The Bloch Waves

In Chapter 3 we have considered the probabiUly density functions describing the electron locations in the atom. It is generally accepted to use the term atomic orbital as a synonym of the probabihty density function. 

However, there are orbitals in some soUds that never have specific values of energy, no matter how many atoms are aggregated. These values are grouped in continued intervals thus forming band gaps. One names these gaps as forbidden bands of energy. 

The delocaUzed electrons in a crystal lattice of a solid have properties of waves. The electron waves interact with atoms of the lattice and scattered waves diffract with each other. The allowed and forbidden ranges of the electron energy arise as a result. The electronic band structure of soUds implies certain intervals of energy of electrons in the crystal lattice. 

The ions in a perfect crystal are arranged in a regular array. The electron moving freely inside a crystal is affected by a potential V(r) with the periodicity of the unit cell, that is 

Interatomic Bonding in Solids Fundamentals,Simulation,andApplications, First Edition. Valim Levitin. 

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Furthermore, the field operator is expanded in the Bloch waves with wave vector k in the band denoted by b as... 

All tensor expressions (35)-(42) involve summation over Bloch waves, i.e. summation over j. For a dynamical diffraction calculation involving AT beams, the number of Bloch waves resulting from Equation (30) equals the number of beams, i.e. N. It should be noted, however, that not all of these Bloch waves will be strongly excited within the crystal and contribute to the electron wave field. The excitation amplitudes of the Bloch waves in the crystal are given by B 0J. Extensive numerical calculations show that in a typical dynamical diffraction calculation, although typically more than... 

This chapter is organized in 6 sections. Section 2 describes the geometry of (CBED). Section 3 covers the theory of electron diffraction and the principles for simulation using the Bloch wave method. Section 4 introduces the experimental aspect of quantitative CBED including diffraction intensity recording and quantification and the refinement technique for extracting crystal stmctural information. Application examples and conclusions are given in section 5 and 6. 

Electron Dynamic Theory - The Bloch wave method... 

Electron dynamic scattering must be considered for the interpretation of experimental diffraction intensities because of the strong electron interaction with matter for a crystal of more than 10 nm thick. For a perfect crystal with a relatively small unit cell, the Bloch wave method is the preferred way to calculate dynamic electron diffraction intensities and exit-wave functions because of its flexibility and accuracy. The multi-slice method or other similar methods are best in case of diffraction from crystals containing defects. A recent description of the multislice method can be found in [8]. 

Fig. 4 shows an example of simulated CBED pattern using the Bloch wave method described here for Si [111] zone axis and electron accelerating voltage of 100 kV. The simulation includes 160 beams in both ZOLZ and HOLZ. Standard numerical routine was used to diagonalize a complex general matrix (for a list of routines freely available for this purpose, see [23]). The whole computation on a modem PC only takes a few minutes. 

Figure 4. A simulated CBED pattern for Si[l 11] zone axis at 100 kV using the Bloch wave...

Accurate measurements of low order structure factors are based on the refinement technique described in section 4. Using the small electron probe, a region of perfect crystal is selected for study. The measurements are made by comparing experimental intensity profiles across CBED disks (rocking curves) with calculations, as illustrated in fig. 5. The intensity was calculated using the Bloch wave method, with structure factors, absorption coefficients, the beam direction and thickness treated as refinement parameters. 

The penetration in the absence of absorption is governed by the extinction distance,. This is the depth at which the intensity drops by a factor 1/e in a perfect crystal, and also the depth at which the Bloch wave in the crystal changes phase by a factor 2. ... 

Equation (8.4.2) suggests that a wavefunction uk(r) needs to be found by standard quantum-chemical means for only the atoms or molecules in the one direct-lattice primitive unit cell. For each of the Avogadro s number s worth of fermions in a solid, the factor exp(ik R) in Eq. (8.4.2) provides a new quantum "number," the wavevector k, that guarantees the fermion requirement of a unique set of quantum numbers. The Bloch waves were conceived to explain the behavior of conduction electrons in a metal. 

Some supplementary remarks to the theory of Penn might be appropriate here. There are additional effects which are of relevance if a more quantitative theory of the photoemission process from an adsorbate-covered surface is envisaged. The first point is that the Anderson model as applied to chemisorption is a clearly oversimplified model to describe real metal-adsorbate systems. Besides overlap effects due to the nonorthogonality of the states k) and a), there are several interaction effects which are neglected in the Hamiltonian, Eq.(5). The adsorbed atom, for instance, may act as a scattering centre for the metal electrons and thus modify the Bloch wave functions characteristic of the free substrate. This can be accounted for by adding a term... 

As in the time-dependent case, the relevant sites on the driver wire are those directly coupled to the A wire of the DA system or to the B wire of the DBA system (e.g., sites 1 and 2 of Fig. 4a, b). The driving conditions on these sites are represented by the Bloch wave ... 

For inorganic crystals, the typical extinction of a strong reflection is on the order of a few tens of nanometers. For a crystal of more than a few nanometers thick, multiple scattering must be taken into account for the interpretation of diffraction intensities. There are several approaches to EMS. For perfect crystals, the Bloch wave approach is most useful, which is based on the Bloch wave method formulated by Hans Bethe in his thesis. The diffraction intensity in the Bloch wave theory is given by... 

To verify the syimnetry and identify the sensitivity of CBED to CO syimnetry, dynamic simulations using the Bloch wave method were examined to see the difference between the two models. The atomic positions within the unit cells for the two models from RadaeUi etal. are very close to each other. To avoid the possible pseudo-symmetry generated in the Bi-stripe model, dynamic simulations from the CO stmctures described by the Wigner-crystal model and Bistripe model are calculated and compared for the thickness of 300 nm in Figure 15(d) and (e). The difference between the two simulations is that the G-M lines exist in four (303) reflections and 2n +, 0, 0) reflections simulated by the Wigner-crystal model, as they are in the experimental CBED patterns, but do not show up in four (303) reflections simulated by the Bi-stripe model. 

The Bom-von Karman contour condition demonstrates that the Bloch wave vector of free electrons in a cubic lattice is, according to Sommerfeld, constituted only by real components. The number of k values (k = p/h) admitted in a primitive cell of a reciprocal lattice is equal to the number of sites in the crystal. The linear momenta operator, p, is... 

In the late 1980s, Feibelman developed his Green s function scattering method using LDA with pseudopotentials to describe adsorption on two-dimensionally infinite metal slabs [175]. based on earlier work by Williams et al [176]. The physical basis for the technique is that the adsorbate may be considered a defect off which the Bloch waves of the perfect substrate scatter. The interaction region is short-range because of screening by the electron gas of the metal. Feibelman has used this technique to study, for example, the chemisorption of an H2 molecule on Rh(OOl) [177]. S adatoms on Al(331) [178] and Ag adatoms on Pt(l 11)... 

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