Method tight-binding

Within the tight-binding (TB) approach. Slater and Roster [64] described the linear combination of atomic orbitals (LCAO) method as an eflRcient scheme for calculation of the electronic structure of periodic solids. As this method is computationally much less demanding than other methods such as the plane-wave methods, it has been extensively employed to calculate electronic structures of various metals, semiconductors, clusters and a number of complex systems such as alloys and doped systems. The calculation of the electronic structure requires solving the Schrodinger equation with the TB Hamiltonian given by 

To calculate the eigenvalue spectra of nanocrystals, we need to know the e s for the various orbitals and the t s for the interactions. These are evaluated by performing a TB-fit, with a prudent choice of the basis orbitals and the interactions, to the band structure of the bulk solid obtained from first principle calculations such as LAPW, LMTO or the pseudopotential methods. 

The final results of the electronic structure of the nanocrystals depend on the type of the orbital basis chosen to build the TB Hamiltonian. The first-principle band structure calculation of the bulk material gives a good indication of the choice of the basis set and the interactions. For example, the density of states (DOS) and the partial DOS (PDOS) for the bulk system clearly illustrate the various orbitals involved in bonding at any given energy. The character of various bands in the band dispersions can also be analyzed to obtain similar, but even more detailed information. Thus, one can appropriately select the orbital basis to perform the TB 

The calculation of the eigenvalue spectrum starts with the generation of the nanocrystal in real space by specifying the coordinates of the atoms. The nano- 

All the TB models discussed so far are based on the nearest neighbor interactions. Recently, Sapra et al. proposed [73] the sp d tight-binding model with cation-anion nearest neighbor and anion-anion next nearest neighbor (NNN) interactions for the A B semiconductor compounds with A = Zn, Cd, Hg and B = S, Se, Te. The model was chosen after a careful analysis of the bulk band structures of these compounds obtained from the linearized muffin tin orbital (LMTO) method as described earlier in this section. These calculations were car- 

The Harris functional provides not only relatively accurate non-self-consistent estimates of energies and structures, it also provides a basis from which the success of other non-self-consistent approximate methods can be understood. One of the more widely used of these, the tight binding method, is discussed in the next section. 

Tight binding expressions give the total energy for a system of atoms as a sum of eigenvalues e of a set of occupied non-self-consistent, one-electron molecular orbitals plus some additional analytic function A of relative atomic distances 

The general idea behind this expression is to introduce approximate quantum mechanics through the eigenvalue calculation, and then apply corrections for these approximations needed to obtain reasonable total energies through the analytic function. 

A pair-additive sum over atomic distances is often assumed for the analytic function 

Several approaches have been used for determining functional forms for the pair sum Eq. [12]. Once the Hamiltonian matrix elements had been specified, Chadi, for example, used a near-neighbor harmonic interaction for covalent materials where the force constants and minimum energy distances were fit to bulk moduli and lattice constants, respectively. This expression was then used to predict energies and bond lengths for surfaces and related structures. More recently. Ho and coworkers have fit the pair sum to the universal binding energy relation. This reproduces not only lattice constant and bulk modulus, but also ensures reasonable nonlinear interatomic interactions that account for properties like thermal expansion. 

---

Elstner M, Porezag D, Jungnickel G, Eisner J, Flaugk M, Frauenheim Th, Suhai S and Seifert G 1998 Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties Phys. Rev. B 58 7260... 

Bowler D R, Aoki M, Goringe C M, Florsfield A P and Pettifor D G 1997 A comparison of linear scaling tight-binding methods Modeiiing Simuiation Mater. Sc/. 5 199... 

Within the Slater-Koster appro.ximation, we can easily test the validity of the approximations made in eqn (7) based on the graphene model. In Fig. 5 we depict the band gaps using the empirical tight-binding method for nanotube radii less than 1.5 nm. The non-metallic nanotubes n m) are shown in the... 

Right after the discovery of MWCNT [4] several reports on the electronic structures of CNT were almost immediately reported based on rather simple tight-binding method or its equivalent [1-3,5,6]. The most interesting and important features therein [1-3] were that CNT will become either metallic or semiconductive depending on the configuration of CNT, that is. 

G. Seifert, Phys. Rev. B, 58, 7260 (1998). Self-consistent-charge Density-Functional Tight-binding Method for Simulations of Complex Materials Properties. 

Formulation of the Tight Binding Method in the Notation of Second Quantization... 

Figure 5 Density of states of Ni V clusters with N — 5, 6, and 7, calculated by the tight binding method sp (dashed lines) and d (continuous lines). Positive and negative values correspond to up and down J, spins, respectively. The Fermi level is at the energy zero. Adapted with permission from Ref. 45.

One of the first ways used to calculate approximate solutions of (6) was the tight-binding method Here we consider an array of N potential wells, of which two are shown in Fig. 1.1, and suppose that in each well individually the electrons can have a number of bound states with energies WQiWl9...,Wn and with wave functions < 0, ,..., 4>n. When the electron is allowed to move from one well to another, a band containing N states is formed from each bound state of a single well. The approximate wave function describing this motion is... 

It should be emphasized that in metals the d-states, for which tight-binding functions may be used, lie above the zero of the muffin-tin potential The reason why the tight-binding method can still be used is the following. The radial part of the Schrodinger equation is... 

Frauenherm, T.. Seifert, G., Elstner, M., Hajnal, Z., Jungnickel. G.. Porezag. D.. Suhai. S.. and Scholz. R. 2000. A Self-consistent Charge Density-functional Based Tight-binding Method for Predictive Materials Simulations in Physics, Chemistry and Biology , Phys. Stat. Sol. B. 217. 41. 

Fig. 4 a, b Energy bands of a orientationally ordered simple-cubic solid C60 and b of the fee solid C60 shown in the same simple-cubic Brillouin zone. The generalized tight-binding method is used [20]... 

FIGURE 6.4 (a) The re-bands of a 2D graphene layer derived on the basis of the tight-binding method. The... 

Electronic structure methods for studies of nanostructures can be divided broadly into supercell methods and real-space methods. Supercell methods use standard k-space electronic structure techniques separating periodically repeated nanostructures by distances large enough to neglect their interactions. Direct space methods do not need to use periodic boundary conditions. Various electronic structure methods are developed and applied using both approaches. In this section we will shortly discuss few popular but powerful electronic structure methods the pseudopotential method, linear muffin-tin orbital and related methods, and tight-binding methods. 

---