Electronic properties of a large quantum dot at a finite temperature

The physical properties of a two-dimensional parabolic quantum dot composed of large number of interacting electrons are numerically determined by the Thomas–Fermi (TF) method at a finite temperature. Analytical solutions are given for zero temperature for comparative purposes. The exact solution of the TF equation is obtained for the non-interacting system at finite temperatures. The effect of the number of particles and temperature on the properties are investigated both for interacting and non-interacting cases. The results indicate that the effect of e–e interaction on the density profile shows different temperature dependencies above and below a certain temperature T_c.     

Solutions of the Lane–Emden and Thomas–Fermi Equations

A solution of the Lane–Emden equation is obtained based on the approach developed for the Thomas–Fermi equation. The solution is obtained for small values of the independent variable. Based on the solution obtained, analytical expressions are suggested for arbitrary values of the independent variable.     

A semiclassical approach to the ground state and density oscillations of quantum dots

A semiclassical Thomas–Fermi method, including a Weizsäcker gradient term, is implemented to describe ground states of two-dimensional nanostructures of arbitrary shape. Time-dependent density oscillations are addressed in the same spirit using the corresponding semiclassical time-dependent equations. The validity of the approximations is tested, both for ground state and density oscillations, compared with the available microscopic Kohn–Sham solutions.     

Majorana Transformation for Differential Equations

We present a method for reducing the order of ordinary differential equations satisfying a given scaling relation (Majorana scale-invariant equations). We also develop a variant of this method, aimed to reduce the degree of nonlinearity of the lower order equation. Some applications of these methods are carried out and, in particular, we show that second-order Emden–Fowler equations can be transformed into first-order Abel equations. The work presented here is a generalization of a method used by Majorana in order to solve the Thomas–Fermi equation.     

Langevin Equation of Collective Modes of Bose–Einstein Condensates in Traps

A quantum Langevin equation for the amplitudes of the collective modes in Bose–Einstein condensate is derived. The collective modes are coupled to a thermal reservoir of quasi-particles, whose elimination leads to the quantum Langevin equation. The dissipation rates are determined via the correlation function of the fluctuating force and are evaluated in the local-density approximation for the spectrum of quasi-particles and the Thomas–Fermi approximation for the condensate.I take great pleasure in dedicating this paper to Gregoire Nicolis on the occasion of his sixtieth birthday.     

Growth by molecular beam epitaxy and electrical characterization of GaAs nanowires

We report on structural and electrical properties of GaAs nanowires (NWs) grown by molecular beam epitaxy (MBE) on GaAs and SiO₂ substrates using Au as growth catalyst. Au–Ga particles are observed on the top of the NWs by transmission electron microscopy (TEM). In most of the observed cases, individual particles contain two Au–Ga compositions, in particular orthorhombic AuGa and β′ hexagonal Au₇Ga₂. The wires grown on GaAs are regularly shaped and tidily oriented on both (1 0 0) and (1 1 1)B substrates. TEM also reveals that the NWs have a wurtzite lattice structure. Electrical transport measurements indicate that nominally undoped NWs are weakly n-type while both Be- and Si-doped wires show p-type behaviour. The effect of the lattice structure on impurity incorporation is briefly discussed.     

On the magnetic ordering implemented by the s–d exchange interaction

The RKKY perturbation scheme for the spin s–d exchange model is reexamined in the case of a finite number of electrons per impurity. The distribution of impurities is assumed to be random. The free electron gas representing the unperturbed system in the RKKY approach is replaced by a mean-field system h_r asymptotically equivalent to the reduced s–d model H_r which includes part of the s–d exchange. Below the transition temperature T_c of h_r the lowest energy levels of the s–d exchange Hamiltonian H_K resulting from second-order perturbation theory prove to be ordered in the same manner as those of h_r in the limit of weak s–d exchange, favouring the same definite parallel alignment of any pair of impurity spins. It follows therefore that in the low temperature, weak coupling regime, the s–d model exhibits ferromagnetic ordering of impurities. At temperatures above T_c there is no ordering and impurity–impurity exchange has the RKKY form. These observations are consistent with the presence of a low-temperature ferromagnetic phase in numerous compounds with s–d exchange interaction.     

Conductance of metallic nanoribbons with defects

The conductances of two typical metallic graphene nanoribbons with one and two defects are studied using the tight binding model with the surface Green’s function method. The weak scattering impurities, U ～ 1 eV, induce a dip in the conductance near the Fermi energy for the narrow zigzag graphene nanoribbons. As the impurity scattering strength increases, the conductance behavior at the Fermi energy becomes more complicated and depends on the impurity location, the AA and AB sites. The impurity effect then becomes weak and vanishes with the increase in the width of the zigzag graphene nanoribbons (150 nm). For the narrow armchair graphene nanoribbons, the conductance at the Fermi energy is suppressed by the impurities and becomes zero with the increase in impurity scattering strength, U > 100 eV, for two impurities at the AA sites, but becomes constant for the two impurities at the AB sites. As the width of the graphene nanoribbons increases, the impurity effect on the conductance at the Fermi energy depends sensitively on the vacancy location at the AA or AB sites.     

Investigation of Diffusion of Magnesium in Lithium Fluoride Crystals by the SIMS Method

Using secondary ion mass spectroscopy, diffusion of magnesium impurity in lithium fluoride is investigated. A temperature dependence of the magnesium diffusion coefficients within the temperature interval 870–1073 K is established, which is described by an expression of the following type: D = 2.8·10^–3·exp(–1.5/kT). Combining the data on self-diffusion of cations with the results of the ion conductivity measurements, estimation is made of the thermodynamic parameters characterizing the jump of a diffusant. The calculated frequencies of the impurity ion jump to the cation vacancy are reported.     

Effect of the dielectric-constant mismatch and magnetic field on the binding energy of hydrogenic impurities in a spherical quantum dot

Within the effective mass approximation and variational method the effect of dielectric constant mismatch between the size-quantized semiconductor sphere, coating and surrounding environment on impurity binding energy in both the absence and presence of a magnetic field is considered. The dependences of the binding energy of a hydrogenic on-center impurity on the sphere and coating radii, alloy concentration, dielectric-constant mismatch, and magnetic field intensity are found for the GaAs–Ga_1−xAl_xAs–AlAs (or vacuum) system.     

The Magnetisation of Large Atoms¶in Strong Magnetic Fields

In this paper we study the asymptotic form of the magnetisation and current of large atoms in strong constant magnetic fields. We prove that the Magnetic Thomas–Fermi theory gives the right magnetisation/current for magnetic field strengths which satisfy B≤Z ^4/3. Received: 24 April 2000 / Accepted: 21 August 2000     

The polaron effect on the binding energy of a shallow donor impurity in quantum-well wires in an electric field

Using a variational technique, the effect of electron-longitudinal optical (LO) phonon interaction on the ground and the first few excited states of a hydrogenic impurity in a semiconductor quantum wire of rectangular cross section under an external electric field is studied theoretically for the impurity atom doped at various positions. The results for the binding energy as well as polaronic correction are obtained as a function of the size of the wire, the applied uniform electric field and the position of the impurity. It is found that the presence of optical phonons changes significantly the values of the impurity binding energies of the system. Taking into account the electron–LO phonon interaction the 1s→2p_y and 1s→2p_z transition energies are calculated as a function of applied electric field for different impurity positions.     

Impurity-related polarizability and photoionization-cross section in double quantum wells under electric fields and hydrostatic pressure

Double quantum well heterostructures are quite important for the exploration of correlated electron states in two-dimensional systems. By using the variational procedure, within the effective-mass and parabolic-band approximations, the effects of both electric field and hydrostatic pressure on the shallow-donor-impurity related polarizability and photoionization cross-section in GaAs–Ga_1−xAl_xAs double asymmetric quantum wells are presented. The electric field is considered to be applied along the growth direction. It is found that the impurity binding energy and polarizability can be tuned by means of an applied external electric field or hydrostatic pressure in asymmetric double quantum wells, a behavior which could be used in the design and construction of semiconductor devices. The photoionization cross-section magnitude increases as the pressure and applied electric field are increased, except beyond the Γ–X crossover in the barrier material, where a decrease of the photoionization cross-section is expected due the smaller confinement of the impurity wave function.     

Origin of the blueshift in the infrared absorbance of intersubband transitions in N/GaN multiple quantum wells

A self-consistent calculation of the subband energy levels of n-doped quantum wells is studied. A comparison is made between theoretical results and experimental data. In order to account for the deviations between them, the ground-state electron–electron exchange interactions, the ground-state direct Coulomb interactions, the depolarization effect, and the exciton-like effect are considered in the simulations. The agreement between theory and experiment is greatly improved when all these aspects are taken into account. The ground-to-excited-state energy difference increases by 8 meV from its self-consistent value if one considers the depolarization effect and the exciton-like effect only. It appears that the electron–electron exchange interactions account for most of the observed residual blueshift for the infrared intersubband absorbance in Al_xGa_1-xN/GaN multiple quantum wells. It seems that electrons on the surface of the k-space Fermi gas make the main contribution to the electron–electron exchange interactions, while for electrons further inside the Fermi gas it is difficult to exchange their positions.     

Pseudogap in the electronic density of states of superconducting glasses

Two loop order results for the electronic density of states (DOS) of both Ising- and XY-type superconducting glass phases are presented. Maximal depression of the DOS by critical fluctuation corrections to its finite mean field value is found to occur at the Fermi energyE _F . The theory is renormalizable and exponentiation of the loop expansion yields (E _F)=0 for the DOS in two dimensions and for arbitrarily small disorder. It is also concluded that the DOS assumes nonzero values, when either |E–E _F|, or magnetic field H, or temperature T become nonzero. A pseudogap (atT=0,H=0) or a smeared gap (forH0 orT0) is thus obtained with a DOS-minimum at the Fermi level. A rough estimate of the effective width is given. In three dimensions, the DOS atE _F vanishes like (n _c ^I –n)^1/2 for the Ising-type glass and like (n _c ^XY –n)¹ for the XY-type glass as the impurity concentrationn exceeds characteristic values. At second loop order, the calculation contains a nontrivial indication of a lower critical dimension two.     

The Hall coefficient and other transport properties of dilute alloys of copper and silver

The low field Hall coefficient of a number of polycrystalline foils of dilute (2%) alloys of copper and silver has been measured in the temperature range 1.5–50°K, and at room temperature. The alloys chosen wereCu-Au andAg-Au (uncharged impurity),Cu Ge andAg-Sn (charged impurity), andCu-Ni andAg-Pd (transition metal impurity). At 20°K and below, the Hall coefficients of the different copper alloys differ widely from each other,Cu-Ge giving the highest (negative) values (up to twice the room temperature value for pure copper), andCu-Au the lowest (down to 0.7 of this value). There are also significant concentration dependences. The silver alloys show corresponding but smaller changes. A relationship, due to Tsuji, gives the Hall coefficient as a function of the Fermi velocityν and the mean curvature 1/ϱ of the Fermi surface, for the case of an isotropic relaxation time. The integrals over the Fermi surface have been numerically estimated using the known Fermi surface and electron velocities. For both Cu and Ag the results agree with the experimental room temperature values, which we take as evidence thatτ(k) for phonon scattering is here close to isotropic. On the other hand, to account for the Hall coefficients of the alloys, it is necessary to assume that the relaxation timeτ varies over the Fermi surface. It is seen that in Cu and Ag the neck regions contribute relatively little toR since both 1/ϱ andν are small there. The main change inR in different alloys arises from the variation in the relative weighting given to the belly regions by different kinds of impurity scattering. A closer analysis shows that the bulges in the Fermi surface of copper in the 〈100〉 directions contribute relatively heavily because of their high positive curvature. The anisotropy ofτ deduced from the Hall coefficient is compared with that deduced from other measurements.     

Effect of boundaries and impurities on electron–phonon dephasing

Electron scattering from boundaries and impurities destroys the single-particle picture of the electron–phonon interaction. We show that quantum interference between ‘pure‘ electron–phonon and electron–boundary/impurity scattering may result in the reduction as well as to the significant enlargement of the electron dephasing rate. This effect crucially depends on the extent, to which electron scatterers, such as boundaries and impurities, are dragged by phonons. Static and vibrating scatterers are described by two dimensionless parametersq_Tl and q_TL, where q is the wavevector of the thermal phonon, l is the total electron mean-free path, L is the mean-free path due to scattering from static scatterers. According to the Pippard ineffectiveness condition , without static scatterers the dephasing rate at low temperatures is slower by the factor 1 / ql than the rate in a pure bulk material. However, in the presence of static potential the dephasing rate turns out to be 1 / qL times faster. Thus, at low temperatures electron dephasing and energy relaxation may be controlled by electron boundary/impurity scattering in a wide range.     

Algebraically Self-Consistent Quasiclassical Approximation on Phase Space

The Wigner–Weyl mapping of quantum operators to classical phase space functions preserves the algebra, when operator multiplication is mapped to the binary * operation. However, this isomorphism is destroyed under the quasiclassical substitution of * with conventional multiplication; consequently, an approximate mapping is required if algebraic relations are to be preserved. Such a mapping is uniquely determined by the fundamental relations of quantum mechanics, as is shown in this paper. The resultant quasiclassical approximation leads to an algebraic derivation of Thomas–Fermi theory, and a new quantization rule which—unlike semiclassical quantization—is non-invariant under action transformations of the Hamiltonian, in the same qualitative manner as the true eigenvalues. The quasiclassical eigenvalues are shown to be significantly more accurate than the corresponding semiclassical values, for a variety of 1D and 2D systems. In addition, certain standard refinements of semiclassical theory are shown to be easily incorporated into the quasiclassical formalism.     

Inelastic magnetic neutron scattering from inner filled shells of impurity atoms in metals

The differential cross section of inelastic magnetic neutron scattering by paired electrons of inner filled s ² shells of impurity atoms in metals is first derived taking into account single electron excitations from the inner shell of the impurity to the free states of the metal. The energy-angle distributions of the scattered neutrons are calculated depending on the effective electronic mass near the Fermi surface.     

The lack ofO(A −1)-contributions in the Iwamoto-Yamada-expansion of the energy expectation value per particle for extended Fermi systems

It is shown that the Iwamoto-Yamada expansion of the energy expectation value per particle of a Fermi system does not contain contributions of the orderO(A ^–1). The consequences which can be drawn are very fundamental for the radial distribution function and the liquid structure function.