Brillouin zone cubic lattice

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... 

A theoretical interpretation relating the valence electron concentration and the structure was put forward by H. Jones. If we start from copper and add more and more zinc, the valence electron concentration increases. The added electrons have to occupy higher energy levels, i.e. the energy of the Fermi limit is raised and comes closer to the limits of the first Brillouin zone. This is approached at about VEC = 1.36. Higher values of the VEC require the occupation of antibonding states now the body-centered cubic lattice becomes more favorable as it allows a higher VEC within the first Brillouin zone, up to approximately VEC = 1.48. 

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... 

In particular, let us consider the band structure along where kr = (0,0,0) and kx = (2n/a)(l, 0,0) with a the edge length of the face-central cubic unit celL (Note that the X point for fee is In/a not nfa like for simple cubic.) In this direction the two lowest free-electron bands correspond to Ek = (H2/2m)k2 and k+I = (H2/2m)(k + g)2 respectively. The term g is the reciprocal lattice vector (2n/a)(2,0,0) that folds-back5 the free-electron states into the Brillouin zone along so that Ek and k+l... 

The NFE behaviour has been observed experimentally in studies of the Fermi surface, the surface of constant energy, F, in space which separates filled states from empty states at the absolute zero of temperature. It is found that the Fermi surface of aluminium is indeed very close to that of a spherical free-electron Fermi surface that has been folded back into the Brillouin zone in a manner not too dissimilar to that discussed earlier for the simple cubic lattice. Moreover, just as illustrated in Fig. 5.7 for the latter case, aluminium is found to have a large second-zone pocket of holes but smaller third- and fourth-zone pockets of electrons. This accounts very beautifully for the fact that aluminium has a positive Hall coefficient rather than the negative value expected for a gas of negatively charged free carriers (see, for example, Kittel (1986)). 

Figure 16.12. Brillouin zones, with symmetry points marked, of (a) the primitive cubic Bravais lattice and (b) the cubic close-packed or fee Bravais lattice.

Because of the translational symmetry of the reciprocal lattice (Section 16.3) and the definition of the Brillouin zone (BZ), the BZ faces occur in pairs separated by a reciprocal lattice vector. For example, the cubic faces of the first BZ of the simple cubic (sc) lattice occur in pairs separated by the reciprocal lattice vectors b (2rc/a)[[l 0 0]] (see eq. (16.3.27)). In general, for every k vector that terminates on a BZ face there exists an equivalent vector k (Figure 17.1) such that... 

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...

The symmetry of the lattice will impose distinct shapes on the Brillouin zones (which by definition are the Wigner-Seitz cells of the reciprocal lattice) for each type of symmetry. Figure 8.11 shows the first Brillouin zone for a face-centered cubic structure. 

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). 

For a simple cubic metal the first zone is given by hkl = 100 (and equivalent values 010, 001, 100, 010 and 001). In the reciprocal lattice 100 is the point on the (reciprocal) x axis at a distance 1 a from the origin. The Brillouin zone is now produced as the locus of the general condition (Fig. 26a, drawn there for 110) ... 

The Brillouin Zone and symmetry points for the zincblendc (diamond, or face-ccntcred cubic) lattice. The view at the right is along a [110] axis that at the left has been tilted. 

We saw in detail in Section 16-D, and in Fig. 16-6 in particular, how nearly-free-electron bands are constructed. The diamond structure has the translational symmetry of the face-centered cubic structure, so the wave number lattice, the Brillouin Zone, and therefore the nearly-free-electron bands for the diamond structure are identical to the face-centered cubic nearly-free-electron bands and are those shown in Fig. 3-8,c. The bands that are of concern now are redrawn in Fig. 18-l,b. The lowest energy at X, relative to the lowest energy at F, is (fi /2m) x InjaY as shown clearly it is to be identified with the lower level... 

Fig. 2 Dispersion relation of the orbital interaction in (7) represented in the Brillouin zone for a simple cubic lattice...

Under the imposed periodic boundary conditions, the wave component exp(zkr) again has to satisfy exp(z yfy) = 1 (/ = 1,2,3), with the same implications for the possible values of k as above, that is, Eq. (4.75) for a cubic lattice. Furthermore, Eqs (4.82) and (4.83) imply that for any reciprocal lattice vector G the wavevectors k and k - - G are equivalent. This implies that all different k vectors can be mapped into a single unit cell, for example the first Brillouin zone, of the reciprocal lattice. In particular, for a one-dimensional lattice, they can be mapped into the range —(jt/a)... (ji/a. The different values of k are then k = (27r/rz)(n/2V ), where N = L/a (L is the length that defines the periodic boundary conditions) is chosen even and where the integer n takes the N different values n = - N/2-, - N/2 + 1,..., N/2 - 1. 

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. 

Fig. 1.6 Brillouin zone for face-centered cubic lattices, with high symmetry points labelled. (After ref. [6])...

It now remains to construct a mesh in the irreducible wedge of the Brillouin zone and define the corresponding tetrahedra. This is done in the subroutine TGEN, and we shall briefly illustrate the procedure with the example of a simple cubic lattice. 

First Brillouin Zone of the Body-Centered Cubic Lattice. The Figure is a Regular Rhombic Dodecahedron... 

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