Bloch wave

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... 

Figure 2. Calculated CBED rocking curves for Si[ 110], a primary beam energy of 193.35 keV and a crystal thickness of 369nm. The three curves shown in the figure were calculated using 80 Bloch waves (circle+solid line) 20 Bloch waves (star solid line) and 5 Bloch waves (dotted line) and the curves correspond to the line of Figure 1 along A-D.

By now, resolution approaches physical limits given by the width of the Is Bloch wave state that acts as a resolution limiting cross section". It is a priori not clear if a further increase of resolution is scientifically beneficial even though some information gain may still be possible for heavier elements with scattering cross sections around 0.5 A. However, there are other equally important aspects that require considerations. 

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. 

Expressed formally, the wave must be matched in amplitude at the surface and in phase velocity parallel to the crystal strrface. This implies that the tangential components of D and H must be continuous across the strrface, and the components in the crystal strrface of the wavevectors inside and outside the strrface must be the same. If n is a rmit vector normal to the crystal strrface, whatever the values of k o or the resttlting Bloch wave inside the crystal, then... 

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. ... 

Figure 4.18 The effect of spherical incident waves on the excitation of Bloch waves, (a) Reciprocal space the divergent incident beam has wavevectors ranging from P j O to P 2 O. (b) Real space energy is distributed throughout the Borrmann fan ABC. The beams generated outside the crystal are indicated...

In Chapter 3 we went as far as we could in the interpretation of rocking curves of epitaxial layers directly from the features in the curves themselves. At the end of the chapter we noted the limitations of this straightforward, and largely geometrical, analysis. When interlayer interference effects dominate, as in very thin layers, closely matched layers or superlattices, the simple theory is quite inadequate. We must use a method theory based on the dynamical X-ray scattering theory, which was outlined in the previous chapter. In principle that formrrlation contains all that we need, since we now have the concepts and formtrlae for Bloch wave amplitude and propagatiorr, the matching at interfaces and the interference effects. 

As one can see, the operator has a property of the wave operator (it transforms the projection of the exact wave function into the exact wave function), however, it should be stressed that the operator converts just one projected wave function into the corresponding exact wave function so we will denote it as a state-specific wave operator in contrast to the so-called Bloch wave operator [46] that transforms all d projections into corresponding exact states. From definition (11) it is iimnediately seen that the state-specific wave operators obey the following system of equations for a = 1,..., d... 

At crystalline surfaces, there are three types of wavefunctions as shown in Fig. 4.1. (1) The Bloch states are terminated by the surface, which become evanescent into the vacuum but remain periodic inside the bulk. (2) New states created at the surfaces in the energy gaps of bulk states, which decay both into the vacuum and into the bulk, the so-called surface states. (3) Bloch states in the bulk can combine with surface states to form surface re.sonances, which have a large amplitude near the surface and a small amplitude in the bulk as a Bloch wave. 

The vacuum tails of Bloch waves are relatively straightforward to understand. Comparing with other experimental methods, STM is more sensitive to the surface states, both the sample and the tip. Therefore, we will spend more time to explain the concept of surface states, from a theoretical point of view and an experimental point of view. 

Here, gi is the length of the primitive reciprocal lattice vector. The constants a, p, and are determined either by considering leading Bloch waves or by fitting with first-principles calculations. The method of Harris and Liebsch is used extensively in the treatment of atom scattering data. With some modifications, the method of Harris and Liebsch is also applicable to calculate STM images. We will discuss it in detail in Chapter 5. 

If at the Fermi level, the only surface Bloch wave of the material is a sinusoidal function with Bloch vector q. 

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