The local density theory is developed by Hohenberg, Kohn and Sham is extended to the spin polarized case. A spin dependent one- electron potential pertinent to ground state properties is obtained from calculations of the total energy per electron made with a 'bubble' (or random phase) type of dielectric function. The potential is found to be well represented by an analytic expression corresponding to a shifted and rescaled spin dependent Slater potential. To test this potential the momentum dependent spin susceptibility of an electron gas is calculated. The results compare favourably with available information from other calculations and from experiment. The potential obtained in this paper should be useful for split band calculations of magnetic materials.

https://iopscience.iop.org/article/10.1088/0022-3719/5/13/012/pdf

A local exchange-correlation potential for the spin polarized case: I

The possibilities of the Hohenberg-Kohn-Sham local density theory are explored in view of recent advances in the theory of the interacting electron gas. The authors discuss and provide numerical data for the effective exchange-correlation potentials mu xc for ground state problems and Vxc for excitation spectra. The potential mu xc(r) is written as a factor beta ( rho (r)), which depends on the local density rho (r), times the Kohn-Sham-Gaspar potential mu x=-e2kF/ pi . The factor beta , which is represented by a simple analytical formula, varies between 1 for very high densities and about 1.5 for very low densities, where mu xc thus approaches the Slater approximation. The potential Vxc is given in the form of a table. The explicit predictions of the approximate theories are mapped in their linearized versions. The theory is also developed for the case when the ion cores are treated separately, and the local density approximation is applied only to the valence electrons.

http://iopscience.iop.org/article/10.1088/0022-3719/4/14/022/pdf

Explicit local exchange-correlation potentials

The d1 layer metals TaS2, TaSe2, NbSe2, in all their various polytypic modifications, acquire, below some appropriate temperature, phase conditions that their electromagnetic properties have previously revealed as ‘anomalous’. Our present electron-microscope studies indicate that this anomalous behaviour usually includes the adoption, at some stage, of a superlattice. The size of superlattice adopted often is forecast in the pattern of satellite spotting and strong diffuse scattering found above the transition. Our conclusions are that charge-density waves and their concomitant periodic structural distortions occur in all these 4d1/5d1 dichalcogenides. We have related the observed periodicities of these CDW states to the theoretical form of the parent Fermi surfaces. Particularly for the 1T octahedrally coordinated polytypes the Fermi surface is very simple and markedly two-dimensional in character, with large near-parallel walls. Such a situation is known theoretically to favour the formation of...

Charge-density waves and superlattices in the metallic layered transition metal dichalcogenides

The properties of quasi-two-dimensional semiconductor quantum dots are reviewed. Experimental techniques for measuring the electronic shell structure and the effect of magnetic fields are briefly described. The electronic structure is analyzed in terms of simple single-particle models, density-functional theory, and "exact" diagonalization methods. The spontaneous magnetization due to Hund's rule, spin-density wave states, and electron localization are addressed. As a function of the magnetic field, the electronic structure goes through several phases with qualitatively different properties. The formation of the so-called maximum-density droplet and its edge reconstruction is discussed, and the regime of strong magnetic fields in finite dot is examined. In addition, quasi-one-dimensional rings, deformed dots, and dot molecules are considered. (Less)

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Electronic structure of quantum dots

Publisher Summary: This chapter discusses two major developments that have taken place over the past decade. First is the enormous wealth of energy band calculations that have had tremendous success in explaining the properties of specific solids, but in which the connection with first principles is not always apparent. Second is the spectacular progress of many-body theory applied to the solid state that has given a number of new results, although often of a rather general and formal nature, such as to provide the justification and a formal basis for a one-electron theory. The electron gas problem is treated in some detail here. The problem of the crystal potential is given due attention. It discusses the potential from the ion cores as well as from the valence electrons, and suggests schemes that incorporate essential exchange and correlation effects for the valence electrons. An energy band calculation that properly includes the effects of exchange and correlation describes the elementary excitations called quasi particles. Quasi-particle properties are usually discussed using the remarkable Landau theory of the Fermi liquid. This chapter gives a brief presentation of the theory and reviews the present status of calculations of the Fermi liquid parameters and how they are determined from experiments. (Less)

Effects of Electron-Electron and Electron-Phonon Interactions on the One-Electron States of Solids

It is shown rigorously that the paramagnetic state of an electron gas is never the Hartree-Fock ground state, even in the high-density---or weak-interaction---limit. The paramagnetic state is always unstable with respect to formation of a static spin density wave. The instability occurs for spin-density waves having a wave vector $Q\ensuremath{\approx}2{k}_{F}$, the diameter of the Fermi sphere. It follows that the (Hartree-Fock) spin susceptibility of the paramagnetic state is not a monotonic decreasing function with increasing $Q$, but rather a function with a singularity near $Q=2{k}_{F}$. Rather convincing experimental evidence that the antiferromagnetic ground state of chromium is a large-amplitude spin density wave state is summarized. A number of consequences of such states are discussed, including the problem of detecting them by neutron diffraction.

Spin Density Waves in an Electron Gas

The influence of electron-electron correlation on exchange instabilities of a metal is examined. The employment of screened interactions does not constitute a proper treatment. Correlation effects suppress ferromagnetic instabilities, as is well known, but they need not supress instabilities of the spin-density-wave type. On the contrary, it is shown that correlation enhances exchange instability of the charge-density-wave type. For either type, the wave vector of such a state adjusts so that the Fermi surface makes critical contact with the energy gaps introduced by the instability. This circumstance optimizes the correlation energy. The observed conjunction of the long-period-superlattice periodicity with the Fermi surface in order-disorder alloys is probably an example of this phenomenon. It is suggested that charge-density-wave ground states are likely in simple metals having weak Born-Mayer ion-ion interactions, such as the alkali metals. The intensity of Bragg reflection satellites caused by a concomitant positive-ion modulation is computed.

Exchange and Correlation Instabilities of Simple Metals

The wave-vector-dependent spin susceptibility of an electron gas is calculated taking dynamically screened electron interactions into account. A quasiparticle approach is developed to deal with the self-energy and screening effects of the electron correlations, and the energy and mass of a quasiparticle at the Fermi surface are calculated for a wide density range. The susceptibility is found using a linearized self-consistent-field treatment of the quasiparticles and numerical solution of the resulting integral equation. It does not show singular or anomalous behavior at large wave vector as in the Hartree-Fock approximation. The density dependence of the zero-wave-vector susceptibility is found to be in better agreement with experiment than that of previous calculations. Statically screened interactions are also used in a series of calculations, and it is shown that a long- but finite-range interaction can lead to anomalies in the susceptibility at large wave vectors.

Electron-Gas Spin Susceptibility

Determination of the Fermi-surface anisotropy of alkali metals by the de Haas-van Alphen effect has provided reliable numerical values for electronic energy gaps at Brillouin-zone boundaries. These determinations allow a quantitative comparison of interband optical absorption power with theory. Serious discrepancies with standard theory are found. For Na, the theoretical absorption is too small by a factor of 4, whereas for Rb and Cs it is too large by a factor greater than 3. We observe that interband matrix elements must include not only the direct interaction with the photon field, but also the Hartree-Fock potential arising from collective motions of all electrons responding to that field. The interband matrix elements are calculated using time-dependent, self-consistent perturbation theory. If only the Hartree term arising from collective motion is added, matrix elements are reduced by a few percent. However, inclusion of the exchange potential profoundly alters the magnitude of the matrix element. For Na the theoretical absorption power is enhanced by a factor of 4, and for Rb and Cs it is suppressed by a factor of 6 or more. This apparent success of the time-dependent, Hartree-Fock method indicates that exchange interactions can contribute unexpectedly to electronic processes in solids.

Collective Effects in Interband Optical Absorption

The 0.5-G splitting of the conduction-electron spin resonance in potassium (at 4200 G) observed by Walsh, Rupp, and Schmidt can be quantitatively explained providing the conduction electrons are in a charge-density-wave (CDW) ground state. The Fermi-surface distortion caused by the CDW energy gap leads to an anisotropic conduction-electron $g$ factor depending on the angle between H and the wave vector Q of the CDW. The extremal values of $g$, corresponding to Q \ensuremath{\perp} H and Q\ensuremath{\parallel}H, differ by $(\frac{3V}{8{E}_{F}})\ensuremath{\Delta}g$, where $\ensuremath{\Delta}g=\ensuremath{-}0.0025$ is the observed $g$ shift. $V$ is the observed threshold energy of the Mayer-El Naby optical-absorption anomaly, and ${E}_{F}$ is the Fermi energy. The predicted maximum splitting is 0.56 G. Interpretation of the data requires the sample to have a macroscopic domain structure, caused by thermal stress and plastic flow when the potassium-Parafilm sandwich is cooled to He temperature. The orientation of Q in stress-free regions should be parallel to H. In regions of high stress, Q is presumed perpendicular to the surface, and therefore approximately perpendicular to H.

A. W. Overhauser

Papers

Spin Density Waves in an Electron Gas

Exchange and Correlation Instabilities of Simple Metals

Electron-Gas Spin Susceptibility

Collective Effects in Interband Optical Absorption

Splitting of Conduction-Electron Spin Resonance in Potassium