Structure, band

Detailed reviews of the band structure of interstitial carbides were made by Schwarz,0 l Calais,[ Neckel,l J Ivanovsky, and Redinger. The band structure can be summarized as follows. The electronic energy spectra of these carbides are similar and contain bands of C2s, C2p-Md,s, and Md,s,p states (M = metal). When the valence concentration (VEC) in the elementary cell of the carbide is 8 (TiC, ZrC, and HfC), the Fermi level is found in the region of the density-of-state minimum between p-d and fif-like bands. With VEC 8 (carbides of Groups V and VI), the Fermi level is in the low-energy region of the metal states band. 

1 he wave vector may also be treated as a label for the irreducible reprt sentation of the translation group. 

In oiner worus, k ueiemnnes wnicn ureuuciuie represenuuion we are ueaung with (Appendix C on p. 903). This means that k tells us which permitted rhythm is exhibited by the coefficients at atomic orbitals in a particular Bloch function (permitted, i.e. assuring that the square has the symmetry of the crystal). There are a lot of such rhythms, e.g., all the coefficients equ each other k = 0), or one node introduced, two nodes, etc. The FBZ represents a set of such k, which corresponds to all possible rhythms, i.e. non-equivalent Bloch functions. In other words the FBZ gives us all the possible symmetry orbitals that can be formed from an atomic orbital. 

The longer the k, the more nodes the Bloch function has it = 0 means no nodes, at the boundary of the FBZ there is the maximum number of nodes. 

The Hamiltonian H we were talking about represents an effective one-electron Hamiltonian. From Chapter 8, we know that it may be taken as the Fock operator. A crystal represents nothing but a huge (quasi-intinite) molecule, and assuming the Born-von Karman condition, a huge tyclic molecule. This is how we will get the Hartree-Fock solution for the crystal - by preparing the Hartree-Fock solution for a cyclic molecule and then letting the number of unit cells N go to infinity. 

Electronic Motion in the Mean Field Periodic Systems 

FIGURE 5. The cr-symmetry orbitals of a long, all-trans polysilane chain showing backbone orbitals (left), orbitals of the SiH bonds (right), and the result of their mutual interaction (center). (Reprinted from Ref. 63.) 

Stronger effects than those of hyperconjugation can be expected for substituents with stronger interacting power. The effect of aryl substituents will depend on their orientation relative to the plane of the silicon backbone.63 

FIGURE 6. Room temperature PL, EX, and ABS of (a) PDMS powder, (b) PDMS film evaporated on PTFE layer, and (c) on KBr cleaved surface. (Reprinted from Ref. 5.) 

Completely networked polysilanes show a variation of n and k with the degree of cross-linking compared to parent linear polysilanes. 

To have free-electron energy levels, the potential in which the electrons move must be constant (V = - V0) in a crystalline metal there are ion cores arranged in a regular array or lattice, which 

the free-electron wave functions will be seriously disturbed when d = mir/ k. The corresponding values of k, 

The band structure (widths of bands, energy gaps) will obviously depend on the arrangement of the atoms as well as on 

For a periodic lattice, it can be shown (Bloch theorem) that the solutions to the one-electron Schrodinger equation are of the 

Here uf = u exp(277ig r) is, like w, periodic with the period of the lattice, and k = k - 27rg is a reduced wave vector. Repeating this as necessary, one may reduce k to a vector in the first Brillouin zone. In this reduced zone scheme, each wave function is written as a periodic function multiplied by elkr with k a vector in the first zone the periodic function has to be indexed, say ujk(r), to distinguish different families of wave functions as well as the k value. The index j could correspond to the atomic orbital if a tight-binding scheme is used to describe the crystal wave functions. 

The volume contribution by the metallic 5f-5f and the covalent cation 5f-N2p bonds to a virial-theorem formulation of the equations of state of a series of light actinide nitrides was calculated in the self-consistent linear muffin tin orbital (LMTO), relativistic LMTO, and spin-polarized LMTO approximations [46]. The results for ThN give the same lattice spacing in all three approximations higher by ca. 3% than the experimental value, which discrepancy is attributed to the assumed frozen core ions [47]. 

These energy patterns are brought together in Fig. 17.13 which shows plots along two directions in the Brillouin zone. These are the directions corresponding to the x and y axes of Fig. 17.12 but, because the Brillouin zone is in reciprocal space, they are called x and y, respectively, to 

The crystal equivalent is called the Fermi Surface. This may be pictured for the Tp example discussed above and which formed the subject of Figs. 17.12 and 17.13 such a picture is given in Fig. 17.15 for the case of an incompletely filled band. Notice particularly the way that the Fermi surface depends on direction in k space. In some directions the Fermi surface shown in Fig. 17.15 is quite close to k = 0 in others it is close to the zone surface. When the Fermi surface lies within a band a conductor results (travelling waves help to explain how electron migration occurs under the influence of an electrical field). When the Fermi surface occurs at the top of a band, the band is full and an insulator results. When there is a small gap between the top of a full band and the bottom of an empty one, a semiconductor results semiconductors are usually classified in terms of their band gap. Two important semiconductors. Si and GaAs, have band gaps of 1.1 and 1.4 eV respectively for comparison that of diamond is 6.0 eV. 

So far we have restricted our discussion to the A atoms and considered those things that can modify the pattern originally derived, one in which the occupancy of a band is determined solely by the occupancy of the orbital from which it is derived. Of course, everything that has been said about the band structure derived from the orbitals of A hold for the orbitals of B also. Next it has to be recognized that there can be interaction between the bands derived from the orbitals of A and B—this is the equivalent of bonding in an isolated AB molecule. In molecules, interaction only occurs between orbitals of the same symmetry. So, too, in solids. Interaction between crystal orbitals can only occur when the crystal molecular orbitals (which are 

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Other methods for detennining the energy band structure include cellular methods. Green fiinction approaches and augmented plane waves [2, 3]. The choice of which method to use is often dictated by die particular system of interest. Details in applying these methods to condensed matter phases can be found elsewhere (see section B3.2). 

Figure Al.3.14. Band structure for silicon as calculated from empirical pseudopotentials [25],...

The structure in the reflectivity can be understood in tenns of band structure features i.e. from the quantum states of the crystal. The nonnal incident reflectivity from matter is given by... 

It is possible to identify particular spectral features in the modulated reflectivity spectra to band structure features. For example, in a direct band gap the joint density of states must resemble that of critical point. One of the first applications of the empirical pseudopotential method was to calculate reflectivity spectra for a given energy band. Differences between the calculated and measured reflectivity spectra could be assigned to errors in the energy band... 

Figure Al.3.24. Band structure of LiF from ab initio pseudopotentials [39],...

Simple metals like alkalis, or ones with only s and p valence electrons, can often be described by a free electron gas model, whereas transition metals and rare earth metals which have d and f valence electrons camiot. Transition metal and rare earth metals do not have energy band structures which resemble free electron models. The fonned bonds from d and f states often have some strong covalent character. This character strongly modulates the free-electron-like bands. 

Is 2s 2p 3s 3p 3d 4s. If the 3d states were truly core states, then one might expect copper to resemble potassium as its atomic configuration is ls 2s 2p 3s 3p 4s The strong differences between copper and potassium in temis of their chemical properties suggest that the 3d states interact strongly with the valence electrons. This is reflected in the energy band structure of copper (figure Al.3.27). 

At a surface, not only can the atomic structure differ from the bulk, but electronic energy levels are present that do not exist in the bulk band structure. These are referred to as surface states . If the states are occupied, they can easily be measured with photoelectron spectroscopy (described in section A 1.7.5.1 and section Bl.25.2). If the states are unoccupied, a teclmique such as inverse photoemission or x-ray absorption is required [22, 23]. Also, note that STM has been used to measure surface states by monitoring the tunnelling current as a fiinction of the bias voltage [24] (see section BT20). This is sometimes called scamiing tuimelling spectroscopy (STS). 

Figure B3.2.1. The band structure of hexagonal GaN, calculated using EHT-TB parameters detemiined by a genetic algorithm [23]. The target energies are indicated by crosses. The target band structure has been calculated with an ab initio pseudopotential method using a quasiparticle approach to include many-particle corrections [194].

The pseudopotential is derived from an all-electron SIC-LDA atomic potential. The relaxation correction takes into account the relaxation of the electronic system upon the excitation of an electron [44]- The authors speculate that ... the ability of the SIRC potential to produce considerably better band structures than DFT-LDA may reflect an extra nonlocality in the SIRC pseudopotential, related to the nonlocality or orbital dependence in the SIC all-electron potential. In addition, it may mimic some of the energy and the non-local space dependence of the self-energy operator occurring in the GW approximation of the electronic many body problem [45]. 

The main drawback of the chister-m-chister methods is that the embedding operators are derived from a wavefunction that does not reflect the proper periodicity of the crystal a two-dimensionally infinite wavefiinction/density with a proper band structure would be preferable. Indeed, Rosch and co-workers pointed out recently a series of problems with such chister-m-chister embedding approaches. These include the lack of marked improvement of the results over finite clusters of the same size, problems with the orbital space partitioning such that charge conservation is violated, spurious mixing of virtual orbitals into the density matrix [170], the inlierent delocalized nature of metallic orbitals [171], etc. 

Starrost F, Bornhoidt S, Soiterbeck C and Schattke W 1996 Band-structure parameters by genetic aigorithm Phys. Rev. B 53 12 549 Phys. Rev. B 54 17 226E... 

Terakura K, Qguchi T, Williams A R and Kubler J 1984 Band theory of insulating transition-metal monoxides Band-structure calculations Phys. Rev. B 30 4734... 

Yussouff M 1987 Fast self-consistent KKR method Electronic Band Structure and Its Applications (Lecture Notes in Physics vol 283) ed M Yussouff (Berlin Springer) pp 58-76... 

Cortona P 1991 Self-consistently determined properties of solids without band structure calculations Phys. Rev. B 44 8454... 

A second constraint is that the relative order of the critical energies at = 0 and fc = I is invariant to the presence or absence of the potential V(4>) [H]-Equation (A.6) shows that the free motion band structure can be folded onto the interval — Consequently, preservation of relative energy orderings at... 

Figure 3. Floquet band structure for a threefold cyclic barrier (a) in the plane wave case after using Eq. (A.l 1) to fold the band onto the interval —I < and (b) in the presence of a threefold potential barrier. Open circles in case (b) mark the eigenvalues at = 0, 1, consistent with periodic boundary conditions. Closed circles mark those at consistent with sign-changing...

As a concrete illustration of the Floquet band structure for a threefold barrier. Section 3.4 of Child [50] contains an explicit analytical form for the matiix u ... 

I be second important practical consideration when calculating the band structure of a malericil is that, in principle, the calculation needs to be performed for all k vectors in the Brillouin zone. This would seem to suggest that for a macroscopic solid an infinite number of ectors k would be needed to generate the band structure. However, in practice a discrete saaipling over the BriUouin zone is used. This is possible because the wavefunctions at points... 

Mujica A and R J Needs 1993. First-principles Calculations of the Structural Properties, Stability, aind Band Structure of Complex Tetrahedral Phases of Germanium ST12 and BC8. Physical Review B48 17010-17017. 

The primary reason for interest in extended Huckel today is because the method is general enough to use for all the elements in the periodic table. This is not an extremely accurate or sophisticated method however, it is still used for inorganic modeling due to the scarcity of full periodic table methods with reasonable CPU time requirements. Another current use is for computing band structures, which are extremely computation-intensive calculations. Because of this, extended Huckel is often the method of choice for band structure calculations. It is also a very convenient way to view orbital symmetry. It is known to be fairly poor at predicting molecular geometries. 

The simplest approximation to the complete problem is one based only on the electron density, called a local density approximation (LDA). For high-spin systems, this is called the local spin density approximation (LSDA). LDA calculations have been widely used for band structure calculations. Their performance is less impressive for molecular calculations, where both qualitative and quantitative errors are encountered. For example, bonds tend to be too short and too strong. In recent years, LDA, LSDA, and VWN (the Vosko, Wilks, and Nusair functional) have become synonymous in the literature. 

The electronic structure of an infinite crystal is defined by a band structure plot, which gives the energies of electron orbitals for each point in /c-space, called the Brillouin zone. This corresponds to the result of an angle-resolved photo electron spectroscopy experiment. 

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