Finite Number Of Electrons Research Articles

Drop model of integer and fractional quantum Hall effects

In this paper, we present a drop model for integer and fractional quantum Hall effects (FQHE). We show that the two-dimensional electron gas breaks up into regions with filling factors [Formula: see text] and [Formula: see text] in the disk geometry, and the formation of drops with a finite number of electrons is possible. Sequences of filling fractions are constructed on the basis of experimental data. For all sequences, there are initial FQHE states, which correspond to a drop with five electrons. The remaining FQHE states are composite states of a drop with five electrons and one or more pairs of electrons.

Electronic and magnetic properties of many-electron complexes in charged InAsxP1−x quantum dots in InP nanowires

We present here a microscopic theory of electronic complexes in charged $\mathrm{In}{\mathrm{As}}_{x}{\mathrm{P}}_{1\ensuremath{-}x}$ quantum dots in InP nanowires with a hexagonal cross section and determine the potential use of an array of such quantum dots as a synthetic spin chain for the possible construction of a topological qubit. The single-particle energies and wave functions are obtained by diagonalizing a microscopic atomistic tight-binding Hamiltonian of multiple quantum dots in the basis of $s{p}^{3}{d}^{5}{s}^{*}$ local atomic orbitals for a given random distribution of arsenic (As) vs phosphorus (P) atoms. The conduction band electronic states are found grouped into $s, p$, and $d$ quantum dot shells. For a double dot, the electronic shells can be understood in terms of interdot tunneling despite the random distribution of As atoms in each quantum dot. The single- and double-dot structures were charged with a finite number of electrons. The many-body Hamiltonian including Coulomb electron-electron interactions was constructed using single atomistic particle states and then diagonalized in the space of many-electron configurations. For a single dot filled with ${N}_{e}=1--7$ electrons, the ground state of a half-filled $p$-shell configuration with ${N}_{e}=4$ was found with total electronic spin $S=1$. The low-energy spectrum obtained using exact diagonalization of a Hamiltonian of a charged double dot filled with ${N}_{e}=8$ electrons, i.e., half-filled $p$ shells in each dot, was successfully fitted to the Hubbard-Kanamori and antiferromagnetic Heisenberg spin-1 Hamiltonians. The atomistic simulation confirmed the potential of InAsP/InP quantum dots in a nanowire for the design of synthetic spin chains.

Space-Charge Limited Current From a Finite Emitter in Nano- and Microdiodes

We simulate numerically the classical charge dynamics in a microscopic, planar, vacuum diode with a finite emitter area and a finite number of electrons in the gap. We assume electrons are emitted under space-charge limited conditions with a fixed potential applied to the diode. The Coulomb interaction between all electrons is included using the method of molecular dynamics. We compare our results to the conventional two-dimensional Child-Langmuir and explain how it is limited in applicability for sub-micron diameter emitters. Finally, we offer some simple relations for understanding space-charge limited flow from very small emitters.

The Einstein–Infeld–Hoffmann legacy in mathematical relativity II. Quantum laws of motion for singularities of spacetime

We report on recent developments toward a relativistic quantum-mechanical theory of motion for a fixed, finite number of electrons, photons and their anti-particles, as well as its possible generalizations to other particles and interactions.

To the Theory of Inhomogeneous Electron Gas

Nonrigorous character of the density-functional theory for inhomogeneous electron gas based on the hypothesis assuming the existence of a universal density functional is demonstrated. A single-particle density matrix must be determined to calculate the ground-state energy of a finite system with a finite number of electrons. A single-particle Green function can be used to unambiguously determine the ground-state energy of an inhomogeneous electron system that satisfies the thermodynamic limit.

Energy in a finite two-dimensional spinless electron gas

We study the properties of a finite two-dimensional electron gas system in the Hartree-Fock approximation. We obtain exact analytical expressions for the energy in a finite two-dimensional fully spin-polarized (spinless) system of electrons interacting with a Coulomb potential immersed in a finite square region uniformly filled with a neutralizing positive charge. The difficult two-electron integrals over the finite square domain are reduced to simple compact expressions involving analytic auxiliary functions. We provide results for the potential energy of systems with a finite number of electrons and show how the energy slowly converges towards its thermodynamic limit bulk value.

Electron gas model for inhomogeneous systems

Recently it has been shown that very accurate exchange and correlation potentials may be predicted from the properties of the one- and two-particle density matrices for a system with a finite number of electrons. The charge density dependence of the exchange potential thus derived is the same as that of a free electron gas (homogeneous system). In order to take into account the inhomogeneity effects, we have carried out an expansion of the charge density around the center of the Fermi hole. This leads to an exchange potential of the same form as that of the Xαβ theory. The value of β predicted for a modified Wigner form of the pair correlation function of electrons with parallel spin, f↑ (1,2), when the number of electrons goes to infinity is 0.0020, in agreement with the theoretical value found by Liberman. Exchange, correlation, and total energies of atoms Li through Ar predicted from several models, corresponding to different forms of f↑ (1,2), are in very good agreement with the exact values.

Shot noise in electron‐beam lithography and line‐width measurements

Electron-beam lithography is used extensively in nanoscience and technology for making masks for the semiconductor industry and, on a limited scale, for maskless lithography: that is, writing the patterns directly on the chip. We expect the latter application to extend in the years to come. Control of the dimensions of the written structures is essential in the semiconductor industry. For 45 nm generation, which is presently under development and should reach production at the end of the decade, the required control over the line width is between 1.5 and 5 nm, depending on the application. One of the factors of influence on the line-width control is the statistics in the number of electrons illuminating the resist. This effect gives line edge roughness, or in other words a lack of control over the local position of a resist edge. This has long been recognized and often discussed. Recently, we developed an analytic model for the line edge position variation, which we shall illustrate and expand in this paper. The model, supported by Monte Carlo simulations, demonstrates that the line-width variation is inversely proportional to the dose used for the illumination of the resist. This makes it impossible to increase the lithography throughput by developing ultrasensitive resists. For 45 nm features written with a typical resolution of 30 nm, a 30 microC/cm2 resist gives 3 nm line-width variation over line segments of 45 nm long. The line width is usually measured in an adapted critical dimension scanning electron microscope (CD-SEM). This measurement needs to be more precise than the result of the lithography step, so the requirements are typically sub-nm. Apart from all the problems to avoid systematic errors, this measurement also suffers from statistical variations, resulting from the finite number of electrons used for the measurement. In this paper we shall derive an estimate for that variation with a similar model as used for the shot noise effect in the lithography step. One of the conclusions is that for the most precise measurements of the line width it is not advisable to tune the CD-SEM for the best resolution. It is better to allow a larger probe size of the electron beam because that can be accompanied by a much larger current and thus a decrease in the noise level.

Negative-U properties for a quantum dot

We present the first findings of the quantum dot that exhibits the negative-U properties caused by a net attraction between the Coulombically repulsive carriers. Therefore, the finite number of electrons (or holes) inside the dot is partitioned into pairs with the second bound more strongly than the first. The negative-U properties for a quantum dot appear to result from twice the amplitude of the Coulombic oscillations with negative (or positive) variations in the sign of the gate voltage. This identification is made possible in the studies of the resonant tunnelling through the quantum dot strongly coupled to the quantum wire that is prepared by the split-gate method inside the silicon quantum well divided by the ferroelectric δ-barriers on the Si(100) surface. The negative-U double quantum dot is found to exhibit the room-temperature operation of single-hole memory such as the hysteresis in the current–voltage characteristics revealed by varying the gate voltage.

Effects of nonequilibrium electron distribution and electron-electron interaction observed in spin-split cyclotron resonance of InAs/AlSb single quantum wells at high magnetic fields

We have observed two remarkable phenomena in spin-split cyclotron resonance of InAs/AlSb single quantum wells under short-pulse high magnetic fields produced by the single-turn coil technique. The first phenomenon is a hysteretic phenomenon, which was observed with a rapid sweep of the magnetic field in increasing and decreasing directions (\ensuremath{\sim}100 T in a few microseconds). The sweep-rate dependence of the cyclotron resonance spectra revealed that the hysteretic phenomenon is attributable to the nonequilibrium electron distribution in the lowest Landau levels $(N=0)$ with plus and minus spins induced by short-pulse high magnetic fields. At low temperatures of around 15 K, a disagreement in intensity between the two different spins is observed in such a way that the total intensity is conserved. The hysteresis is attributed to a slow spin-flip relaxation between the two spin states of the $N=0$ Landau level with a time constant of 1 \ensuremath{\mu}s comparable with the field pulse duration. At high temperatures of around 100 K, a different type of hysteresis is found, where the absorption intensity of the minus spin is smaller in the down sweep than in the up sweep. The sweep-rate dependence shows that the up sweep does not change significantly, but the minus spin decreases with increasing the sweep rate in the down sweep. This is attributed to a slow electron transfer from donor levels in AlSb to a Landau level in InAs. The second phenomenon is the electron-electron interaction effect on the spin-split cyclotron resonance, which displays an intriguing dependence on the filling factor \ensuremath{\nu} and the temperature at around 45 T, where $0.3<~\ensuremath{\nu}<~1.2.$ With increasing temperature, the absorption intensity of the down spin increases, exceeding the amount expected from the Boltzmann distribution, while that of the up spin shows the opposite behavior. For samples with different carrier concentrations, the spin splitting at $\ensuremath{\nu}=0.7$ is smaller than that at $\ensuremath{\nu}=0.4$ by 1 T at around 50 K. These features can be explained qualitatively by a recent theory by Asano and Ando that calculates the effect of the electron-electron interaction on cyclotron resonance spectra by numerical diagonalization for a finite-size system with a finite number of electrons.

Edge magnetoplasmon excitations in a quantum dot in high magnetic fields

We investigate the collective magnetoplasmon excitations in a quantum dot containing finite number of electrons in the high magnetic field limit. We consider the electrons in the lowest Landau level and neglect mixing between the higher Landau levels. The dispersion relation of these edge modes are estimated following the energy weighted sum-rule approach. In this finite size system the edge magnetoplasmon modes have different multipolarities (or angular momemtum k). Their dependence on the magnetic field and on the system size is investigated. With increasing magnetic field, energy of these collective modes decreases and in the bulk limit they become gapless. We also consider the breathing mode of a dot in the presence of a strong magnetic field, and the energy of this mode approaches the cyclotron frequency ?\(\).

Thermodynamic properties of a small superconducting grain

The reduced BCS Hamiltonian for a metallic grain with a finite number of electrons is considered. The crossover between the ultrasmall regime, in which the level spacing d is larger than the bulk superconducting gap $\ensuremath{\Delta}$ and the small regime, where $\ensuremath{\Delta}\ensuremath{\gtrsim}d,$ is investigated analytically and numerically. The condensation energy, spin magnetization, and tunneling peak spectrum are calculated analytically in the ultrasmall regime, using an approximation controlled by $1/\mathrm{ln}N$ as a small parameter, where N is the number of interacting electron pairs. The condensation energy in this regime is perturbative in the coupling constant \ensuremath{\lambda} and is proportional to $\mathrm{dN}{\ensuremath{\lambda}}^{2}={\ensuremath{\lambda}}^{2}{\ensuremath{\omega}}_{D}.$ We find that also in a large regime with $\ensuremath{\Delta}>d,$ in which pairing correlations are already rather well developed, the perturbative part of the condensation energy is larger than the singular, BCS part. The condition for the condensation energy to be well approximated by the BCS result is found to be roughly $\ensuremath{\Delta}>\sqrt{d{\ensuremath{\omega}}_{D}}.$ We show how the condensation energy can, in principle, be extracted from a measurement of the spin magnetization curve and find a reentrant susceptibility at zero temperature as a function of magnetic field, which can serve as a sensitive probe for the existence of superconducting correlations in ultrasmall grains. Numerical results are presented, which suggest that in the large N limit the $1/N$ correction to the BCS result for the condensation energy is larger than \ensuremath{\Delta}.

Collective excitations in quantum dots

We investigate different types of collective excitations in a quantum dot containing finite number of electrons at zero magnetic field. To estimate the excitation energies analytically we follow the energy weighted sum-rule approach. We consider the most general multipole excitation with angular momentum l, and the breathing mode excitation (monopole excitation) of a large quantum dot, for three different types of effective electron–electron interaction. These are the logarithmic interaction, the short-range pseudopotential and the Coulomb interaction. The ground-state density of the many-body system is calculated within Thomas–Fermi approximation. The analytical results for the collective excitation energies and their dependence on the system size and other external parameters are discussed in detail.

Hypernetted-chain treatment and the extended shadow wave functions for fractional quantum Hall hierarchical states

The hypernetted-chain theory is applied to study hierarchical states in the fractional quantum Hall effect. It is noted that a class of wave functions introduced by Girvin [Phys. Rev. B 29, 6012 (1984)] and MacDonald, Aers, and Dharma-wardana [Phys. Rev. B 31, 5529 (1985)], based on charge-conjugation procedures, is of the extended shadow wave-function type. The correlation energy, pair distribution function, and static structure function have been calculated in the thermodynamic limit at various filling factors. The results obtained agree with those of previous calculations performed with a finite number of electrons.

Improved Transport Equations Including Correlations for Electron–Phonon Systems: Comparison with Exact Solutions in One Dimension

We study transport equations for quantum many-particle systems in terms of correlations by applying the general formalism developed in an earlier paper to exactly soluble electron-phonon models. The one-dimensional models considered are the polaron model with a linear energy dispersion for the electrons and a finite number of electrons and the same model including a Fermi sea (Tomonaga-Luttinger model). The inclusion of two-particle correlations shows a significant and systematic improvement in comparison with the usual non-Markovian equations in Born approximation. For example, the improved equations take into account the renormalization of the propagation by the self-energies to second order in the coupling.

Multi-anyon ions and photoluminescence spectrum of fractional quantum Hall systems

Abstract Quasiparticles of fractional quantum Hall effect states are anyons carrying fractional charge and obeying fractional statistics. Anyons can interact with charged impurities on or near the plane of the electron layer to form bound states. We study a system consisting of a finite number of electrons confined to a spherical surface interacting with a localized valence-band hole separated from the surface by a distance d. A number of ionic states, consisting of several quasielectrons bound to the localized hole, can be identified from the dependence of the energy spectrum on Landau level filling factor v e. The photoluminescence spectrum resulting from electron-hole recombination is studied as a function of separation d and filling factor v e at a finite temperature. The multipeaked structure of the general spectrum supports the hypothesis of stable anyonic ions.

On the applicability of the screened‐Coulomb exchange model in Kohn–Sham density functional studies

AbstractThe screened‐Coulomb exchange model used first in the Xα method and later on in Kohn–Sham density functional theory is reexamined. Based on the well‐elaborated framework of the local spin‐density approximation, we show that this model is not well suited when systems with a finite number of electrons are concerned, because it does not respect the pair‐density sum rule. A proper modification of the model is proposed by reformulating it in terms of a screened‐exchange hole and ensuring the sum rule for this hole. As a result, it is shown how the static screened exchange in finite systems is accompanied by a conjugate antiscreened one. The possible consequences of this effect on the application of the screened‐Coulomb exchange model are discussed. © 1994 John Wiley & Sons, Inc.

The influence of Coulomb interaction on transport through mesoscopic two-barrier structures

We investigate the influence of the Coulomb interaction on the energy spectrum of a finite number of electrons in a geometrically confined quantum mechanical system. The spectrum is calculated numerically using the Slater determinants of the one-electron states as basis set. It is found to be dominated by the Coulomb repulsion when the system is large. Coulomb and exchange matrix elements for a given combination of four one-electron states are of the same order of magnitude. As a consequence, the energy difference between the ground states of the (N+1)- and theN-electron system is an order of magnitude smaller than each of the matrix elements, although being much larger than the separation of the one-electron energy levels. We discuss the importance of the interaction effects for the explanation of the recently observed resonant behavior of the electronic transport through quantum dots.

Kinetic and exchange energy density functionals for finite systems

The authors propose a new model for the Fermi correlation function for a pair of electrons of the same spin in a system with a finite number of electrons. Using this correlation function, they have derived new expressions for the exchange and kinetic energy density functionals, depending explicitly on the number of electrons. The numerical results obtained for several atomic systems show good agreement, thus demonstrating the accuracy of prediction of the proposed model.

Image processing for electron energy loss microanalysis

Over the years biology and materials science have profited greatly from the imaging potential of electron microscopy for structural and morphological analysis. Gross structures, grains, subgrains or subcellular organelles, mitochondria and the Golgi were fairly easy to interpret at a glance. However, as more high resolution detail was sought in the form of macromolecular structure, or dislocations or even atomic images, the wave nature, the finite number of electrons and specimen destruction from the high energy of the latter all made interpretation of the images more difficult. Image simulation, image analysis and image processing have helped greatly to extract relevant information and have assisted in our understanding not only of the structure per se but also of the interaction between the structure and the probing electron.