As discussed in Section 3.2.3, time reversal (i.e. E(k)=E(−k)) allows us to consider just half of the Brillouin zone in describing the electronic structure of any 1D periodic system. The other half can be obtained from the first by application of this type of symmetry. The fraction of the Brillouin zone
Fig. 9.5
Brillouin zone for the orthorhombic reciprocal lattice of Fig. 9.1b.
(a) Brillouin zone for a rectangular lattice. (b) Determination of the irreducible part by looking at the set ofk points equivalent by symmetry to a generalkpoint.
that may generate the whole Brillouin zone by application of the different symmetry operations of the system is known as theirreducible Brillouin zone.
Time reversal is a general property associated with the imposed boundary conditions and this kind of symmetry therefore also applies to any 2D or 3D system. However, the portion of the Brillouin zone that must be considered in actual computations is usually considerably smaller once the symmetry properties of the system have been fully exploited. [4, 5] This can save a lot of computational effort in studying real materials and we must therefore briefly consider this point here.
Let us consider for instance the case of a rectangular lattice witha andb lattice vectors. In this example the reciprocal space is also a rectangular lattice, with lattice vectorsa∗ andb∗, and the Brillouin zone is a rectangle centred on one of the lattice points (see Fig. 9.6). The rectangular lattice possesses a symmetry plane (σh) that is the plane of the lattice,C2axes perpendicular to this plane, symmetry planes perpendicular to thea(σa) andbdirections (σb), and inversion centres (i). In other words, the operations of theD2hsymmetry group. What is the effect of these symmetry operations on a Bloch orbital associated with a general point (ka,kb) of the Brillouin zone?
The Bloch orbital associated with thiskpoint is:
√1 N
n
m=−n+1 n
p=−n+1
exp(i2πmka)exp(i2πpkb)(φ)m,p (9.18)
It can be graphically represented as in Fig. 9.7a. Here, on top of each lat- tice point, the appropriate phase factor (exp(i2πmka)ãexp(i2πpkb)) is shown, wheremandp are the two indices characterising any direct lattice point. Let us first consider theC2axis. Under the effect of this symmetry element the (ka, kb) point of the Brillouin zone moves to the (–ka, –kb) point. The Bloch orbital associated with the (–ka, –kb) point is:
√1 N
n
m=−n+1 n
p=−n+1
exp(−i2πmka)exp(−i2πpkb)(φ)m,p (9.19) and can be represented as shown in Fig. 9.7b. The Bloch orbitals asso- ciated with the two k points are different but completely equivalent. The BO(−ka,−kb)can be obtained from the BO(ka,kb)by application of theC2
Fig. 9.7
Schematic representation of the Bloch orbitals BO(ka,kb)(a) and BO(−ka,−kb)(b) of the rectangular lattice.
Fig. 9.8
Schematic representation of the Bloch orbitals BO(−ka,kb)(a) and BO(ka,−kb)(b) of the rectangular lattice.
symmetry operation. In a similar way, the Bloch orbitals corresponding to the (–ka,kb) and (ka,–kb) points of the Brillouin zone (see Fig. 9.8) can be obtained from the initial one (Fig. 9.7a) by application of the σb and σa symmetry operations, respectively.
In general, when a set of k points of the Brillouin zone can be generated from each other by application of the symmetry operations of the lattice, the Bloch orbitals for all these k points can be obtained from just one of them by using the appropriate symmetry operations. Thus, the Bloch orbitals for all thesekpoints are equivalent and, of course, degenerate in energy.
Consequently, to determine the irreducible part of the Brillouin zone for a given system (i.e. the minimum number of k points for which the crystal orbitals are needed to generate the crystal orbitals associated with the different kpoints of the full Brillouin zone), one just needs to consider how a generalk point of the Brillouin zone is transformed under application of all the symmetry operations of the system. In addition, since the time-reversal operation is, by construction, a general property of the Bloch orbitals, the inversion symmetry should always be used even if it does not occur in the system under considera- tion. As illustrated in Fig. 9.6b, in the case of the rectangular lattice, application of all the symmetry operations to a general(ka,kb)point leads to a set of four equivalentkpoints. Consequently, the area of the irreducible Brillouin zone is one-quarter the area of the full Brillouin zone.
Several points that lie on faces, edges, or vertexes of the Brillouin zone are usually given special labels and the centre is always referred to as . These points are usually high-symmetry points. For instance, in the case of the
Brillouin zones that are a small portion of the Brillouin zone. For instance, the volume of the irreducible Brillouin zones of simple cubic or hexagonal systems with the full symmetry properties of the lattice are481 and241 of the volume of the Brillouin zone.
Crystal orbitals may be classified according to their behaviour with respect to the symmetry operations of the lattice. This may be very useful in generating the band structure (i.e. determination of avoided or allowed band crossings, etc.), studying optical transitions, etc. As discussed in Chapter 6, the crystal orbitals for most of the points of the Brillouin zone do not exhibit the full sym- metry properties of the lattice. Thus, it is important to know which symmetry elements are appropriate for classifying the crystal orbitals at different points of the Brillouin zone.
A crystal orbital is an eigenfunction of a symmetry operator if, under application of this operator, the same crystal orbital multiplied by a constant is generated. As noted above, when twokpoints of the Brillouin zone are related by a symmetry operation of the lattice, the crystal orbitals of the twokpoints may be generated from each other by application of this symmetry operation.
Thus, because of the presence of thekvector in the phase factors, a crystal orbital can only be an eigenfunction of symmetry operators leaving thekpoint unaltered or transforming it into a translationally equivalent one (the crystal orbitals corresponding tok points separated by one reciprocal lattice vector formally differ by a constant exp(i2π)=1).
For instance, consider thesband of a rectangular lattice with one atom per cell. The general expression for a crystal orbital of this system is given in Fig. 9.7a. In the case of thepoint, wherekaandkbare 0, the crystal orbital is simply a collection ofs orbitals with the same sign at every lattice point (Fig. 9.9a). This crystal orbital is, for example, symmetric with respect to the C2axis going through one of the lattice points. For the Y point, where kais 0 andkbis 1/2, the crystal orbital is a collection of sorbitals with the same sign along theadirection but alternating positive and negative signs along the b direction (Fig. 9.9b). This crystal orbital is also symmetric with respect to the sameC2axis. In contrast, it is clear from Fig. 9.7a that a general crystal
(a) (b)
Fig. 9.9
Schematic representation of the Bloch orbitals BO()(a) and BO(Y)of the rectangular lattice.
orbital with kaand/orkb values other than 0 or 1/2 is neither symmetric nor antisymmetric with respect to this axis. This is not surprising since thepoint is left unaltered by theC2rotation and the Y point evolves under the action of theC2axis into a point (ka= 0,kb= –1/2), which is equivalent to Y under a translation,b∗. In contrast, a generalkpoint of the Brillouin zone that does not coincide with the , X, Y, or M points is not left unaltered nor transformed into a translationally equivalent point by theC2rotation operation. The reader may use the general crystal orbital in Fig. 9.7a to verify that the crystal orbitals for the , X, Y, and M points are symmetric or antisymmetric with respect to the symmetry operations of the D2hgroup, those for a general point along the →X,→Y, X→M, and Y→M lines can be classified according to the symmetry operations of theC2vsubgroup and those for all other points of the Brillouin zone according to the symmetry operations of theCssubgroup.