Brillouin zone irreducible

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. 

Understanding how symmetry is used to reduce the number of k points for which calculations are actually performed can help in understanding how long individual calculations will take. But overall convergence is determined by the density of k points in the full Brillouin zone, not just the number of k points in the irreducible Brillouin zone. 

DOS = Density of states BO = Bloch orbital IBZ = Irreducible Brillouin zone BZ = Brillouin zone PZ = Primitive zone COOP = Crystal orbital overlap population CDW = Charge density wave MO = Molecular orbital DFT = Density functional theory HF = Hartree-Fock LAPW = Linear augmented plane wave LMTO = Linear muffin tin orbital LCAO = Linear combination of atomic orbitals. 

As is obvious from the preceding discussion, calculation of the electronic energy levels for a periodic sohd is no more complex than calculation of the electronic energy levels of a molecule. The only difference is that in the case of a periodic solid the calculations have to be carried out for a large number of k values, sampling the first BZ of the solid, that is, from —Ttja to r/a for a ID system. However, in this case, because ofthe general relationship e, (/c) = e, (—fc), calculations can be restricted to the k values Q [Pg.1289]

Thus, the irreducible Brillouin zone in reciprocal space is the equivalent of the asymmetric unit in real space. 

Figure 4-10. Band structure and DOS profiles of Cao Ceo 90 9 along high-symmetty directions of the irreducible Brillouin zone. The valence-band maximum is taken as the energy zero.

The sums over k are performed over all Brillouin zone vectors, but can be reduced to sums on the irreducible Brillouin zone by taking advantage of the space group of the lattice . 

Figure 3.7. Left the symmetry operations of the two-dimensional square lattice the thick straight lines indicate the reflections (labeled ax,ay,ai,as) and the curves with arrows the rotations (labeled C4, C2, C4). Right the Irreducible Brillouin Zone for the two-dimensional square lattice with full symmetry the special points are labeled F, S, A, M, Z, X.

Wang wa used. The total energies were converged to 0.1 mRy/atom. The number of k points was chosen so as to correspond to 120 points in the irreducible wedge of the Brillouin zone of the fee structure, the energy cut-off was 16 Ry. We have tested various values of these parameters and it turned out that the present choice is sufficient to achieve desired uniform accuracy for all structures. For each structure the total energy was minimized with respect to the lattice constant. These interaction parameters correspond to the locally relaxed parameters and are denoted by superscript CW. 

The orbitals <]) j(k r) are Bloch functions labeled by a wave vector k in the first Brillouin zone (BZ), a band index p, and a subscript i indicating the spinor component. The combination of k and p. can be thought of as a label of an irreducible representation of the space group of the crystal. Thequantity n (k)is the occupation function which measures... 

In the case of a perfect crystal the Hamiltonian commutes with the elements of a certain space group and the wave functions therefore transform under the space group operations accorc g to the irreducible representations of the space group. Primarily this means that the wave functions are Bloch functions labeled by a wave vector k in the first Brillouin zone. Under pure translations they transform as follows... 

The wurtzite-structure optical phonons at the / -point of the Brillouin zone belong to the following irreducible representation [32]... 

TABLE 1 Some details of first principles calculations of impurities in Ill-nitrides. Columns list the treatment of the 3d shell if specified (nice = non-linear core correction), size and symmetry (c = cubic, w = wurtzite) of the super cell and the points considered in the irreducible wedge of the Brillouin zone (BZ). 

First, the irreducible part of the Brillouin zone now varies from k = 0 to k = Tr/d = tt/2d. Indeed, doubling the parameter of the unit cell in real space halves the size of the Brillouin zone (or the reciprocal-space unit cell). Second, recall that orbital interactions are additive and that the final MO diagram (or band structure) is just the result of the sum of all the orbital interactions. Within each individual H2 unit the interactions simply correspond to the bonding (a) and antibonding (a ) MOs of each individual H2 unit. There are three types of interactions involving the MOs of different H2 units interactions between all the a orbitals interactions between all the a orbitals and interactions between the a and the a orbitals. Since all the an orbitals are equivalent by translational symmetry, their interaction is described by the Bloch function ... 

Simplification is necessary. We pick certain crucial points in k space, situated on the surface of the irreducible part of the first Brillouin zone, and see how the bands vary between the points. Recall we did a similar thing in the one-dimensional systems by focusing attention on the k points 0 and /d. Remember at the k = 0 point a repeat unit function is taken ++-1-+ in the CO whereas at k = ir/d it is... 

Fig. 2. Band structure and total DOS of bulk V2O5. The energy bands are shown for characteristic paths connecting high symmetry points of the irreducible part of the orthorhombic Brillouin zone (BZ) which is included at the bottom. All energies e(k) are taken with respect to that of the highest occupied state. The DOS is given in states per unit volume and per eV.

Wurtzite ZnO structure with four atoms in the unit cell has a total of 12 phonon modes (one longitudinal acoustic (LA), two transverse acoustic (TA), three longitudinal optical (LO), and six transverse optical (TO) branches). The optical phonons at the r point of the Brillouin zone in their irreducible representation belong to Ai and Ei branches that are both Raman and infrared active, the two nonpolar 2 branches are only Raman active, and the Bi branches are inactive (silent modes). Furthermore, the Ai and Ei modes are each spht into LO and TO components with different frequencies. For the Ai and Ei mode lattice vibrations, the atoms move parallel and perpendicular to the c-axis, respectively. On the other hand, 2 modes are due to the vibration of only the Zn sublattice ( 2-low) or O sublattice ( 2-high). The expected Raman peaks for bulk ZnO are at 101 cm ( 2-low), 380 cm (Ai-TO), 407 cm ( i-TO), 437 cm ( 2-high), and 583 cm ( j-LO). 

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