We report calculations of electron inelastic mean free paths (IMFPs) for 50–2000 eV electrons in a group of 27 elements (C, Mg, Al, Si, Ti, V, Cr, Fe, Ni, Cu, Y, Zr, Nb, Mo, Ru, Rh, Pd, Ag, Ta, W, Re, Os, Ir, Pt, Au and Bi). This work extends our previous calculations (Surf. Interface Anal. 11, 57 (1988)) for the 200–2000 eV range. Substantial variations were found in the shapes of the IMFP versus energy curves from element to element over the 50–2000 eV range and we attribute these variations to the different inelastic scattering properties of each material. Our calculated IMFPs wee fitted to a modified form of the Bethe equation for inelastic electron scattering in matter; this equation has four parameters. These four parameters could be empirically related to several material parameters for our group of elements (atomic weight, bulk density and number of valence electron per atom). IMFPs and those initially calculated was 13%. The modified Bethe equation and our expressions for the four parameters can therefore be used to estimate IMFPs in other materials. The uncertainties in the algorithm used for our IMFP calculation are difficult to estimate but are believed to be largely systematic. Since the same algorithm has been used for calculating IMFPs, our predictive IMFP formula is considered to be particularly useful for predicting the IMFP dependence on energy in the 50–2000 eV range and the material dependence for a given energy.

Calculations of electorn inelastic mean free paths. II. Data for 27 elements over the 50–2000 eV range

The review introduces the reader to a theoretical method describing the transport properties of dilute alloys on an ab initio basis. The calculations start from density or spin density functional theory using a Green function formalism to investigate the underlying electronic structure of the ideal and perturbed system on equal footing. The residual resistivity is calculated solving the quasiclassical Boltzmann equation. The theory is outlined and various methods and approximations developed to solve the transport equation are reviewed and compared with respect to accuracy and validity. It will be demonstrated that the theory is able to quantitatively account. The success and limitations of these calculations are discussed for a large variety of systems, non-magnetic and ferromagnetic, in comparison with experimental results. It will be shown that these calculations confirm empirical rules and concepts, elucidate the microscopic processes behind the trends and can be used to make a theoretical material design.

Transport properties of dilute alloys

We present recent advances in numerical implementations of hybrid functionals and the GW approximation within the full-potential linearized augmented-plane-wave (FLAPW) method. The former is an approximation for the exchange–correlation contribution to the total energy functional in density-functional theory, and the latter is an approximation for the electronic self-energy in the framework of many-body perturbation theory. All implementations employ the mixed product basis, which has evolved into a versatile basis for the products of wave functions, describing the incoming and outgoing states of an electron that is scattered by interacting with another electron. It can thus be used for representing the nonlocal potential in hybrid functionals as well as the screened interaction and related quantities in GW calculations. In particular, the six-dimensional space integrals of the Hamiltonian exchange matrix elements (and exchange self-energy) decompose into sums over vector–matrix–vector products, which can be evaluated easily. The correlation part of the GW self-energy, which contains a time or frequency dependence, is calculated on the imaginary frequency axis with a subsequent analytic continuation to the real axis or, alternatively, by a direct frequency convolution of the Green function G and the dynamically screened Coulomb interaction W along a contour integration path that avoids the poles of the Green function. Hybrid-functional and GW calculations are notoriously computationally expensive. We present a number of tricks that reduce the computational cost considerably, including the use of spatial and time-reversal symmetries, modifications of the mixed product basis with the aim to optimize it for the correlation self-energy and another modification that makes the Coulomb matrix sparse, analytic expansions of the interaction potentials around the point of divergence at k = 0, and a nested density and density-matrix convergence scheme for hybrid-functional calculations. We show CPU timings for prototype semiconductors and illustrative results for GdN and ZnO.

https://iopscience.iop.org/article/10.1088/0953-8984/24/29/293201/pdf

Hybrid functionals and GW approximation in the FLAPW method

Methods for the preparation of nano-sized metal particles are considered. Theoretical approaches to the analysis of electron behaviour in systems with quantum-size effects are outlined. The results of experimental investigations of the thermodynamic and optical characteristics of small metal particles and thin films are presented. The data of experiments on the electron dynamics in single nanoparticles and nanoparticle ensembles are described. The self-assembly of nanoparticles to form supported ordered structures is considered. The bibliography includes 343 references.

Quantum-size colloid metal systems

A generalized density functional theory (DFT) is proposed based on a generalized Hohenberg-Kohn theorem with the pair density as the key quantity and the kinetic energy as a universal functional of the pair density. It is assumed that there exists an effective interaction potential which, via the corresponding two-particle (2P) Schrodinger equation, generates 2P orbitals (geminals) from which follows the pair density, just as in the conventional DFT the density follows from 1P orbitals (as the solutions of an effective 1P Schrodinger equation). According to three different representations (natural spectral resolutions) of the pair density or the cumulant pair densities in terms of geminals, three versions of a pair DFT (PDFT) are formally sketched. Also considered are the relation between electron correlation and particle-number fluctuations in fragments of the system and the use of the pair density for an estimation of such fluctuations. © 1996 John Wiley & Sons, Inc.

Attempts toward a pair density functional theory

Ergebnisse in der Elektronentheorie der Metalle

1. One-Particle Picture of Metal Electrons. Electron gas. Band structure. Many-particle problems. 2. Methods of Band Structure Calculations. Tight binding method. Pseudopotential method. Multiple scattering picture. APW method. KKR method. H-NFE-TB method. Coherent potential approximation. 3. Results of Band Structure Calculations. Simple metals. Transition metals. Compounds and alloys. 4. Experimental Determination of Band Structure. Dynamics of lattice electrons. Experimental determination of the Fermi surface. Photo-electron and X-ray spectroscopy. 5. Further Electronic Properties. Density of states, heat capacity, transition temperature of superconductivity, magnetic susceptibility, and transport quantities. Dielectric function, inelastic electron and photon scattering, and optical reflection. Lattice properties. Electron-phonon-interaction. Electronic properties of localized defects. Other properties. 6. Surfaces, Thin Layers, Quasi-One- and Quasi-Two-Dimensional Conductors. Peculiarities of surfaces and thin films. Ideal surfaces of metals. Systems with low-dimensional structure. Peculiarities of low-dimensional systems. Peierls instability. Electronic excitations in the Peierls system. References. Solid state physics in general. Recent monographs and proceedings. Special literature quoted in the text. Summaries. Index of abbreviations. Natural constants and atomic units. Characteristic quantities of the electron gas. Subject index.

Gerd Lehmann

Papers

Ergebnisse in der Elektronentheorie der Metalle

Electronic properties of metals