtained simulation results show that the influence of the external fields leads to significant changes in the coefficients of nonlinear optical rectification, second and third harmonic generation in the system. As a result, it has been seen that the amplitude and position of the peaks of nonlinear optical rectification, second and third harmonic coefficients can be controlled by changing the applied external fields.

Analysis of electron transport in the nano-scaled Si, SOI and III-V MOSFETs: Si/SiO2 interface charges and quantum mechanical effects

The ITRS predicts that the scaling of planar CMOS (Complementary Metal Oxide Semiconductor) technology will continue till the 22 nm technology node [1] and a possible extension beyond is appealing [2]. In this work, we investigate the effect of electron confinement [3] in nanoscaled transistor channels of 25 nm surface channel Si and 32 nm SOI (Silicon on Insulator) and 15 nm IF (Implant Free) III-V MOSFETs using a self-consistent solution of 1 D Poisson - Schrödinger equation [4,5]. For simulat ion and development with accuracy of nano-scaled of 25 nm gate length Si MOSFET (Metal Oxide Semiconductor Field Effect Transistor), 32 nm SOI Implant Free (IF) MOSFET, and 15nm Implant Free III-V MOSFET transistors, we investigated the bandstructure and quantum confinement effects occurring near the oxide-semiconductor interface inmetal-Oxide-Semiconductor (MOS) structure of Si MOSFET device. These investigation have been carried out using a selfconsistent solution of 1D Poisson-Schrödinger equation across the channel of conventional Si / SOI / III-V MOSFET Transistors. To solve self-consistently 1D Poisson-Schrödinger equations across the channel of a conventional Si, SOI, and an Implant Free III-V MOSFETs to determine the conduction and valence band profiles, electron density, electron sheet density, eigenstate and eigenfunctions in these structures. We present the simulat ion results of conduction band profile, electron density (classical and quantum mechanical), eigenstate and eigenfunctions for Si, SOI and III-V MOSFET structures at two different bias voltages of 0.5 V and 1.0 V. For comparison, we calculate the electron sheet density (quantum mechanically) as a function of the applied gate voltages.

Morse quantum well modulated by nonresonant intense laser field: Binding energy and optical absorption related to shallow donor impurities

Binding energies of the lowest two s-symmetric impurity states and linear, third order nonlinear, and total optical absorption coefficients corresponding to the 1s → 2s transitions for electrons in a Morse quantum well are investigated. Influence of a non-resonant intense laser field is considered for different values of the structure parameter, impurity position, and laser field intensity. In order to find the eigenvalues and eigenfunctions of the unbound electron confined within the Morse quantum well under the intense laser field, the time independent Schrödinger equation is solved by using the effective mass and parabolic band approximations. The impurity binding energy is determined by means of a variational procedure, using hydrogen-like trial wave functions, and the optical response is treated with the use of a two-level approach in the density matrix expansion. Our results show that it is possible to design suitable devices described by a Morse-like confinement pattern in the desired wavelength by changing the mentioned parameters.

Excited Electron–Hole States in Molecular Chains

The eigenvalues and the eigenfunctions of molecular excitons, charge-transfer excitons, and electron–hole pairs have been found in the approximation of electron and hole transfer between the lowest unoccupied and highest occupied orbitals in a rigid molecular chain of identical photosensitive molecules, the recognized model of organic solar cells. It has been shown that as the Coulomb binding energy decreases, the wave functions become superposition of functions of the increasing number of sites and the decay time, determined by electron or hole transitions, is shorter that the transfer time of the exciton as a whole, so that energy transfer and charge transfer become interrelated processes.

Coherent Transport of Electron Excitations in Organic Solar Cells

The mechanisms of formation and coherent transport of free and bound electron excited states in organic solar cells are considered. In the model of a photocell (one-dimensional chain of photosensitive molecules in a uniform electric field of the p–n junction), the energy eigenvalues and the eigenfunctions of molecular excitons, charge-transfer excitons (CTE), and electron–hole pairs are determined. It is assumed that processes of transport between adjacent sites dominate in the case of the Coulomb interaction between the electron and the hole constituting a CTE. With decreasing Coulomb coupling energy, the CTE wavefunctions become superpositions of localized functions of the increasing number of sites. The decay time determined by independent transitions of the electron and the hole in this case becomes shorter than the transport time of the CTE as a whole. It is shown that autoionization of molecular excitons and small-radius CTEs in a strong electric field of a nanosize chain induces the mixing of states of these excitons as well as of electron–hole pairs, which substantially increases the quantum yield of the photoeffect.

Electronic structure of tetra(4-aminophenyl)porphyrin studied by photoemission, UV–Vis spectroscopy and density functional theory

The valence and conduction bands of a thin film of tetra(4-aminophenyl)porphyrin (TAPP) are investigated by direct and inverse photoemission as well as by comparison to density functional theory (DFT) calculations. By projecting the electronic eigenfunctions onto the molecular framework it was possible to interpret the origin of each spectroscopic feature. Although the majority of the photoemission spectrum is attributed to the unsubstituted tetraphenylporphyrin (TPP) parent molecule, several features are clearly due to the amino substitution. Substitution also has important consequences for the energy positions of the frontier orbitals and therefore on the low-energy electronic excitations. The measured electronic transport energy gap (Eg=1.85eV) between the highest occupied molecular orbital (HOMO) and lowest unoccupied (LUMO) in TAPP is found to be significantly reduced with respect to TPP. Moreover, an increased energy separation between the two highest occupied states (HOMO and HOMO−1) is found both experimentally and by DFT calculations. Such evidence is attributed to an increased HOMO orbital destabilization due to an enhanced electron-donor character of the phenyl substituents upon amino functionalization. Finally, the above findings together with further time-dependent DFT calculations are used to interpret the effect of the amino groups on the UV–Vis absorption spectrum, namely an overall red-shift of the spectrum and remarkable intensity changes within the Q band.

Structure of electron eigenfunctions in the quantum theory of axial channeling

The transverse motion of fast charged particles within a crystal in the continuous potential field of atomic chains may be quantized. In the present study, the eigenfunctions of the discrete spectrum of the transverse motion of an electron in the field of a single atomic chain or a pair of parallel chains are determined (with [110] chains in a silicon crystal used as an example). The differences between eigenfunctions in the cases corresponding to regular and chaotic motion within the classical limit are demonstrated.

Electron-acoustic phonon field induced tunnel scattering

Theory of electron-acoustic single phonon scattering has been reconsidered. It is assumed that the non-degenerate semiconductor has a spherical parabolic band structure. In the basis of the reconsideration there is a phenomenon of the tilting of semiconductor bands by the perturbing potential of an electric field. In this case, electron eigenfunctions are not plane waves or Bloch functions. In low-field regime, the expressions for electron intraband transition probability and scattering time are obtained under elastic collision approximation. Dependencies of scattering time on electron energy and uniform electric field are analyzed. The results of corresponding numerical computations for n-Si at 300 K are presented. It is established that there is no fracture on the curve of electron scattering time dependence on the electron energy.

Compton scattering in strong magnetic fields: Spin-dependent influences at the cyclotron resonance

The quantum electrodynamical (QED) process of Compton scattering in strong magnetic fields is commonly invoked in atmospheric and inner magnetospheric models of x-ray and soft gamma-ray emission in high-field pulsars and magnetars. A major influence of the field is to introduce resonances at the cyclotron frequency and its harmonics, where the incoming photon accesses thresholds for the creation of virtual electrons or positrons in intermediate states with excited Landau levels. At these resonances, the effective cross section typically exceeds the classical Thomson value by over 2 orders of magnitude. Near and above the quantum critical magnetic field of 44.13 TeraGauss, relativistic corrections must be incorporated when computing this cross section. This paper presents formalism for the QED magnetic Compton differential cross section valid for both subcritical and supercritical fields, yet restricted to scattered photons that are below pair creation threshold. Calculations are developed for the particular case of photons initially propagating along the field, mathematically simple specializations that are germane to interactions involving relativistic electrons frequently found in neutron star magnetospheres. This exposition of relativistic, quantum, magnetic Compton cross sections treats electron spin dependence fully, since this is a critical feature for describing the finite decay lifetimes of the intermediate states. The formalism employs both the Johnson and Lippmann (JL) wave functions and the Sokolov and Ternov (ST) electron eigenfunctions of the magnetic Dirac equation. The ST states are formally correct for self-consistently treating spin-dependent effects that are so important in the resonances. Relatively compact analytic forms for the cross sections are presented that will prove useful for astrophysical modelers.

Mapped Finite Element Discrete Variable Representation

Efficient numerical solver for the Schrödinger equation is very important in physics and chemistry. The finite element discrete variable representation (FE-DVR) was first proposed by Rescigno and Mc-Curdy [Phys. Rev. A 62, 032706 (2000)] for solving quantum-mechanical scattering problems. In this work, an FE-DVR method in a mapped coordinate was proposed to improve the efficiency of the original FE-DVR method. For numerical demonstration, the proposed approach is applied for solving the electronic eigenfunctions and eigenvalues of the hydrogen atom and vibrational states of the electronic state 3Σg+ of the Cs2 molecule which has long-range interaction potential. The numerical results indicate that the numerical efficiency of the original FE-DVR has been improved much using our proposed mapped coordinate scheme.

Two electrons in a cylindrical quantum dot under constant magnetic field

We present the results obtained for the problem of two electrons in a cylindrical quantum dot with finite step potential in the presence of orthogonal magnetic field. The method we adopted is linear variational theory, where the basis states are constructed from single electron eigenfunctions of the harmonic oscillator potential. We show how the two electron energy levels vary with the magnetic field for various quantum numbers. Magnetization of the system is then calculated after determining its free energy at a non-zero temperature. Finally, we also plot the electron density and pair correlation function for various quantum numbers and field strengths.

Computation of the Electronic Flux Density in the Born–Oppenheimer Approximation

A molecule in the electronic ground state described in the Born–Oppenheimer approximation (BOA) by the wave function ΨBO = Φ0χ0 (where Φ0 is the time-independent electronic energy eigenfunction and χ0 is a time-dependent nuclear wave packet) exhibits a nonzero nuclear flux density, whereas it always displays zero electronic flux density (EFD), because the electrons are in a stationary state. A hierarchical approach to the computation of the EFD within the context of the BOA, which utilizes only standard techniques of quantum chemistry (to obtain Φ0) and quantum dynamics (to describe the evolution of χ0 on the ground-state potential energy surface), provides a resolution of this puzzling, nonintuitive result. The procedure is applied to H2(+) oriented parallel with the z-axis and vibrating in the ground state (2)Σg(+). First, Φ0 and χ0 are combined by the coupled-channels technique to give the normally dominant z-component of the EFD. Imposition of the constraints of electronic continuity, cylindrical symmetry of Φ0 and two boundary conditions on the EFD through a scaling procedure yields an improved z-component, which is then used to compute the complementary orthogonal ρ-component. The resulting EFD agrees with its highly accurate counterpart furnished by a non-BOA treatment of the system.

The optical phonon effect of quantum rod qubits

The Hamiltonian of a quantum rod with an ellipsoidal boundary is given by using a coordinate transformation in which the ellipsoidal boundary is changed into a spherical one. Under the condition of strong electron—longitudinal optical phonon coupling in the rod, we obtain both the electron eigenfunctions and the eigenenergies of the ground and first-excited state by using the Pekar-type variational method. This quantum rod system may be used as a two-level qubit. When the electron is in the superposition state of the ground and first-excited states, the probability density of the electron oscillates in the rod with a certain period. It is found that the oscillation period is an increasing function of the ellipsoid aspect ratio and the transverse and longitudinal effective confinement lengths of the quantum rod, whereas it is a decreasing function of the electron—phonon coupling strength.

Multifractal analysis of the electronic states in the Fibonacci superlattice under weak electric fields

Influence of the weak electric field on the electronic structure of the Fibonacci superlattice is considered. The electric field produces a nonlinear dynamics of the energy spectrum of the aperiodic superlattice. Mechanism of the nonlinearity is explained in terms of energy levels anticrossings. The multifractal formalism is applied to investigate the effect of weak electric field on the statistical properties of electronic eigenfunctions. It is shown that the applied electric field does not remove the multifractal character of the electronic eigenfunctions, and that the singularity spectrum remains non-parabolic, however with a modified shape. Changes of the distances between energy levels of neighbouring eigenstates lead to the changes of the inverse participation ratio of the corresponding eigenfunctions in the weak electric field. It is demonstrated, that the local minima of the inverse participation ratio in the vicinity of the anticrossings correspond to discontinuity of the first derivative of the difference between marginal values of the singularity strength. Analysis of the generalized dimension as a function of the electric field shows that the electric field correlates spatial fluctuations of the neighbouring electronic eigenfunction amplitudes in the vicinity of anticrossings, and the nonlinear character of the scaling exponent confirms multifractality of the corresponding electronic eigenfunctions.

Electronic energy levels, wavefunctions and potential landscape of nanostructures probed by magneto-tunnelling spectroscopy

We create electrostatically induced quantum dots by thermal diffusion of interstitial Mn out of a p-type (GaMn)As layer into the vicinity of a GaAs quantum well. This leads to the formation of deep, approximately circular and strongly confined dot-like potential minima in a large mesa diode structure. The minima are formed without need for advanced lithography or electrostatic gating. Using fields of up to 30 T, magnetotunnelling spectroscopy of an individual dot reveals the symmetry of the electronic eigenfunctions and, for the approximately circular dots, a rich spectrum of Fock-Darwin-like states with an orbital angular momentum component |lz| ranging from 0 up to 11. We find that a small fraction of the dots has elongated potential minima, giving rise to quenching of the orbital angular momentum of the electronic eigenstates. By developing a model to describe the diffusion of the Mn interstitial ions, we determine the electrostatic potential landscape in the quantum well and hence the distribution of dot shapes and sizes. This is in a good agreement with our experimental data.

Effect of wave-function localization on the time delay in photoemission from surfaces

We investigate streaking time delays in the photoemission from a solid model surface as a function of the degree of localization of the initial-state wave functions. We consider a one-dimensional slab with lattice constant alatt of attractive Gaussian-shaped core potentials of width σ . The parameter σ/alatt thus controls the overlap between adjacent core potentials and localization of the electronic eigenfunctions on the lattice points. Small values of σ/alatt � 1 yield lattice eigenfunctions that consist of localized atomic wave functions modulated by a “Bloch-envelope” function, while the eigenfunctions become delocalized for larger values of σ/alatt 0.4. By numerically solving the time-dependent Schr¨ odinger equation, we calculate photoemission spectra from which we deduce a characteristic bimodal shape of the band-averaged photoemission time delay: as the slab eigenfunctions become increasingly delocalized, the time delay quickly decreases near σ/alatt = 0.3 from relatively large values below σ/alatt ∼ 0.2 to much smaller delays above σ/alatt ∼ 0.4. This change in wave-function localization facilitates the interpretation of a recently measured apparent relative time delay between the photoemission from core and conduction-band levels of a tungsten surface.

Coupled-Channels Quantum Theory of Electronic Flux Density in Electronically Adiabatic Processes: Fundamentals

The Born-Oppenheimer (BO) description of electronically adiabatic molecular processes predicts a vanishing electronic flux density (j(e)), <j(e, NCM) (x,t) >=1/2∫dR[Δ(b) (x;R) - Δ(a) (x;R)]<j(b,a) (R,t)> even though the electrons certainly move in response to the movement of the nuclei. This article, the first of a pair, proposes a quantum-mechanical "coupled-channels" (CC) theory that allows the approximate extraction of j(e) from the electronically adiabatic BO wave function . The CC theory is detailed for H(2)(+), in which case j(e) can be resolved into components associated with two channels α (=a,b), each of which corresponds to the "collision" of an "internal" atom α (proton a or b plus electron) with the other nucleus β (proton b or a). The dynamical role of the electron, which accommodates itself instantaneously to the motion of the nuclei, is submerged in effective electronic probability (population) densities, Δ(α), associated with each channel (α). The Δ(α) densities are determined by the (time-independent) BO electronic energy eigenfunction, which depends parametrically on the configuration of the nuclei, the motion of which is governed by the usual BO nuclear Schrödinger equation. Intuitively appealing formal expressions for the electronic flux density are derived for H(2)(+).