Crystal Structures

Pna2₁ structure, and Pbn2₁ nomenclature.

The observed crystal structure for most of the II-IV-N₂ compounds is known as the β-NaFeO₂ structure. It is orthorhombic, and the standard space group is usually denoted Pna2₁. However, significant confusion occurs in the literature because of the ambiguity of the assignment of lattice constants as a, b, and c, and this same crystal structure may be referred to as Pbn2₁. We explain below the relation between the two nomenclatures.

To obtain a systematic convention it is useful to relate the orthorhombic structure to the underlying wurtzite structure. This relation is shown in Fig. 1. The vectors a₁, a₂ define the hexagonal wurtzite unit cell. The vectors a, b, are the orthorhombic structure lattice vectors. The third lattice vector c is normal to the page for both the orthorhombic and the wurtzite structures.

The ordering of the cations lowers the symmetry from wurtzite to orthorhombic (Fig. 1). Here, in Fig. 1, only the cations are shown. The nitrogen atoms sit on top of the cations along the c axis. The red and green spheres denote the two types of cations, A and B, belonging to the group II and group IV valences, respectively. The solid colored spheres are located in the bottom plane along c. The shaded spheres are shifted by c/2 normal to the page.

The relation between the orthorhombic Pbn2₁ and the wurtzite structure, if distortions from the wurtzite atom positions are neglected for the orthorhombic ordering, is a=2a₁, b= a₁+2a₂. These orthorhombic axes are the so-called ortho-hexagonal axes, which one would use to describe the hexagonal structure with a Cartesian coordinate system x||a, y||b, z||c. Here we have b= √ 3 a_w=1.732 a_w, a=2 a_w, c=c_w= √(8/3) a_w. In the orthorhombic structure, the b/a and c/a ratios will in general differ slightly from √ 3 a_w and √(8/3) a_w, and the angle between a₁ and a₂ will differ slightly from 120 degrees. Note that in this convention a>b>c and thus it is easy to check whether the a,b,c designations one finds in the literature conform to this choice or not, and eventually, if necessary, make the appropriate interchanges.

Now, how do we label the space group?

The symmetry elements are also indicated in Fig. 1. The black ellipses with little end dashes indicate the two-fold screw axes along the z (or c) axis, the vertical mirror planes are orthogonal to x=a and are labeled here as σ_x or b, and the horizontal mirror planes are labeled σ_y or n. In the notation of the International Tables of X-Ray Crystallography, a mirror plane b means a glide plane with a non-primitive translation by b/2. In this case, the glide plane is the bc plane and the glide is by b/2 along the y direction. As an example, the glide by b/2 would shift the red shaded atom in the lower upward pointing triangle to the one in the middle upper triangle. The mirror plane σ_y is of a different kind: it has a shift in both a and c directions by half a lattice vector each. Thus, the glide translates the filled red atom on the left hand corner to the shaded red atom in the center triangle. This translation involves a mirroring plus a horizontal shift by a/2 and a shift by c/2 out of the plane. The two-fold screw axis relates, for example, the red filled atom on the left hand corner to the red shaded one in the center of the lower left triangle: we rotate by 180 degrees and shift by c/2.

Now that we see the symmetry operations, we can explain the space group notation. For this choice of axes the correct notation is Pbn2_1. The "P" signifies that we have a primitive lattice, rather than a side-centered, body-centered or face-centered Bravais lattice. The "b" means that perpendicular to the a-axis there is a b-type glide mirror plane (single glide with glide along b in the bc plane). The "n" signifies there is a n-type glide mirror plane (which has two glides) perpendicular to b, and the 2₁ means a two-fold screw axis along c.

The space group Pna2₁ would be the correct notation if we had interchanged a and b. One often finds this notation in the literature. The n-type glide mirror plane would then be perpendicular to the first axis or the a-axis. The single glide type glide plane would then be perpendicular to b and would thus be an ac glide plane with a glide along a by a/2; hence "a" instead of "b" in the notation.

Again, although the correct notation with this choice of axes is, strictly speaking, Pbn2₁, one still often finds the notation Pna2₁ because this notation is the generic space group name listed in the International Tables of X-ray Crystallography, corresponding to space group No. 33. Its Schoenflies notation is C_2v⁹. Its page in the Tables of X-ray Crystallography can be found here.

The atoms in the II-IV-N₂ compounds are in Wyckoff positions 4a, which means that for each type of atom there are 4 symmetry-equivalent positions: x,y,z; -x,-y, z+1/2; -x+1/2, y+1/2, z; and x+1/2, -y+1/2, z+1/2. Thus, to specify the crystal structure fully, we need to provide the lattice constants a, b, c, and the positions x,y,z of the group II atom, the group IV atom, and the two nitrogen atoms N_II and N_IV, which sit on top of the group II and the group IV atoms, respectively.

For the ideal Pbn2₁ structure, the x,y,z are given by

Fig.1 : Relationship between the orthorhombic and wurtzite structures, and symmetry elements.

Monoclinic and wurtzite-disordered.

Although the above described structure is the only ordered structure thus far reported experimentally for any of the II-IV-N₂ compounds, one also sometimes finds mention of the wurtzite structure or a monoclinic structure. This situation can occur when one cannot distinguish the A and B atoms or sites, in principle either because the atoms are too close in atomic scattering factor, as are for example Zn and Ge in X-ray or electron diffraction but not in neutron diffraction, or because the group II and group IV atoms are intermixed on the A and B sites. Although one then strictly has, on average, a wurtzite structure, one can sometimes see the deviation in bond angles from 120 degrees; therefore, for example, ZnGeN₂ was at first indexed as a monoclinic material, in which a₁ and a₂ have slightly different lengths and make an angle slightly different from 120 degrees. This situation occurred not because of disorder on the cation sublattice but because the ordering of the Zn and Ge atoms could not be detected, while the bond angle distortions were resolved.

Pmc2₁ and 1D-disordered structures.

More recently [1], a different ordered structure was proposed, which corresponds to the space group Pmc2₁ (space group No. 26, Schoenflies notation C_2v²). Its page in the Tables of X-Ray Crystallography is here . This structure is shown in Fig. 2. Keeping the x-axis the same as before for Pbn2₁, and the b axis along the ortho-hexagonal y-axis, we now have b>a because a=a_w. The vertical mirror plane is now an ordinary mirror plane and is therefore denoted "m". The horizontal mirror plane has a shift by c/2 and is perpendicular to b, and the two fold screw axes are along c as before. So, this notation corresponds to the choice of axes as shown below.

While this structure was found to have almost the same energy as the Pbn2₁ phase in ZnSnN₂, it has thus far not been identified experimentally for any of the heterovalent ternary nitrides.

However, it was realized [2] that the Pbn2₁ and Pmc2₁ structures can be viewed simply as different stackings of rows of alternating A and B atoms. Therefore, one can also envision random mixings of these two stackings, as well as polytypes. The potential for random stackings provides an alternative model for disorder in these materials. Note that in both of these structures each nitrogen atom is bonded to exactly two group II and two group IV cations, and thus charge neutrality is conserved (the octet rule is preserved) in each nearest-neighbor tetrahedron. [2]

Fig. 2 : Pmc2₁ structure and symmetry elements. From [3].

Chalcopyrite

The structures described above are all based on the hexagonal wurtzite lattice. These structures are composed of two interpenetrating hexagonally close-packed lattices: one for the cations, and one for the anions. These are the commonly found structures of all group-III nitrides, with the exception of c-BN. One might also envision structures based on a cubic close packing, or fcc lattice. We then have the zincblende structure for III-V materials, and the chalcopyrite structure for the II-IV-V₂ materials. Some theoretical papers consider the relative stability of these cubic close-packed-based structures to the hexagonal ones, but the cubic-based structures were thus far found to have higher energy in the case of the nitrides, and have not been observed. This situation does not exclude the possibility that the cubic-based structures could be stabilized by hetero-epitaxial growth on cubic substrates, as was indeed found to be possible for some III-N materials.

[1] L. Lahourcade, N. C. Coronel, K. T. Delaney, S. K. Shukla, N. A. Spaldin, and H. A. Atwater, Adv. Mater. 25, 2562 (2013).

[2] P. C. Quayle, E. W. Blanton, A. Punya, G. T. Junno, K. He, L. Han, H. Zhao, J. Shan,W. R. L. Lambrecht, and K. Kash, Phys. Rev. B91, 205207 (2015).

[3] M. Hagemann, C. Bhandari and W. R. L. Lambrecht, Solid State Commun. 233, 46-49 (2016).