Cubic Materials

The low-indexed surfaces, which we will consider the main zinc blende planes, differ in terms of their composition, that is, the amount of cations and anions within the plane. In the (011) direction, the planes consist of the same number of cations and anions, which means that the 011 surfaces are stoichiometric. In the (001) and (111) directions, single atomic planes are either populated by cations or 

In Table 13.1, the covalent radii, the electronegativity values, and the electron configuration for the most important group II, III, V, and VI elements are Hsted. 

---

The higher-order bulk contribution to the nonlmear response arises, as just mentioned, from a spatially nonlocal response in which the induced nonlinear polarization does not depend solely on the value of the fiindamental electric field at the same point. To leading order, we may represent these non-local tenns as bemg proportional to a nonlinear response incorporating a first spatial derivative of the fiindamental electric field. Such tenns conespond in the microscopic theory to the inclusion of electric-quadnipole and magnetic-dipole contributions. The fonn of these bulk contributions may be derived on the basis of synnnetry considerations. As an example of a frequently encountered situation, we indicate here the non-local polarization for SFIG in a cubic material excited by a plane wave (co) ... 

As an illustration, we consider the case of SFIG from the (111) surface of a cubic material (3m. syimnetry). More general treatments of rotational anisotropy in centrosymmetric crystals may be found in the literature [62. 63 and M]- For the case at hand, we may detennine the anisotropy of the radiated SFl field from equation Bl.5.32 in conjunction with the fonn of -)from table Bl.5.1. We fmd, for example, for the p-in/p-out and s-... 

D. L. Woodraska, J. A. Jaszczak. A Monte Carlo simulation method for [111] surfaces of silicon and other diamond-cubic materials. Surf Sci 574 319, 1997. 

In table 2 and 3 we present our results for the elastic constants and bulk moduli of the above metals and compare with experiment and first-principles calculations. The elastic constants are calculated by imposing an external strain on the crystal, relaxing any internal parameters (case of hep crystals) to obtain the energy as a function of the strain[8]. These calculations are also an output of onr TB approach, and especially for the hep materials, they would be very costly to be performed from first-principles. For the cubic materials the elastic constants are consistent with the LAPW values and are to within 1.5% of experiment. This is the accepted standard of comparison between first-principles calculations and experiment. An exception is Sr which has a very soft lattice and the accurate determination of elastic constants is problematic. For the hep materials our results are less accurate and specifically in Zr the is seriously underestimated. ... 

The calculation of the magnetic anisotropy of non-cubic materials requires an expansion up to 1 /c . Except in the case of fully relativistic calculations, the expansion is never carried out consistently and only the spin-orbit perturbation is calculated to second order (or to infinite order), without taking account of the other terms of the expansion. In this section, we shall follow Gesztesy et al. (1984) and Grigore et al. (1989) to calculate the terms H3 and H. Hz will be found zero and H4 will give us terms that must be added to the second order spin-orbit calculation to obtain a consistent semi-relativistic expansion. 

We now need to define a collection of atoms that can be used in a DFT calculation to represent a simple cubic material. Said more precisely, we need to specify a set of atoms so that when this set is repeated in every direction, it creates the full three-dimensional crystal stmcture. Although it is not really necessary for our initial example, it is useful to split this task into two parts. First, we define a volume that fills space when repeated in all directions. For the simple cubic metal, the obvious choice for this volume is a cube of side length a with a corner at (0,0,0) and edges pointing along the x, y, and z coordinates in three-dimensional space. Second, we define the position(s) of the atom(s) that are included in this volume. With the cubic volume we just chose, the volume will contain just one atom and we could locate it at (0,0,0). Together, these two choices have completely defined the crystal structure of an element with the simple cubic structure. The vectors that define the cell volume and the atom positions within the cell are collectively referred to as the supercell, and the definition of a supercell is the most basic input into a DFT calculation. 

The shape of the curve in Fig. 2.1 is simple it has a single minimum at a value of a we will call g0. If the simple cubic metal exists with any value of a larger or smaller than a0, the total energy of the material could be reduced by changing the lattice parameter to a0. Since nature always seeks to minimize energy, we have made a direct physical prediction with our calculations DFT predicts that the lattice parameter of our simple cubic material is a0. 

The simple cubic crystal structure we discussed above is the simplest crystal structure to visualize, but it is of limited practical interest at least for elements in their bulk form because other than polonium no elements exist with this structure. A much more common crystal stmcture in the periodic table is the face-centered-cubic (fee) structure. We can form this structure by filling space with cubes of side length a that have atoms at the corners of each cube and also atoms in the center of each face of each cube. We can define a supercell for an fee material using the same cube of side length a that we used for the simple cubic material and placing atoms at (0,0,0), (0,g/2,g/2), (g/2,0,g/2), and (g/2,g/2,0). You should be able to check this statement for yourself by sketching the structure. 

In Chapter 2 we mentioned that a simple cubic supercell can be defined with lattice vectors a, = a or alternatively with lattice vectors a = 2a. The first choice uses one atom per supercell and is the primitive cell for the simple cubic material, while the second choice uses eight atoms per supercell. Both choices define the same material. If we made the second choice, then... 

Figure 4.5 shows another surface that can be defined in a face-centered cubic material this time the highlighted plane intercepts the x, y, and z axes at 1, 1, and 1. The reciprocals of these intercepts are 1/1, 1/1, and 1/1. 

This is identical in form to the equation for V (z) in an isotropic material. But of course R hkl (8) is quite different from any isotropic reflectance function. For a cubic material an anisotropy factor can be defined as... 

Fig. 1.13 Electron density of states N(E) in a cubic material F denotes the Fermi energy (a) normal metal (b) semimetal (c) insulator, (d) n-type degenerate semiconductor.

The first-order coefficients of magnetostriction of cubic materials are then... 

Methods to calculate coherency stresses in anisotropic materials, and an example calculation for cubic materials, have been published [17]. 

For molybdenum trifluoride, it has been established that the cubic material contains oxide impurity, and the pure compound has been prepared by the reduction of the pentafluoride (38, 39) or hexafluoride (37) with molybdenum metal and has a rhombohedral unit cell. 

In this regard, for the (100) reflection of a powdered cubic material, some crystallites will be oriented in such a way that the (100) reflection will occur, and for others in such a position the (010) or (001) reflection will occur, but since dm = dmo = dm, these reflections contribute to the same peak [21],... 

MCM-48, a cubic material, exhibits an x-ray diffraction pattern consisting of several peaks that can be assigned to the Ia3d space group [117]. The structure of MCM-48 has been proposed to be bicontinuous with a simplified representation of two, infinite, three-dimensional, mutually intertwined, unconnected network of rods [118]. 

On a continuum level, the Heisenberg exchange energy of a cubic material is... 

The most common substrates for the growth of cubic P-GaN have been GaAs and 3C-SiC, discussed in Datareviews A7.7 and A7.8 respectively. There have been some structural studies of P-GaN films grown on Si (001) [8] and MgO [1]. The major defects in the cubic material are stacking faults along the 111 planes and perfect edge dislocations at the interface [1,8-11]. 

The precise description of the effect of particle size on an infinitely sharp diffraction peak for correctly only cubic materials takes the form of Equation (11) when using Equation (10) for the definition of a peak via its intensity steps (W) and with A(20) being the range of integration over the peak. 

The object of this experiment is to determine the crystal structure of a solid substance from x-ray powder diffraction patterns. This involves determination of the symmetry classification (cubic, hexagonal, etc.), the type of crystal lattice (simple, body-centered, or face-centered), the dimensions of the unit cell, the number of atoms or ions of each kind in the unit cell, and the position of every atom or ion in the unit cell. Owing to inherent limitations of the powder method, only substances in the cubic system can be easily characterized in this way, and a cubic material will be studied in the present experiment. However, the recent introduction of more accurate experimental techniques and sophisticated computer programs make it possible to refine and determine the structnres of crystals of low syimnetiy from powder diffraction data alone. 

Example 2.2 Calculate the stress distribution required to obtain the deformation specified in Example 2.1 in a cubic material. 

Solution In Example 2.1 it was found that 5 = Aa. In reduced notation, 5 = Aa, while S2 through Sa are all zero. Using the reduced notation and the stiffness matrix for a cubic material [1] ... 

---