Brillouin cubic crystals

In fignre A1.3.9 the Brillouin zone for a FCC and a BCC crystal are illustrated. It is a connnon practice to label high-synnnetry point and directions by letters or symbols. For example, the k = 0 point is called the F point. For cubic crystals, there exist 48 symmetry operations and this synnnetry is maintained in the energy bands e.g., E k, k, k is mvariant under sign pennutations of (x,y, z). As such, one need only have knowledge of (k) in Tof the zone to detennine the energy band tlnoughout the zone. The part of the zone which caimot be reduced by synnnetry is called the irreducible Brillouin zone. 

Figure 4.13 First Brillouin zone of the body-centered cubic crystal lattice, k, ky,kz are the axes of a Cartesian coordinate system in fe-space. The symmetry points and symmetry lines are indicated. See Table 4.1 for details.

Figures 4.13 and 4.14 present the Brillouin zones for cubic crystal lattices. One can see the Brillouin zone for hexagonal close-packed crystal lattice in Figure 4.15.

Table 4.1 Points and directions of high symmetry in the first Brillouin zones, fee is the face-centered cubic crystal lattice bcc is the body-centered cubic crystal lattice hep is the hexagonal close-packed crystal lattice.

The sets of special points for numerical integration over the Brillouin zone of cubic crystals are given in Tables 4.1-4.3. They are obtained by symmetrical increasing of... 

Fig. 7. (a) RBj-cubic crystal structure. Large spheres without pattern and small spheres with pattern show the R atoms and the B atoms, respectively, (b) Brillouin zone of the simple cubic crystal lattice. 

Fig. 45. (a) NaCl type cubic crystal structure of the RX compounds. Spheres with and without pattern show the X and R atoms, respectively (b) Brillouin zone of the face-centered cubic crystal lattice. 

Fig. 2.10 Schematic representation of the bulk (shaded area) and sur-foce (dashed line) phonon dispersions for a (110) surface of a cubic crystal. The symmetry lines for the first 2D Brillouin zone are shown in Fig. 2.7a.

Surface modes can be clearly identified in the dispersion relations ft>(qn) when they appear in regions where no bulk bands appear. Similar to the identification of surface electronic states, the projected bulk modes form the bulk phonon bands in the surface Brillouin zone, as shown in Figure 9.46. In the bulk case, there are three acoustic phonon bands and 3(S-1) optical phonon bands, with S as the number of atoms in the primitive unit cell of the bulk crystal. Along high-symmetry directions in the bulk, such as the (100) or (111) directions in cubic crystals, the phonons can be classified either as transverse or longitudinal, depending on whether or not their displacements are perpendicular or parallel to the direction of the 3D wave vector. 

Figure 2.5 (a) The method for construeting the Brillouin zone for a square planar lattice and the first Brillouin zone for (b) a faee-eentered eubic crystal and (c) a body-centered cubic crystal. For the fee erystal the diamond-shaped faees of the Brillouin zone are along cube axes, [100]-type direetions, while the hexagonal faees are along [111] cube diagonals. For discussion of the [100], [111], and other erystal indiees, see Chapter 4. 

The region within which k is considered (—n/a first Brillouin zone. In the coordinate system of k space it is a polyhedron. The faces of the first Brillouin zone are oriented perpendicular to the directions from one atom to the equivalent atoms in the adjacent unit cells. The distance of a face from the origin of the k coordinate system is n/s, s being the distance between the atoms. The first Brillouin zone for a cubic-primitive crystal lattice is shown in Fig. 10.11 the symbols commonly given to certain points of the Brillouin zone are labeled. The Brillouin zone consists of a very large number of small cells, one for each electronic state. 

First Brillouin zone for a cubic-primitive crystal lattice. The points X are located at k = it/a in each case... 

Thus, the reciprocal lattice of a simple cubic lattice is also simple cubic. It is shown in Fig. 5.7 in the xy plane, where it is clear that the bisectors of the first nearest-neighbour (100) reciprocal lattice vectors from a closed volume about the origin which is not cut by the second or any further near-neighbour bisectors. Hence, the Brillouin zone is a cube of volume (2n/a)2 that from eqn (2.38) contains as many allowed points as there are primitive unit cells in the crystal. The second, third, and fourth zones can... 

Fig. 6. Energy of an electron in a crystal plotted against wave number kx for a simple-cubic lattice with Brillouin-zone boundaries at kx = (7r/a), zt(2ir/a). Solid line is for reduced wave number — (w/a) < kx < (w/a). Heavy dashed line is for kx defined in range — o < kx < . Light dashed line is solution for a constant potential. Quantum number m defines the band. Effective mass of interest is...

First Brillouin zone for a face-centered cubic (FCC) crystal (with body-centered Brillouin zone). The point T is at the origin (0, 0, 0) the Miller60 indices of three faces are shown. The points L and X are at the centers of a hexagonal and a square face, respectively, while points U and K bisect sides, and point W is at the vertex of three adjacent sides. 

In the reciprocal space in three dimensions the zones, instead of line sections, become geometrical bodies determined by the values of the three integers h, k, and l instead of by m. They are just such polyhedra (Brillouin zones) in the reciprocal lattice as the faces of a crystal in ordinary space which correspond to the same hkl (e.g. 100 is the cube, 111 the octahedron, etc. in the cubic system). 

The band structure of solids has been studied theoretically by various research groups. In most cases it is rather complex as shown for Si and GaAs in Fig. 1.5. The band structure, E(kf is a function of the three-dimensional wave vector within the Brillouin zone. The latter depends on the crystal structure and corresponds to the unit cell of the reciprocal lattice. One example is the Brillouin zone of a diamond type of crystal structure (C, Si, Ge), as shown in Fig. 1.6. The diamond lattice can also be considered as two penetrating face-centered cubic (f.c.c.) lattices. In the case of silicon, all cell atoms are Si. The main crystal directions, F —> L ([111]), F X ([100]) and F K ([110]), where Tis the center, are indicated in the Brillouin zone by the dashed lines in Fig. 1.6. Crystals of zincblende structure, such as GaAs, can be described in the same way. Here one sublattice consists of Ga atoms and the other of As atoms. The band structure, E(k), is usually plotted along particular directions within the Brillouin zone, for instance from the center Falong the [Hl] and the [HX)] directions as given in Fig. 1.5. 

Here the = (mi x + m2 y + m3 z)a are the positions of the ions. We see immediately that there are as many values of k as there are chlorine ions these correspond to the conservation of chlorine electron states. We also see that the wave functions for states of different k are orthogonal to each other. Values for k run almost continuously over a cubic region of wave number space, — n/a < k < nia, — nja Brillouin Zone, here cubic, depends upon the crystal structure.) For a macroscopic crystal the Af, are very large, and the change in wave number for unit change in is very tiny. Eq. (24) is an exact solution of Eq. (2-2) however, we will show it for only the simplest approximation, namely, for the assumption that the s,) are sufficiently localized that we can neglect the matrix element Hji = (sj H s,) unless (1) two states in question are the same (/ = j) or... 

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