Crystal lattice periodic potential

In a perfect crystal with periodic potentials, electron wave functions form delocalized Bloch waves [46]. Impurities and lattice defects in disordered... 

Th.e refinements of the theory, which have been worked out in particular by Houston, Bloch, Peierls, Nordheim, Fowler and Brillouin, have two main objects. In the first place, the picture of perfectly free electrons at a constant potential is certainly far too rough. There will be binding forces between the residual ions and the conduction electrons we must elaborate the theory sufficiently to make it possible to deduce the number of electrons taking part in the process of conduction, and the change in this number with temperature, from the properties of the atoms of the substance. In principle this involves a very complicated problem in quantum mechanics, since an electron is not in this case bound to a definite atom, but to the totality of the atomic residues, which form a regular crystal lattice. The potential of these residues is a space-periodic function (fig. 10), and the problem comes to this— to solve Schrodinger s wave equation for a periodic poten-tial field of this kind. That can be done by various approximate methods. One thing is clear if an electron... 

Where b is Planck s constant and m and are the effective masses of the electron and hole which may be larger or smaller than the rest mass of the electron. The effective mass reflects the strength of the interaction between the electron or hole and the periodic lattice and potentials within the crystal stmcture. In an ideal covalent semiconductor, electrons in the conduction band and holes in the valence band may be considered as quasi-free particles. The carriers have high drift mobilities in the range of 10 to 10 cm /(V-s) at room temperature. As shown in Table 4, this is the case for both metallic oxides and covalent semiconductors at room temperature. 

Another special case of weak heterogeneity is found in the systems with stepped surfaces [97,142-145], shown schematically in Fig. 3. Assuming that each terrace has the lattice structure of the exposed crystal plane, the potential field experienced by the adsorbate atom changes periodically across the terrace but exhibits nonuniformities close to the terrace edges [146,147]. Thus, we have here another example of geometrically induced energetical heterogeneity. Adsorption on stepped surfaces has been studied experimentally [95,97,148] as well as with the help of both Monte Carlo [92-94,98,99,149-152] and molecular dynamics [153,154] computer simulation methods. 

Even a molecularly smooth single-crystal face represents a potential energy surface that depends on the lateral position x, y) of the water molecule in addition to the dependence on the normal distance z. One simple way to introduce this surface corrugation is by adding the lattice periodicity. An example of this approach is given by Berkowitz and co-workers for the interaction between water and the 100 and 111 faces of the Pt crystal. In this case, the full (x, y, z) dependent potential was determined by a fit to the full atomistic model of Heinzinger and co-workers (see later discussion). 

If V(R) is small with respect to T in (11) the periodic potential V(R) of the ionic cores is that of a lattice of scattering centres. The electronic excitation, described by is a scattered electron wave of high kinetic energy, traveling through the crystal. Many... 

The electrical properties of many solids have been satisfactorily explained in terms of the band theory . Briefly, the motion of an electron detached from its parent atom but free to move in a periodically varying potential field, such as that existing between atoms on a crystal lattice, is expressed in terms of a wave function (Boch Function). This particular... 

In equation 3, ran is the effective mass of the electron, h is the Planck constant divided by 2/rr, and Eg is the band gap. Unlike the free electron mass, the effective mass takes into account the interaction of electrons with the periodic potential of the crystal lattice thus, the effective mass reflects the curvature of the conduction band (5). This curvature of the conduction band with momentum is apparent in Figure 7. Values of effective masses for selected semiconductors are listed in Table I. The different values for the longitudinal and transverse effective masses for the electrons reflect the variation in the curvature of the conduction band minimum with crystal direction. Similarly, the light- and heavy-hole mobilities are due to the different curvatures of the valence band maximum (5, 7). 

Gerasimov et al. have reported that poly-p-PDA Et is obtained quantitatively at 170 - 4.2 K and that the activation energy is 1600 300 eal/mol at 170 - 100 K and close to zero (<20 cal/mol) at 90 — 4.2 K, respectively. From the outstanding reactivity of p-PDA Et at an extremely low temperature, the barrier to the reaction in the monomer crystals has been attributed to the force of the crystal lattice and classified into the region of negative values of the potential energy. In addition the observed induction period at 4.2 K has been attributed to the growth period of crystal defects (see Sect. IV.a.) In the case of DSP, quantitative conversion of monomer to polymer crystals has been achieved by photoirradiation at — 60°C26). 

Despite advanced methods of crystal growth, sohds still contain lattice defects at concentrations around lO -lO " cm, and typical concentrations of defects and impurities in commercial samples are around 10 cm . The latter concentration is usually much greater than the concentration of photogenerated free charge carriers in solids under moderate photoexcitation. Consequently, defects are expected to play an important role in photoexcitation and relaxation processes in heterogeneous systems. In fact, defects create a local distortion of the periodic potential in the solid s lattice. 

Fig. 10.—The potential in a crystal lattice as a periodic function of position in space.

Bloch electrons in a perfect periodic potential can sustain an electric current even in the absence of an external electric held. This infinite conductivity is limited by the imperfections of the crystals, which lead to deviations from a perfect periodicity. The most important deviation is the atomic thermal vibration from the equilibrium position in the lathee however, electric perturbations can also promote this type of vibration. A quantitative treatment of the external electric perturbation of a crystal, therefore, starts with the observation of the change in the lattice vibrations [1] ... 

The analysis of the electronic levels in a periodic potential can be first treated in 1-dimension. The advantage is obtaining an exact solution to the problem and not an approximated one as in the generalization. Suppose that we have a periodic crystal lattice of a periodic potential energy of the form... 

We now wish to extend the dimer model to encompass an infinitely large three-dimensional crystal lattice this is very similar to the transition from the covalent bonding of two atoms to the band structure of a metal or a semiconductor with delocalised states. Starting from the more or less sharp energy levels in the two-body system, we arrive at a band of energy states whose width depends on the interactions of the individual molecules or the overlap of the molecular orbitals in the lattice. We must then take the interaction of an excited molecule with aU the other molecules in the crystal and with the periodic lattice potential into account The levels and E in the dimer model of Fig. 6.7 are transformed into a more or less broad band of energy levels. These are the excitonic bands of the crystal, which we shall treat in this section. 

A crystal is a potentially endless, three-dimensional, periodic discontinuum built up by atoms, ions or molecules. Because of the periodicity, every object is regularly repeated in three-dimensional space that means every unit cell has exactly the same orientation with all molecules in the same conformation as in the cells to its left, right, top, bottom, front, and back. However, an ideal crystal does not exist in most real crystals there are several lattice defects and/or impurities. Frequently, parts of molecules (or in some extreme cases whole molecules) are found in more than one crystallographically independent orientation. One can distinguish three cases ... 

With a large magnification and an objective aperture containing a large number of beams, as seen on the electron diffraction pattern shown in fig. 18 (the circle being the aperture projection), it is possible to image the crystal lattice. Thus, a periodic potential imaging inside a crystal can be obtained with a resolution of about 1.4 A. The TEM presented in fig. 19 shows a succession of lamellar microdomains of polytypic materials. The different structures have been... 

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