Hubbard Approximation Research Articles

The Hubbard sub-band structure and the cohesive energy of narrow band systems

The Hubbard sub-band structure in a narrow band has been obtained in an analytical form in the third Hubbard approximation, using the Lorentzian form for the density of states for non-interacting electrons. The imaginary part of the self-energy signifying the lifetime of the interacting electrons is analysed. The contribution to the cohesive energy from electrons in a narrow non-degenerate band is then obtained analytically. Certain aspects of the band and the atomic contributions to the cohesive energy are clarified. The results are compared with those obtained earlier. Possible origins of the asymmetry of the experimental data on the heat of formation of transition-metal compounds for particles and holes are discussed.

The ground-state energy of an electron-hole liquid in a strong magnetic field

The behaviour of electron-hole systems such as Bi and Bi1-xSbx in a strong magnetic field has been investigated using simple models. The ground-state energy is numerically calculated using the Hartree-Fock, random-phase and Hubbard approximations. The correlation energy of the system is shown to have the properties inherent in a quasi-one-dimensional and one-component spin system. The effect of electron correlation on ultrasonic attenuation is discussed.

On the Magnetism in Narrow‐Band Conductors

AbstractThe problem of the double exchange interaction mechanism in narrow‐band magnetic conducting crystals in terms of the Vonsovskii s‐f exchange model is considered taking into account the Coulomb interaction between conduction electrons, using Hubbard approximation. The values of the Néel paramagnetic temperature and of effective paramagnetic moment, as function of concentration of current carriers, are determined. A comparison of the theoretical expressions with the experimentally found Gd concentration dependence of the Curie paramagnetic temperature of solid solutions Eu1–xGdxSe helps to obtain a correlation between the conduction band width, the magnitude of the intraatomic exchange integral, and the value of the Coulomb interaction of electrons on a site. The variation of the effective magnetic moment value in the Eu1–xLaxS series, as a function of La concentration is discussed.

Density functional approximation for the quasiparticle properties of simple metals. I. Theory and electron gas calculations

Calculations of the self-energy of the electron gas for energies near the Fermi energy are presented. The results of these calculations are required to quantify a previously suggested approximation for the self-energy of real metals. Two separate types of improvement on the random-phase approximation (RPA) for the self-energy are considered. The first type is similar to the Hubbard approximation while the second type explicitly evaluates terms of second and third order in the renormalised interaction expansion for the self-energy. Local exchange-correlation potentials from these and other recent calculations and parameterised in a form that will be useful in bandstructure calculations. Results for the correlation contribution to the ground-state energy are also obtained and inciate that RPA gives results that are about 25% too large at the lowest metallic densities.

Interaction between a plasma and a strong electromagnetic wave

The interaction between a high density plasma and a strong electromagnetic wave are studied by using the quantum- mechanical two times Green's function method. The equations of motion for Green's function of a many component plasma in an electromagnetic wave with several frequencies and polarization vectors are obtained as a set of simultaneous equations. The equations of motion are solved in the case of a plasma which is composed of electrons and heavy ions, and the dispersion relation and the density correlation functions for electrons and ions are formulated in closed form. The quantum-mechanical dispersion relation is similar to the type of expression obtained by Silin for the classical plasma. The effects of exchange Coulomb interactions, which become appreciable in high density plasmas, on the formulation are studied for the case of Fermion particles by using Hubbard's approximation.

The Falikov-Kimball model: An intermediate solution

We have solved the Falicov-Kimball model of a metal-insulator transition for a cubic substance, in the simplest of Hubbard's approximations. Our calculations cover the one band and split band cases, and show no transition or metallic state.

Electron correlations in inversion layers

The two-dimensional electron gas (2DEG) is a widely used model for the electrons in the inversion layer of certain MOSFET devices. The sensitivity of a number of physical quantities of the 2DEG to a proper treatment of many-body effects has been examined. This was achieved by calculating the plasmon dispersion, pair correlation function, exchange and correlation energy and the self-energy at the Fermi level using three different approximations to the many-body problem, namely: (i) the RPA; (ii) the Hubbard approximation (HA); and (iii) the self-consistent STLS approximation of Singwi et al. (1968). The RPA and HA were found to be less satisfactory approximations for a 2DEG than for a 3DEG.

Quasi-particle properties in surface quantized states of silicon

A field theoretical approach to surface quantized states in semiconductors is presented. In this system the longitudinal dielectric function is a tensor whose suffices are the subbands. The quasi-particle properties in Si n-channel inversion and accumulation layers, and in an Si p-channel inversion layer are calculated in the extended Hubbard approximation. The results show excellent agreements with the experiments both in the mass and in the level separation between the ground and the first excited subband.

A generalized formula for the energies of alternant molecular orbitals. I. Homonuclear molecules

AbstractUsing the method of alternant molecular orbitals (AMO) it is shown that the energies ofAMO's (Ek), for any alternant homonuclear molecule having a singlet ground state, are connected with the energies of theMO's (ek) obtained by the conventional Hartree–Fock (HF) method by the formula\documentclass{article}\pagestyle{empty}\begin{document}$ E_{k\alpha (\beta )} = \pm \sqrt {\Delta ^2 + e_k ^2 } $\end{document}, where Δ is the correlation correction. The formula is applicable in the semiempiricalLCAOform used in the Pariser–Parr–Pople theory, by Hubbard's approximation of γ integrals.

Effective Mass andgFactor of Interacting Electrons in the Surface Inversion Layer of Silicon

The effective mass and the $g$ factor of quasiparticles near the Fermi surface of a two-dimensional electron gas are calculated in the random-phase approximation and in the Hubbard approximation, and are compared with the experimental results for an inversion layer on a (100) surface of silicon.

Electron correlation in the narrow-band model for disordered ferromagnetic alloys

The ferromagnetism of a random disordered substitutional binary alloy described by the narrow-energy-band model of Hubbard is discussed within the framework of the coherent-potential approximation with strong correlation effects included. The treatment of Coulomb correlations used is an improved version of the first Hubbard approximation which seems to adequately describe those correlation effects that have a major influence on the stability of the ferromagnetic phase for large intra-atomic Coulomb repulsion. Disorder is considered in the site-diagonal terms of the Hubbard model. Numerical illustrations are presented which describe the ferromagnetic solutions in terms of the component magnetizations and component densities of states as functions of electron concentration in the band for several cases of disorder and alloy compositions. In these examples the intra-atomic Coulomb repulsion is assumed to be infinite. Comparisons are made with results obtained using a Coulomb repulsion of the order of the bandwidth treated by the Hartree-Fock approximation. One finds that results are qualitatively similar for small electron concentrations and the examples detail the differences at larger concentrations. Finally a brief discussion with respect to experiment is made.

Calculation of the momentum density in ferromagnetic nickel

A band-structure calculation on the basis of Hubbard's approximation scheme has been made of the momentum-density distribution for positron annihilation and Compton scattering in ferromagnetic nickel. Graphs are shown of the contributions of the various bands for both spin directions, of the total momentum density, and of the spin density observable with polarized positrons. Owing to the symmetry of the wave functions the only pieces of Fermi surface observable with these techniques are the large sheets in the sixth majority- and minority-spin bands, and the minority-spin hole pocket in the third band. A comparison of the predicted angular correlation with experiment shows that the calculation somewhat overestimates the high-momentum region.

Knight shift and NMR line shape in the metallic region of correlated impurity band electrons

Abstract The Knight shift and NMR line shape in the metallic region are calculated on the donor-sphere model, with use of the Hubbard approximation for the electron correlation. The theory is successfully applied to the experimental results by Sasaki, Ikehata and Kobayashi (1973) for Si : P. It is established that the NMR line width is due to the inhomogeneous broadening resulting from the spatial dispersion of the wave function and that, because of the enhancement due to the correlation, the Knight shift depends on the donor impurity concentration very weakly.

Electron Correlation and Metal-Nonmetal Transition in a Disordered Binary System

The metal-insulator transition due to electron correlation and potential fluctuations in a substitutionally disordered $n$-component system, particularly in a binary disordered system, is studied on the basis of the Hubbard model and of the localization-delocalization concept in the Anderson sense. For illustration, the case of $n=1$ is first studied. This is the original Hubbard system of a regular crystal, for which the mobility gap coming from the random distribution of spins is calculated according to the localization function. The existence of the mobility gap alters not only the critical metal-nonmetal (M-NM) density but also the sharpness of the transition in the sense that the critical index of the mobility gap is 1/2, while that of the density-of-states gap is 3/2. It is shown that for reasonably general systems the resonance corrections in the Hubbard approximation modify the effect of the scattering corrections quantitatively rather than qualitatively. It is proved that, when only the scattering correction of the Hubbard theory is taken into account, the problem of treating the effect of electron correlation in an $n$-component alloy with substitutional disorder is reduced to that of treating an independent-electron picture in the coherent-potential approximation for a $2n$-component system. For binary systems with $n=2$, the quasiparticle densities of states are evaluated for various interatomic distances and the localization of the states is examined. Numerical results are discussed with emphasis on the M-NM transition. A possible extension of the method to amorphous systems with topological disorder is mentioned in relation to experiments on the M-NM transition in some metal rare-gas solids.

Nuclear Spin-Lattice Relaxation Time Due to the Orbital Interaction

In the free electron model the spin-lattice relaxation time T 1 (orb) due to the interaction with the orbital motion of conduction electrons diverges if the nuclear Zeeman energy is neglected. Without neglecting it, T 1 (orb) with the field dependence of a form log ( H ext ) is derived for high magnetic fields. The divergence which arises for H ext =0 is removed by taking into account the dipole interaction between nuclear spins. The effect of the electron correlation at high fields is examined by use of the RPA approximation. The Hubbard approximation for the exchange interaction has no effect. Therefore, by use of a reasonable approximation, the result that ( T 1 (orb) ) -1 increases in the presence of the exchange interaction is found. It is seen that the enhancement is small compared with the one for the spin paramagnetism.

The Ground State Energy of the Hubbard Model in Hubbard's Approximation

Abstract We have calculated the ground state energy of the Hubbard model in the approximation of Hubbard's first paper1 . For the neutral model with nearest neighbour interaction the energy resulting from the selfconsistent paramagnetic solution is compared with those ones following from the (ferromagnetic) Hartree-Fock and an (antiferromagnetic) single particle theory. The energy of the latter one turns out to be the best approximation of the true ground state energy of the model for all values of the coupling constant V0 , but the energy derived from Hubbard's approximation, in spite of the absence of magnetic ordering, is a reasonable approximation at least for sufficiently large values of V0.

Charged particle energy loss to electron gas

AbstractThe contribution of the conduction electrons to the stopping power (– dE/dx) of fast charged particles in a crystal has been calculated using different forms of the dielectric function. The various approximate forms of dielectic functions used are: 1. random phase approximation (RPA), 2. Hubbard approximation (both, screened and unscreened), 3. Kleinman approximation, and 4. the dielectric function obtained by variational calculation of Langreth. The intermediate velocity range for the incident particles, namely 2 VF<V<10 VF was investigated and it has been found that in this region the stopping power varies inversely to the particle velocity. Moreover the velocity dependence is found to be insensitive to the form of the dielectric function used in the calculation.

Gibbs-Duhem Relation and Consistency of Hubbard's Approximation

It is proposed that some of the formulae derived from the Gibbs-Duhem relation may serve as a simple test of consistency for approximations employed in solving many-body problems. When two of the intensive parameters are partly developed in t~rms of normalized extensive parameters, such as density, simple connections among the expansion coefficients follow directly from the Gibbs-Duhem relation. As an example chemical potential and external magnetic field are chosen as the two intensive parameters, to apply the relation to the Hubbard model. It is shown that the Hartree-Fock approximation surely satisfies the relation, while Hubbard's first approximation does not. Thus the latter approximation is thermodynamically inconsistent, and its validity should be seriously questioned.

Screening of a Fixed Charge in the Modified Hubbard Approximation

AbstractUsing a modified Hubbard approximation for the static dielectric function, numerical calculations of the screening charge density around a fixed foreign charge in an electron liquid are presented for values of rs corresponding to the metallic density range. The screening parameter has been taken to reproduce the compressibility limit properly. The results are compared with those based on the original Hubbard approximation as calculated by Langer and Vosko, and it is found that the difference is small but not negligible.

Screening of a Fixed Charge in Metals

AbstractUsing a modified Hubbard formula for the dielectric function which has been previously shown to give results for the pair distribution function which are better than those given by the original Hubbard dielectric function, numerical calculations of the screening charge density around a fixed foreign charge in an electron liquid are presented for values of rs corresponding to the metallic density range. The results are compared with those calculated by Langer and Vosko based on the original Hubbard approximation and appreciable differences are found at distances corresponding to the first three shells of ions around the impurity.