Long-range Coulomb Potential Research Articles

Characteristic Spectrum of Very Low-Energy Photoelectron from Above-Threshold Ionization in the Tunneling Regime

We report an experimental and theoretical study of very low-energy photoelectrons in tunneling ionization process from noble gas atoms interacting with ultrashort intense infrared laser pulses. A universal peak structure with electron energy well below 1 eV in the photoelectron spectrum, corresponding to the double-hump structure in the longitudinal momentum distribution, is identified experimentally for all atomic species. Our quantum and semiclassical analysis reveal the role of long-range Coulomb potential in the production of this very low-energy peak structure.

Quantum phase-space analysis of electronic rescattering dynamics in intense few-cycle laser fields

We present a quantum phase-space analysis of the ionization and rescattering dynamics of a one-electron atom in an intense few-cycle laser field. Snapshots of the Wigner function W(x, p) will be analysed for both soft-core Coulomb and short-ranged potentials tuned to yield identical ground state energies and similar energies for the first lowest electronic states. The influence of the long-range Coulomb potential on the rescattering wavepacket in the continuum can be disentangled from that of the induced bound-state polarization. Short- and long-range atomic potentials entail differences in the rescattering dynamics, manifesting themselves in high-order harmonic generation spectra.

Positive and negative Coulomb drag in vertically integrated one-dimensional quantum wires

Electron interactions in and between wires become increasingly complex and important as circuits are scaled to nanometre sizes, or use reduced-dimensional conductors such as carbon nanotubes, nanowires and gated high-mobility two-dimensional electron systems. This is because the screening of the long-range Coulomb potential of individual carriers is weakened in these systems, which can lead to phenomena such as Coulomb drag, where a current in one wire induces a voltage in a second wire through Coulomb interactions alone. Previous experiments have demonstrated Coulomb electron drag in wires separated by a soft electrostatic barrier of width ≳80nm (ref.12), which was interpreted as resulting entirely from momentum transfer. Here, we measure both positive and negative drag between adjacent vertical quantum wires that are separated by ∼15nm and have independent contacts, which allows their electron densities to be tuned independently. We map out the drag signal versus the number of electron sub-bands occupied in each wire, and interpret the results both in terms of momentum-transfer and charge-fluctuation induced transport models. For wires of significantly different sub-band occupancies, the positive drag effect can be as large as 25%.

Numerical treatment of the long-range Coulomb potential with Berggren bases

The Schrodinger equation incorporating the long-range Coulomb potential takes the form of a Fredholm equation whose kernel is singular on its diagonal when represented by a basis bearing a continuum of states, such as in a Fourier-Bessel transform. Several methods have been devised to tackle this difficulty, from simply removing the infinite-range of the Coulomb potential with a screening or cut function to using discretizing schemes which take advantage of the integrable character of Coulomb kernel singularities. However, they have never been tested in the context of Berggren bases, which allow many-body nuclear wave functions to be expanded, with halo or resonant properties within a shell model framework. It is thus the object of this paper to test different discretization schemes of the Coulomb potential kernel in the framework of complex-energy nuclear physics. For that, the Berggren basis expansion of proton states pertaining to the sd-shell arising in the A ~ 20 region, being typically resonant, will be effected. Apart from standard frameworks involving a cut function or analytical integration of singularities, a new method will be presented, which replaces diagonal singularities by finite off-diagonal terms. It will be shown that this methodology surpasses in precision the two former techniques.

Coulomb bridge function and the pair-distribution functions of the two-dimensional electron liquid in the quantum regime

The electron-electron pair-distribution functions (PDFs) of the two-dimensional (2D) electron fluid in the quantum regime (at $T=0$) are calculated using a classical-map-hypernetted-chain scheme and compared with currently available quantum Monte Carlo (QMC) simulations in the coupling range ${r}_{s}=1$ to 50. We iteratively extract the bridge function of the ``equivalent'' classical 2D liquid in the quantum regime. These bridge functions $B(r)$ are relatively insensitive to spin-polarization effects. The structure of the bridge functions changes significantly for ${r}_{s}>6$, suggesting the onset of strongly correlated clusters. The new $B(r)$, appropriate for the long-range Coulomb potential, can be used to replace the hard-sphere $B(r)$ previously used in these calculations. They provide accurate classical representations of the QMC-PDFs even at very strong coupling, and probably at finite-$T$ near $T=0$.

Diffraction at a time grating in above-threshold ionization: The influence of the Coulomb potential

We analyze the photoelectron emission spectrum in atomic above-threshold ionization by a linearly polarized short-laser pulse. Direct electrons can be characterized by both intracycle and intercycle interferences. The former results from the coherent superposition of two different electron trajectories released in the same optical cycle, whereas the latter is the consequence of the superposition of multiple trajectories released in different cycles. In the present article, a semiclassical analytical expression for the complete (both intracycle and intercycle) interference pattern is derived. We show that the recently proposed semiclassical description in terms of a diffraction process at a time grating remains qualitatively unchanged in the presence of the long-range Coulomb potential. The latter causes only a phase shift of the intracycle interference pattern. We verify the predictions of the semiclassical model by comparison with full three-dimensional (3D) time-dependent Schr\odinger equation (TDSE) solutions. One key finding is that the subcycle interference structures originating from trajectories launched within a time interval of less than 1 femtosecond should be experimentally observable also in low-resolution spectra for longer multicycle pulses.

Role of Coulomb focusing on the electron transverse momentum of above-threshold ionization

We have investigated the 2D photoelectron momentum spectra for ATI of atoms exposed to linearly polarized pulses by quantum mechanical calculations. By comparing the 2D momentum spectra for the long-range and short-range Coulomb potentials, the focusing of the electron transverse momentum by the long-range interaction is clearly revealed. Analysis indicates that the Coulomb attraction of the parent core to the returning electron is responsible for the focusing of the electron transverse momentum. Moreover, the strong dependence of the focusing of the electron transverse momentum on the laser wavelength and intensity is discussed.

Stripe Formation Induced by Doping

Formation of charged clusters in two-dimensional systems, are studied by means of Landau theory of phase transitions. We analyzed shapes and sizes of charged cluster in two-dimensional systems. Following previous works, where we proposed that the charge is coupled to the square of the order parameter (Jamei et al. in Phys. Rev. Lett. 94:056805, 2005), we study now the case when the charge density is coupled to the order parameter linearly as suggested Miranda and Kabanov (Physica C 468:358–361, 2008). However, in both cases the long-range Coulomb potential is present and we prove that it is the key factor for the patterns obtained. In particular we show that a stripe phase exists in a wide range of doping.

The Spatial Region of Photoionization in a Strong Laser Field

In the framework of the Keldysh–Faisal–Reiss theory in the velocity gauge we investigate the magnitude of the space region, where photoionization in a strong laser field takes place. We find substantial differences between the short-range and the long-range (Coulomb) potentials, and between linear and circular polarizations of incident radiation. It appears that only for the initial state in the Coulomb potential the region of space, where ionization is held, expands significantly with increasing intensity for a typical optical frequency and non-relativistic but strong circularly polarized laser field. As a result of our considerations, we suggest to modify the idea of Reiss and Krainov of a certain simple Coulomb correction to the Volkov wave function. We show that photoionization rate calculated for the H(1s) atom, using our approach, is in better agreement with other theoretical results for moderately strong circularly polarized laser field.

Carrier wave-packet transport under the influence of charged quantum dot in small-area resonant tunneling diode

It was shown experimentally that charging InAs quantum dots (QDs) embedded in a small-area GaAs/AlAs double-barrier resonant tunneling diode (RTD) effectively modified the carrier transport properties of the RTD. By adopting and comparing the one-dimensional plane-wave and three-dimensional (3D) wave-packet transport theories we show that the electron transports in the QDRTD device are in the form of 3D wave packets, which are strongly affected by the 3D long-range Coulomb potential induced by charged InAs QDs. This explains well experimental data and indicates that the 3D wave-packet transport model is more appropriate for the QDRTD device.

Faddeev-Noble Versus Sasakawa-Sawada Equations for the Nuclear Three-Body Coulomb Problem

We compare the efficiency of two Faddeev-type approaches for the nuclear three-body Coulomb problem. The first one is a modification of Noble’s approach, the second one is due to Sasakawa and Sawada. In an integral-equation formulation both of these methods rely on the same Green’s operators and driving terms. The differences lie in the treatment of the long-range Coulomb potentials. Numerical examples show that the modified Faddeev-Noble approach provides for faster convergence.

Optical and spectral properties of quantum domain walls in the generalized Wigner lattice

We study the spectral properties of a system of electrons interacting through long-range Coulomb potential on a one-dimensional chain. When the interactions dominate over the electronic bandwidth, the charges arrange in an ordered configuration that minimizes the electrostatic energy, forming Hubbard's generalized Wigner lattice. In such strong coupling limit, the low energy excitations are quantum domain-walls that behave as fractionalized charges, and can be bound in excitonic pairs. Neglecting higher order excitations, the system properties are well described by an effective Hamiltonian in the subspace with one pair of domain-walls, which can be solved exactly. The optical conducitivity $\sigma(\omega)$ and the spectral function $A(k,\omega)$ can be calculated analytically, and reveal unique features of the unscreened Coulomb interactions that can be directly observed in experiments.

Local Density of States around Impurity in a Strong Magnetic Field: I. Two-Dimensional System with Parabolic Dispersion

Bound states around an impurity are investigated for a two dimensional electron system in a strong magnetic field. Long-range Coulomb potential and related potentials are considered. Schr\"odinger equation is solved numerically to obtain the bound states. The energy and wave function of these bound states are indirectly observed by the scanning tunneling spectroscopy as local density of states (LDOS). Theoretically obtained LDOS is compared with experiment. Reasonable agreement is obtained.

Analysis of two-dimensional photoelectron momentum spectra and the effect of the long-range Coulomb potential in single ionization of atoms by intense lasers

Two-dimensional (2D) electron momentum distributions and energy spectra for multiphoton ionization of atoms by intense laser pulses, calculated by solving the time-dependent Schr\odinger equation (TDSE) for different wavelengths and intensities, are compared to those predicted by the strong-field approximation (SFA). It is shown that the momentum spectra at low energies between the TDSE and SFA are quite different and the differences arise largely from the absence of a long-range Coulomb interaction in the SFA. We further found that the low-energy 2D momentum spectra from the TDSE exhibit ubiquitous fanlike features where the number of stripes is due to a single dominant angular momentum of the low-energy electron. The specific dominant angular momentum in turn has been found to be decided by the minimum number of photons needed to ionize the atom only. The electron momentum spectra predicted by models modified from the SFA are also examined and found to lack the fanlike features as in the SFA.

Ionization of a hydrogen-like ion by a strong low-frequency linearly polarized laser field

The influence of the long-range Coulomb potential on the ionization rate of a hydrogen-like ion has been investigated within the framework of the quasi-classical method. In a rather strong field, the dependence of the ionization rate on the field intensity does not have a purely power-law character in the multiphoton region. The analytical equations obtained are in agreement with the experimental data on the cross section of multiphoton ionization and give an estimate of the threshold field values in the tunneling region.

Two- and three-charged-particle nuclear scattering in momentum space: A two-potential theory and a boundary condition model

The two- and three-charged-particle nuclear scattering problems are investigated in momentum space. The three-body equations with a short-range nuclear potential, a three-body force potential, and the long-range Coulomb potential are presented in a mathematically rigorous way within a generalized two-potential theory. To remove the serious singularity in the two- and three-body Coulomb problems in momentum space, we have proposed a novel boundary condition to the phase shift that arises from a potential difference, the so-called auxiliary potential (AP), between the Coulomb potential and the screened Coulomb one: ${V}^{\ensuremath{\phi}}={V}^{C}\ensuremath{-}{V}^{R}$. Furthermore, we point out the importance of the off-shell amplitude for the AP, by which one can uniquely obtain the two-body on-shell and off-shell Coulomb amplitude in momentum space. Therefore, this formulation is also useful for atomic systems, which is another benefit. It is recalled that the traditional phase-shift renormalization theory, in which the screened Coulomb amplitude is sandwiched by renormalized phase factors ${e}^{i\ensuremath{\phi}}$, is not consistent with two-potential theory. Some ambiguities or misunderstandings of the traditional methods for handling the Coulomb problem are clarified. Finally, the three-body unitarity relation is proved for the amplitude generated from these three kinds of potential. Moreover, a generalized two-potential theory is presented. It is pointed out that the asymptotic three-body nuclear wave function with the Coulomb potential could not be written by a product of two individual wave functions ${\stackrel{~}{\ensuremath{\psi}}}_{x}(x){\stackrel{~}{\ensuremath{\psi}}}_{y}(y)$ defined in terms of Jacobi coordinates $x$ and $y$, respectively.

Quadratic response theory for spin-orbit coupling in semiconductor heterostructures

This paper examines the properties of the self-energy operator in lattice-matched semiconductor heterostructures, focusing on nonanalytic behavior at small values of the crystal momentum, which gives rise to long-range Coulomb potentials. A nonlinear response theory is developed for nonlocal spin-dependent perturbing potentials. The ionic pseudopotential of the heterostructure is treated as a perturbation of a bulk reference crystal, and the self-energy is derived to second order in the perturbation. If spin-orbit coupling is neglected outside the atomic cores, the problem can be analyzed as if the perturbation were a local spin scalar, since the nonlocal spin-dependent part of the pseudopotential merely renormalizes the results obtained from a local perturbation. The spin-dependent terms in the self-energy therefore fall into two classes: short-range potentials that are analytic in momentum space, and long-range nonanalytic terms that arise from the screened Coulomb potential multiplied by a spin-dependent vertex function. For an insulator at zero temperature, it is shown that the electronic charge induced by a given perturbation is exactly linearly proportional to the charge of the perturbing potential. These results are used in a subsequent paper to develop a first-principles effective-mass theory with generalized Rashba spin-orbit coupling.

Spectral properties of LiTaO3:Pr3+ under high hydrostatic pressure

The high-pressure photoluminescence of theLiTaO3:Pr3+ system obtained for the pressure range from ambient to 232 kbar at 20 K andambient temperatures is presented. The observed pressure red shift of allspectral lines representing the f–f transitions is related to the changes inF2 Racach integral and the spin–orbit coupling with pressure. To analyse the deexcitationkinetics and the dependence of the intensity of the and on the excitation wavelength and pressure we propose a model of aPr3+ trapped exciton witha hole localized at the Pr3+ and an electron at the Rydberg states bounded by the long-range Coulomb potential of thehole.

Theoretical study of phosphorousδ-doped silicon for quantum computing

We present microscopic model calculations of phosphorous $\ensuremath{\delta}$-doped silicon. Using the planar Wannier orbitals [Y. C. Chang and G. Li, Comput. Phys. Commun. 95, 158 (1996)] obtained based on the pseudopotential method, we calculate the electronic structures of a large slab (1000 ML) self-consistently (for doping electrons), taking into account both the long-range Coulomb potential in the direction perpendicular to the doping plane and the short-range interaction due to the $1∕4$ monolayer (ML) phosphor $\ensuremath{\delta}$ doping. Our results show that the Fermi level after doping is about $100\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$ below silicon conduction band minimum (CBM). We also find that the short-range interaction due to P dopants only has a minor influence on the Fermi level.

Effective way to sum over long-range Coulomb potentials in two and three dimensions

I propose a method to calculate the logarithmic interaction in two dimensions and the Coulomb interaction in three dimensions under periodic boundary conditions. This paper considers the case of a rectangular cell in two dimensions and an orthorhombic cell in three dimensions. Unlike the Ewald method, there is no parameter to be optimized, nor does the method involve error functions, thus leading to the accuracy obtained. This method is similar in approach to that of Mol. Simul. 22, 199 (1999)]], but the derivation is considerably simpler and physically appealing. An important aspect of the proposed method is the faster convergence of the Green's function for a particular case as compared to Sperb's work. The convergence of the sums for most parts of the unit cell is exponential, and hence requires the calculation of only a few dozen terms. In a very simple way, we also obtain expressions for the interaction for systems with slab geometries. Expressions for the Madelung constants of CsCl and NaCl are also obtained.