Fermi standard model

Atomic units will be used throughout. The explicit density functionals representing the different contributions to the energy from the different terms of the hamiltonian are found performing expectation values taking Slater determinants of local plane waves as in the standard Fermi gas model. Those representing the first relativistic corrections are calculated in the Appendix. 

We use the standard model [18, 19, 20] for Fermi-liquid leads adiabatically connected to the wire. We assume that the action (3) is applicable for x < L only. At large x the interaction strength K(x—y), Eq. (1), is zero. This model can be interpreted as a quantum wire with electron interaction completely screened by the gates near its ends. Electric fields of external charges are assumed to be screened in all parts of the wire. A simple modification of this model describes electrically neutral leads [20]. All results coincide for our set-up and the model [20]. 

We also note, that the electrostatic potentials resulting from the homogeneous and the Fermi-type models are very closely similar over the full range of r (with the standardization of the models made here), so that the former model, which is much simpler to handle, may serve as a good substitute for the latter model. 

A remark should be made here with respect to the generation and adjustment of the widely used effective core potentials (ECP, or pseudopotentials) [85] in standard non-relativistic quantum chemical calculations for atoms and molecules. The ECP, which is an effective one-electron operator, allows one to avoid the explicit treatment of the atomic cores (valence-only calculations) and, more important in the present context, to include easily the major scalar relativistic effects in a formally non-relativistic approach. In general, the parameters entering the expression for the ECP are adjusted to data obtained from numerical atomic reference calculations. For heavy and superheavy elements, these reference calculations should be performed not with the PNC, but with a finite nucleus model instead [86]. The reader is referred to e.g. [87-89], where the two-parameter Fermi-type model was used in the adjustment of energy-conserving pseudopotentials. 

However the richness of the FISDW was not exhausted a striking result has been the discovery of a reentrance of the metallic state at around 30 teslasan unexpected behavior within the standard model. Few models have been proposed to explain such a behavior. One of them takes explicitely into account interactions between electrons, instead of merely considering the nesting properties of the Fermi surface, as in the standard model. 

Known elementary constituents of matter are quarks and leptons (see Table 12.1). Three families have been discovered. In each family one has two flavors of quarks and one lepton with the associated neutrino. The decay of the free neutron observed in 1932 and described first by the Fermi theory of weak interactions is understood today as the decay of a d-quark (one of three quarks composing the neutron) into a u-quark (which forms the final proton with the unchanged other two quarks) and an electron plus its antineutrino. The particles participating in this process constitute the lightest (first) particle family of the Standard Model. 

To obtain a more quantitative analysis of STM data, three-dimensional wave functions for the tip and the sample are calculated by expHcitly solving the Schrodinger equation for the combined system. In a standard model, commonly referred to as the Tersoff-Hamann modd [16], the tip wave function is approximated by an s-wave function. One can show that, within this model, for small bias voltages, the STM image reflects the SDOS at the Fermi energy at the position of the tip center. 

The density functional theory of Hohenberg, Kohn and Sham [173,205] has become the standard formalism for first-principles calculations of the electronic structure of extended systems. Kohn and Sham postulate a model state described by a singledeterminant wave function whose electronic density function is identical to the ground-state density of an interacting /V-clcctron system. DFT theory is based on Hohenberg-Kohn theorems, which show that the external potential function v(r) of an //-electron system is determined by its ground-state electron density. The theory can be extended to nonzero temperatures by considering a statistical electron density defined by Fermi-Dirac occupation numbers [241], The theory is also easily extended to the spin-indexed density characteristic of UHF theory and of the two-fluid model of spin-polarized metals [414],... 

As for semiconductor/metal contacts, a change in the Fermi level of the liquid phase should result in a different amount of charge transferred across the semicondnctor/liqnid junction. For semiconductor/liquid junctions, the important energetic trends for a series of different liqnid contacts can thns be determined by measuring the solntion redox potential relative to a standard reference electrode system. Within this model, solutions with more positive redox potentials shonld indnce greater charge transfer in contact with n-type semicondnctors. 

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