Friedel Sum Rule Research Articles

Pseudo-quaternary boro-carbides: a two sub-lattice model

We consider recently studied boro-carbides RE(Ni 1− x M x ) 2B 2C, with emphasis in RE=Y and Ho. Ho compounds exhibit in some cases spiral magnetic order. We extended a recent previous work on Heusler compounds to study the specific heat of Y boro-carbides doped with Co and to calculate the coupling between Ho moments in neighbouring planes. The effect of impurity disorder is included in the electronic structure via the Coherent Potential Approximation (CPA) and in the absence of experimental data for the pseudoquaternary specific heat, we restricted ourselves to the dilute limit and determine the impurity potential via Friedel's sum rule. The electron–phonon coupling ( λ( x)) is extracted from the measured superconducting transition temperature, using the Allen–Dines formula. The existent specific heat data for Y compounds is used to fix the parameters of the two sub-lattice model. Finally we study the change, with transition metal concentration, of the angle between moments of the spiral magnetic order.

Velocity-dependent screening in metals

We propose a model to calculate in a quantum-mechanical way the screening and energy loss of swift ions in metals. The model is based on an extension of the Friedel sum rule to finite velocities, which allows us to describe the scattering and nonlinear screening of electrons in the field of the moving ion. The scattering process is formulated according to the partial-wave expansion, assuming an effective scattering potential that is adjusted in a self-consistent way using the phase-shift sum rule. We consider in particular the limits of low and high velocities, where the results of this model agree with those obtained using the density-functional and perturbation theories. We apply the method to the calculation of the mean energy loss (stopping power) on the whole range of velocities.

Electronic energy loss of slow protons channeled in metals

The electronic energy loss for low-velocity protons channeled in the $〈$100$〉$ direction of single-crystal Au and Al is calculated. The proton trajectories are determined by solving the equation of motion. In the proton dynamics two forces are included, a repulsive term arising from the nuclei and core electrons and a friction force, depending on proton velocity, arising from the valence electrons. The repulsive force on the proton is evaluated using a superposition of conservative potentials. The friction coefficient is evaluated by using the local density of valence electrons, and a quantum-mechanical transport cross-section approach with a self-consistent model based on Friedel's sum rule. The results allow us to describe the nonlinear behavior of energy loss with ion velocity observed experimentally in Au, as well as the linear behavior observed in Al.

Surface Characterisation from Adsorbate Knight Shifts

Recent works on catalyst surface characterisation from adsorbate Knight shifts are reviewed. It has been shown how for supported catalysts the proton NMR gives a better estimate of the metal dispersion than the classical volumetric technique where the amount of hydrogen adsorbed is measured to determine the dispersion. It has also been shown that adsorption on multiple sites can be probed by the adsorbate Knight shifts. Again, for bimetallic systems the Knight shifts has been shown to indicate the composition of the surfaces. This has been corroborated by a microscopic theory using Friedel sum rule. The role of promoters on active site blocking, the evaluation of kinetic parameters and the effect of segregation on the adsorption energy may also be found from NMR Knight shifts. The works reviewed here concern generally hydrogen as an adsorbate. However, some work using 2 H , 13 C and 129 Xe NMR are also discussed. Deuterium NMR helps to find the kinetic parameters in exchange reaction and is complementary to the proton NMR in surface studies. Similarly, Carbon-13 NMR helps to understand the organic reactions on metal surfaces as well as the surface structure of catalysts. 129 Xe NMR helps to understand the internal structure of zeolites.

Consistent calculation of the stopping power for slow ions in two-dimensional electron gases

Within the framework of quantum scattering theory, we present a consistent calculation of the stopping power for slow protons and antiprotons moving in two-dimensional electron gases. The Friedel sum rule is used to determine the screening constant in the scattering potential. For the stopping power our results are compared with that of the random-phase approximation dielectric theory and that predicted by the linear Thomas-Fermi potential.

Application of the Friedel sum rule to symmetric molecular conductors

Recent developments in nanotechnology have made it possible to measure the resistance of individual molecular wires self-assembled between two metallic pads. In this paper we present a simple expression for the resistance of a symmetric molecule in terms of a set of M phase shifts, M being the number of conducting channels connecting the molecule to the metallic pads. These phase-shifts are related by a sum rule to ${\mathrm{N}}_{\mathrm{d}}$=${\mathrm{N}}_{\mathrm{even}}$-${\mathrm{N}}_{\mathrm{odd}}$, where ${\mathrm{N}}_{\mathrm{even}}$ and ${\mathrm{N}}_{\mathrm{odd}}$ are the number of electrons occupying the even and odd eigenstates respectively. For many molecular wires of interest all the phase shifts are nearly zero except for one (per spin), allowing us to write a particularly simple relation for the resistance: ${\mathrm{R}}^{\mathrm{\ensuremath{-}}1}$=(${2\mathrm{e}}^{2}$/h)sin $^{2}$(\ensuremath{\pi}${\mathrm{N}}_{\mathrm{d}}$/2). The relations presented here are based on the Friedel sum rule and should be valid even in the presence of interactions.

Studies of Charging Effects on Resonant Tunneling Diodes in Terms of Extended Friedel Sum Rule

The Friedel sum rule is extended to express the electron number accumulated within an asymmetric potential well in terms of the phases of transmission and reflection amplitudes of a double-barrier potential, in the presence of a bias voltage and charging effects. It is applied to study non-equilibrium transport properties in asymmetric double-barrier systems. The current-voltage characteristics depends on the sign of the bias voltage, which is caused by the different accumulation of charges in the quantum well. Our self-consistent calculation shows the hysteresis in the current-voltage characteristics which agrees with the experimental observation.

Studies of Charging Effects on Resonant Tunneling Diodes in Terms of Extended Friedel Sum Rule

The Friedel sum rule is extended to express the electron number accumulated within an asymmetric potential well in terms of the phases of transmission and reflection amplitudes of a double-barrier potential, in the presence of a bias voltage and charging effects. It is applied to study non-equilibrium transport properties in asymmetric double-barrier systems. The current-voltage characteristics depends on the sign of the bias voltage, which is caused by the different accumulation of charges in the quantum well. Our self-consistent calculation shows the hysteresis in the current-voltage characteristics which agrees with the experimental observation.

Scattering treatment of the quasiparticle effective mass

We obtain the quasiparticle effective mass at the Fermi surface of the three-dimensional electron gas by enforcing the Friedel sum rule on the effective potential coming from the Sjölander-Stott theory. Our results indicate that the effective mass is a monotonically decreasing function of r s throughout the whole metallic range, and are in a reasonable agreement with the most recent numerical data. This approach does not explicitly use the Fermi Liquid character of the electron gas.

Quasiparticle effective mass for the two- and three-dimensional electron gas.

We calculate the quasiparticle effective mass for the electron gas in two and three dimensions in the metallic region. We employ the single-particle scattering potential coming from the Sj\"olander-Stott theory and enforce the Friedel sum rule by adjusting the effective electron mass in a scattering calculation. In three dimensions (3D) our effective mass is a monotonically decreasing function of ${\mathit{r}}_{\mathit{s}}$ throughout the whole metallic domain, as implied by the most recent numerical results. In two dimensions (2D) we obtain reasonable agreement with the experimental data, as well as with other calculations based on the Fermi-liquid theory. We also present results of a variety of different treatments for the effective mass in 2D and 3D. \textcopyright{} 1996 The American Physical Society.

Energy loss of slow protons channeled in Au.

The electronic energy loss for low-velocity protons channeled in the 〈100〉 direction of single-crystal Au is calculated. The spatial distribution of valence electronic density in Au is calculated by using the tight-binding linear muffin-tin method [O. K. Andersen and O. Jepsen, Phys. Rev. Lett. 53, 2571 (1984); O. K. Anderson, Z. Pawlowska, and O. Jepsen, Phys. Rev. B 34, 5253 (1986)]. The proton trajectories are determined by numerical integration of the classical equation of motion, and the energy loss is evaluated by using a quantum-mechanical transport cross-section approach with a self-consistent model based on Friedel's sum rule. The results allow us to describe the nonlinear behavior of energy loss with ion velocity observed experimentally. \textcopyright{} 1996 The American Physical Society.

Aharonov-Bohm oscillations in a mesoscopic ring with a quantum dot

We present an analysis of the Aharonov-Bohm oscillations for a mesoscopic ring with a quantum dot inserted in one of its arms. It is shown that microreversibility demands that the phase of the Aharonov-Bohm oscillations changes {\it abruptly} when a resonant level crosses the Fermi energy. We use the Friedel sum rule to discuss the conservation of the parity of the oscillations at different conductance peaks. Our predictions are illustrated with the help of a simple one channel model that permits the variation of the potential landscape along the ring.

FRIEDEL SUM RULE ON A LATTICE

The Friedel Sum Rule is derived for electrons whose motion through the lattice is described by the tight-binding model. The Friedel Sum Rule is a statement of charge neutrality: the screening charge around an impurity cancels the charge from the impurity itself. The answer is given in closed form for a complex of n-impurities.

FRIEDEL OSCILLATIONS ON A SQUARE LATTICE

An impurity in a conductor will cause oscillations in the charge density near the impurity. We calculate these Friedel Oscillations for impurities on a square lattice while using the tight-binding model. Far from the impurity the oscillations are described by a set of phase shifts. These are the same phase shifts which enter the Friedel Sum Rule.

An enhanced perturbational study on spectral properties of the Anderson model

The infinite-U single-impurity Anderson model for rare earth alloys is examined with a new set of self-consistent coupled integral equations, which can be embedded in the large-N expansion scheme (N is the local spin degeneracy). The finite-temperature impurity density of states (DOS) and the spin-fluctuation spectra are calculated exactly up to the order O(1/N2). The presented conserving approximation goes well beyond the 1/N approximation (NCA) and maintains local Fermi-liquid properties down to very low temperatures. The position of the low-lying Abrikosov-Suhl resonance (ASR) in the impurity DOS is in accordance with Friedel's sum rule. For N=2 its shift toward the chemical potential, compared to the NCA, can be traced back to the influence of the vertex corrections. The width and height of the ASR are governed by the universal low-temperature energy scale TK. Temperature and degeneracy N dependence of the static magnetic susceptibility is found to be in excellent agreement with the Bethe ansatz results. Threshold exponents of the local propagators are discussed. The resonant level regime (N=1) and intermediate-valence regime ( mod epsilon f mod < Delta ) of the model are thoroughly investigated as a critical test of the quality of the approximation. Some applications to the Anderson lattice model are pointed out.

Kondo resonance in tunneling phenomena through a single quantum level.

Effects of Coulomb repulsion on the process of resonant tunneling through a single quantum level are studied by applying the quantum Monte Carlo method and the maximum-entropy method to the Wolf model on a one-dimensional chain. In the calculated spectral function there is a sharp Kondo peak near the chemical potential \ensuremath{\mu}, which contributes to the resonance tunneling. Correspondingly, the conductance calculated by using the Friedel sum rule shows the expected transparency, i.e., the transmission probability is almost unity when \ensuremath{\mu} is in the range ${\mathrm{\ensuremath{\epsilon}}}_{0}$${\mathrm{\ensuremath{\epsilon}}}_{0}$+U, where ${\mathrm{\ensuremath{\epsilon}}}_{0}$ is the on-site energy of the single quantum level and U is the Coulomb repulsion.

Beyond the NCA: new results for the spectral properties of the Anderson model

In the framework of direct perturbation theory, a fully self-consistent approximation beyond the well-known NCA is presented for the Anderson model. The resummation of a class of skeleton diagrams up to infinite order in V includes all contributions up to the order O(1/ N 2) ( N = degeneracy of the magnetic state). Qualitative improvements in maintaining local Fermi liquid relations and one-particle spectral properties in comparison to the well-known NCA will be reported. The location and temperature dependence of the AS-resonance for the case N = 2 is found to be rather close to the chemical potential in excellent agreement with Friedel's sum rule; the static magnetic susceptibility exhibits the same N-dependence as the exact Bethe-ansatz solution.

Phase Shifts, the Friedel Sum Rule, Resonant Tunneling through Asymmetric Potentials

Phase shifts are defined for the scattering off an asymmetric potential in a one dimensional space. The Friedel sum rule is derived for such a system, and the resonant tunneling through the asymmetric double-barrier potential is analysed. When the potential goes to symmetric, the newly defined phase shifts do not tend to those defined by Kawabata for the symmetric potential, although the physical quantities expressed in terms of the phase shifts are the same in this limit.

Generalized multiple-scattering theory.

Multiple-scattering theory, as it is applicable to metallic systems, is generalized systematically compared with the usual formulation. The generalization refers to the reference system, which conventionally is free space, but which can be chosen to be much more general. For example, in describing a binary alloy, a defect in a metal, or a finite impurity-cluster model for the latter system, the reference system may be an average medium, the unperturbed metallic host system, or the finite host cluster, respectively. Applications are indicated for the generalized Friedel sum rule, the local density of states, and the transition probability of a Bloch electron. A state in a system is shown to be built up of properly generalized incoming and outgoing waves.

Generalized Friedel sum from Lloyd's formula.

Expressions for the generalized Friedel sum rule are derived which are applicable as a condition for charge neutrality at defects in metallic systems. These defects include the thoroughly studied substitutional impurity as well as the poorly explored interstitial impurity and an atom at the saddle-point position between two vacancies, being typical for substitutional electromigration. The expressions are derived starting from Lloyd's formula for the density of states of an arbitrary collection of scattering potentials.