Second-neighbor hopping

The two-dimensional Hubbard model (1) has been studied extensively. Nesting is excellent in the half-filled band defined by ka a + kb b = -rr and deteriorates gradually as the occupancy moves away from this value. As a consequence [see Eqs. (35) and (36)] the CDW and SDW responses will decrease away from n = 1. There is also a van Hove singularity in the noninteracting electronic density of states at midband. Note that the inclusion of second-neighbor hopping modifies this considerably. 

We are interested in a situation where the extra particles in the lattice are described by a single band Hubbard Hamiltonian coupled to the acoustic phonons of the lattice as given in Equation 12.12 [ 128]. In the latter equation, the first and second terms describe the nearest-neighbor hopping of the extra-particles with hopping amplitudes J, and interactions V, computed for each microscopic model by band-structure calculations for Uj = 0, respectively. The third term is the phonon Hamiltonian. The fourth term is the phonon coupling obtained in lowest order in the displacement... 

Tlie suffices i and J refer to individual atoms and S and Sj to the species of the atoms involved. The summation over j extends over those neighbors of the atom i for which ry, the separation of atoms i and J, is within the cutoff radii of these potentials. The second term in Equation (la) is the attractive many-body term and both V and are empirically fitted pair potentials. A Justification for the square root form of the many-body function is provided in the framework of a second moment approximation of the density of states to the tight-binding theory incorporating local charge conservation in this framework the potentials represent squares of the hopping integrals (Ackland, et al. 1988). 

Point defects in solids make it possible for ions to move through the structure. Ionic conductivity represents ion transport under the influence of an external electric field. The movement of ions through a lattice can be explained by two possible mechanisms. Figure 25.3 shows their schematic representation. The first, called the vacancy mechanism, represents an ion that hops or jumps from its normal position on the lattice to a neighboring equivalent but vacant site or the movement of a vacancy in the opposite direction. The second one is an interstitial mechanism where an interstitial ion jumps or hops to an adjacent equivalent site. These simple pictures of movement in an ionic lattice, known as the hopping model, ignore more complicated cooperative motions. 

When the Hamiltonian (3) acts upon the function, the electrons hop to neighboring unfilled lattice sites. For example, the first electron can hop to the second site along the lattice or to the fifth site. With our enumeration over the... 

Still, there is opposite-spin state left at each site. Hopping to one of the next-neighbor sites requires a spin flip. Therefore, metal-insulator transitions involve intersite magnetic exchange coupling. This is the second effect directly or indirectly involved in conductive properties of these crystals. Magnetic effects are briefly discussed in Sect. 4.2. 

With the assumption that electrons hop only between nearest-neighbor sites, we may write Ht in second quantization as... 

All rate constants kf, kox, n, and are defined per individual process and per second. Zan and Zaa are the average nearest neighbor numbers of active/ inactive and active/active sites, respectively. Introducing Zan and Zaa permits a more detailed representation of the surface structure. The term allows COad hopping between neighboring active and inactive sites. 

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