Reduced first Brillouin zone

Here uf = u exp(277ig r) is, like w, periodic with the period of the lattice, and k = k - 27rg is a reduced wave vector. Repeating this as necessary, one may reduce k to a vector in the first Brillouin zone. In this reduced zone scheme, each wave function is written as a periodic function multiplied by elkr with k a vector in the first zone the periodic function has to be indexed, say ujk(r), to distinguish different families of wave functions as well as the k value. The index j could correspond to the atomic orbital if a tight-binding scheme is used to describe the crystal wave functions. 

The atoms of a crystal vibrate around their equilibrium position at finite temperatures. There are lattice waves propagating with certain wavelengths and frequencies through the crystal [7], The characteristic wave vector q can be reduced to the first Brillouin zone of the reciprocal lattice, 0 < q <7t/a, when a is the lattice constant. 

The system (8) becomes closed if we assume that only k=0, k=3 or k=0, k=2, k=4 modes are induced. In other words the equation system becomes closed for modes in the center and in the middle of the first Brillouin zone and also for modes in the center and in the distance of 1/3 from the first Brillouin zone. In general, such reduction of the system is caused by the existence of Abelian subgroups in the translation group along the corresponding direction and, perhaps, by bush-modes describing in [14, 15]. So, in the case of N=6, k=0, k=3 the reduced system of equations is the following ... 

Therefore k is generally restricted (hence referred to as the reduced wave vector) to a region of k space such that no two points in this region are separated by any vector K. This is a unit cell in reciprocal space, and is referred to as the first Brillouin zone. 

The requirement that — 1 < cos ka< 1 used to determine the allowed bands means that the value for k for each band is not unique in the way implied by the extended band scheme shown in Fig. 4.4(b). Instead, the band structure is more properly represented in a reduced zone scheme with —n/a < k < nja, which corresponds to the first Brillouin zone, as shown in Fig. 4.4(c). 

The d-like band is partially filled up to 5/6. The -ir-like band is occupied up to 1/3. The Peierls instability in the quasi one-dimensional ir-like band causes a superstructure with the period of 3c. This reduces the first Brillouin zone to 1/3... 

Thus these points in a small but well-defined region of k space include all possible irreducible representations of the translation group the vectors of the reciprocal lattice transform points in the Brillouin zone into equivalent points. The Brillouin zone therefore contains the whole symmetry of the lattice, each point corresponding to one irreducible representation, and no two points being related by a primitive translation. The smallest value of k ki, k2, kz) belonging to the rep is called the reduced wave-vector. The set oi reduced wavevectors is called the first Brillouin zone. 

Firstly, as based on the result (3.65) the energetic analysis can be reduced at the behavior of E-k dependence at the level of the first Brillouin zone, see Figure 3.17 (Putz, 2006). 

Regrouping the intervals allowed as solutions of Eq. (3.99) fi-om the Figure 3.23 in terms of energy quantification (3.100), the representation of Figure 3.24 results (Putz, 2006) in its extended version (from the left side) it presents the discontinuities at the wave vectors values k = S = nK I a and respectively the energetic bands in the reduced version at the first Brillouin zone (from the right side). 

FIGURE 3.31 Phenomenological representation for the velocities sign (left) and the electronic motion (right) in the energetic bands reduced at the first Brillouin zone after (Further Readings on Quantum Solid 1936-1967 Putz, 2006). 

We have seen that one-dimensional systems have degenerate COs for equal values of k and so, if we wish to portray only the unique energies of the system, we need consider only the range from 0 to n/a. The analogous situation in two or three dimensions is that symmetrically equivalent positions k give degenerate COs. Hence we need consider only a symmetrically unique portion of the FBZ—the reduced first Brillouin zone (RFBZ). Since the FBZ is symmetric for the reflections mentioned above, the RFBZ is the quadrant shown in Fig. 15-27c. 

First Brillouin zone The primitive cell in the reciprocal lattice containing the point k=0. All physically different electronic states in a crystal can be characterized by a wave vector reduced to the first Brillouin zone. 

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