Crystal structure and symmetry
Symmetry
The prototype for the crystal structure of the II-IV-N_{2} compounds is ^-NaFeO_{2}, which is actually a I-III-VI2 compound. This crystal structure can be viewed as a 2 x %/3 superlattice of wurtzite along ortho-hexagonal axes, as illustrated in Fig. 15.1, showing a projection on the c-plane.
It has sixteen atoms per unit cell. The b/a and c/a can both adjust as well as the internal positions x, y, z of each of the four types of atom, the group II, group IV, and two inequivalent N positions, Njj being on top of the group II and N/y being on top of the group IV element. We choose the a axis to be 2a_{w} and b « V3a_{w} unlike the commonly found choice in the crystallography literature, which interchanges a and b from ours.
The space group is Pbn2_{1} (space group No. 33, or C'^_{v}), meaning that there is a two-fold screw axis along the z direction with translation 1 c, a diagonal
Fig. 15.1. Left: Projection of crystal structure of Zn-IV-N_{2} compounds on c- plane, indicating symmetry elements. Large open circles indicate cations in bottom plane, (light grey group-II, darker grey group IV), small open circles in top plane, filled circles are N atoms above them, as indicated. The symmetry elements are indicated and chosen so that the 2i axes passes through the origin. From (62). Right: 3D view of the crystal structure: small light grey spheres are N, large spheres are cations, dark grey: group-II, lightest grey: group-IV. From (10).
glide plane n perpendicular to b with translations 2(a + c), and an axial glide plane perpendicular to the a-axis with translation 1 b. These symmetry elements are indicated in Fig. 5.1. We note that the space-group denomination in the International Tables of Crystallography (ITC) is Pna2_{i}. Our choice has the a and b axes interchanged. Most of the experimental papers in the literature adopt the ITC choice of axes, but we prefer the interchange of a and b axes because it makes the relation to wurtzite easier to see. This means that in our case, a = 2a_{w }is the longest lattice constant, b « V3a_{w} is the next longest, and c = c_{w}, where the subscript indicates the wurtzite structure. In this setting the origin is chosen to fall on the two-fold screw axis. This simplifies the non-primitive translation vector of the screw axis to be just c/2.
The point group is C_{2v}. The character table for this group is given in Table 15.1. The meaning of this table is easy to understand in this case, even to readers not familiar with group theory. The characters of the irreducible representations (irrep for short) listed as rows, simply describe whether the functions corresponding to this irrep are odd or even under the corresponding symmetry operations in the column headings. The irreducible representations ai, bi, and b_{2} correspond to basis functions transforming like z, x, and y, with x along a, y along b, and z along the c-axes. The a_{2} irreducible representation is even under the two-fold rotation, but odd under both mirror-planes, and corresponds to an xy basis function. These character tables are useful for classifying the vibrational modes at Г as well as the electronic states.
Table 15.1 Character table for the group C_{2v}: first column, basis functions; second column, irreducible representations; first row, classes.
E |
C2z |
^{a}y |
|||
z |
ai |
1 |
1 |
1 |
1 |
x |
bi |
1 |
-1 |
-1 |
1 |
y |
b2 |
1 |
-1 |
1 |
-1 |
a2 |
1 |
1 |
-1 |
-1 |