Quantum Chemistry of Solids / 12-Space Groups and Crystalline Structures

O0.33159975 0.08159975 0.33680050 1.8925 0.94625 1.602160

O–0.33159975 –0.08159975 –0.33680050 –1.8925 –0.94625 –1.60216

These data correspond to the coordinate system origin shifted by (1/8,–1/8,–1/4) (in the units of primitive translations), in comparison with the origin choice made in [19].

The lanthanum cuprate structure (Fig. 2.13) belongs to the symmorphic space group I4/mmm with the body-centered tetragonal lattice and contains 7 atoms in the primitive unit cell.

The Cu atom occupies Wycko position 1a(000), two La atoms and two oxygen atoms O2 occupy Wycko position 2e ±(0, 0, z) with di erent numerical values of the z-parameter for La and oxygen atoms; the two remaining oxygen atoms occupy Wycko position 2c(0, 1/2, 0; 1/2,0,0). The structure is defined by two tetragonal lattice parameters ( lengths of a1 = a2 primitive translations and c – translation vector of the tetragonal unit cell containing two primitive cells) and two z-parameters defining the positions of the La and O2 atoms. The numerical data for this structure are taken from [20]. The primitive translation vector a3(1/2,1/2,1/2) is given in units of a, a, c.

Primitive vectors

a1 = (3.78730, 0.00000, 0.00000) a2 = (0.00000, 3.78730, 0.00000) a3 = (1.89365, 1.89365, 6.64415) Volume = 95.30130

The La2CuO4 structure (see Fig. 2.13) consists of CuO2–La–O–O–La planes repeated along the z-axis (translation vector c).

The electronic structure calculations show (see Chap. 9): the highest occupied band states in the lathanum cuprate are O1 − 2px, 2py states strongly mixed with Cu − 3dx2−y2 states. This result agrees with the hypothesis that the high-Tc conductivity can be explained by CuO2-plane consideration as is done in most theoretical models.

2.3.4 Orthorhombic Structures: LaMnO3 and YBa2Cu3O7

In Table 2.7 we give the information about two orthorhombic structures: lanthanum manganite LaMnO3 (Fig. 2.14) and yttrium cuprate YBa2Cu3O7 (Fig. 2.15).

38 2 Space Groups and Crystalline Structures

Fig. 2.13. La2CuO4 structure

Among the manganese oxides, LaMnO3 is important as it is the parent system in the family of manganites, that show colossal magneto resistance e ects. In the low-temperature phase this crystal belongs to the nonsymmorphic space group N62 with the simple orthorhombic lattice. There are four formula units (20 atoms) in the primitive unit cell. The structure parameters can be found in the literature in two di erent settings used for the nonsymmorphic space group N62 description. In the standard setting Pnma (accepted in [19]) the primitive orthorhombic lattice vectors a, b, c define the following Wycko positions occupied by atoms: La 4c(x, 1/4, z), Mn 4a( 0,0,0;1/2,0,1/2;0,1/2,0;1/2,1/2,1/2), O1 4c(x, 1/4, z;1/2 − x, −1/4, 1/2 + z;−x, −1/4, −z; x + 1/2, 1/4, 1/2 − z) and O2 8d (x, y, z). The Pbnm setting means the cyclic transposition of basic translation vectors a, b, c to c, a, b and the corresponding changes in the Wycko positions coordinates: c(z, x, 1/4) and

Fig. 2.14. Orthorhombic LaMnO3 structure

Table 2.7. Orthorhombic structures.*

Equivalent sets of Wycko positions for space groups [16]

62 Pnma (ab)(c)(d)

47 Pmmm (abcdefgh)(ijklmnopqrst)(uvwxyz)

d(z, x, y). When using the structure data from the literature one has to take into account the setting chosen. The structure is defined by three orthorhombic lattice parameters and 7 internal parameters: two parameters x, z for La and O1 atoms and 3 parameters for O2 atoms. The equivalent description of LMO structure given in Table 2.7 for the fixed setting means in fact the change of the coordinate system origin - moving the Mn atom position from a(000) to b(001/2). The coordinates of c and d points has to be also changed to (x, 1/4, z + 1/2), d(x, y, z + 1/2). In this case, the structure parameters do not change.

40 2 Space Groups and Crystalline Structures

Fig. 2.15. YBa2Cu3O7 structure

Taken from [20] the structure data are given in the Pnma setting. Primitive vectors

a1 = (5.6991, 0.0000, 0.0000) a2 = (0.0000, 7.7175, 0.0000) a3 = (0.0000, 0.0000, 5.5392) Volume = 243.62954930

The Y–Ba–Cu–O systems are known as high-Tc superconductors (Tc =93 K) when oxygen atoms in the YBa2Cu3O7 system are partly replaced by vacancies or fluorine atoms to synthesize ceramic oxides YBa2Cu3O7−x. The atomic structure of this compound is described by symmorphic space group Pmmm, the primitive unit cell of the orthorhombic lattice consists of one formula unit with the atoms distributed over several planes (see Fig. 2.15). The three copper atoms form two nonequivalent groups: Cu1 atom occupies Wycko position 1a(0,0,0), forming Cu–O chains and two Cu2 atoms – Wycko position 2q(0,0,±z)(see Table 2.7), forming CuO2 planes. Two Ba atoms occupy 2t(1/2,1/2,±z) position. The seven oxygens form four nonequivalent atomic systems: O1 – 1e(0,1/2,0), O2 – 2r(0,1/2,±z), O3 – 2s(1/2,0,±z) and O4 – 2q(0,0,±z). The structure requires 8 parameters for its definition: three orthorhombic lattice parameters and five internal parameters of Ba, Cu2 and O2, O3, O4 atomic positions. All the structural data from [20] are the following.

Primitive vectors

a1 = (3.8227, 0.0000, 0.0000) a2 = (0.0000, 3.8872, 0.0000) a3 = (0.0000, 0.0000, 11.6802) Volume = 173.56308

Fig. 2.17. Wurtzite structure

Fig. 2.18. ScMnO3 structure

The graphite structure belongs to the nonsymmorphic space group P63/mmc, the primitive unit cell of the hexagonal lattice contains 4 carbon atoms occupying two nonequivalent Wycko positions: 2a (0,0,0;0,0,1/2) and 2b(0,0,1/4;0,0,–1/4). The structure is described by two hexagonal lattice parameters a and c. The graphite structure consists of atomic layers separated by a distance larger than the interatomic distance in one layer. Therefore, the one-layer approximation is in some cases used for graphite: only two carbon atoms in the primitive cell of the plane hexagonal lattice are included in the structure.

The numerical values of the graphite structure data from [20] are the following:

44 2 Space Groups and Crystalline Structures

Fig. 2.19. Corundum structure

Table 2.8. Hexagonal structures*

Primitive vectors

a1 = (1.22800000, −2.12695839, 0.00000000) a2 = (1.22800000, 2.12695839, 0.00000000) a3 = (0.00000000, 0.00000000, 6.69600000) Volume = 34.97863049

C0.00000000 0.00000000 0.25000000 0.00000000 0.00000000 1.67400000

C0.00000000 0.00000000 0.75000000 0.00000000 0.00000000 5.02200000

C0.33333333 0.66666667 0.25000000 1.22800000 0.70898613 1.67400000

C0.66666667 0.33333333 0.75000000 1.22800000 –0.70898613 5.02200000

The wurtzite structure of ZnS is the hexagonal analog of the zincblende structure of ZnS (see Sect. 3.2.2) and belongs to the nonsymmorphic space group P63mc with a hexagonal lattice. The primitive unit cell contains two formula units, both Zn and S atoms occupy the same Wycko position 2b(1/3, 2/3, z; 2/3, 1/3, z + 1/2) with the di erent values of internal parameter z. Thus, the wurtzite structure is defined by 4 parameters – two lattice and two internal ones. The numerical values of the structure data from [20] are the following:

Primitive vectors

a1 = (1.91135000, −3.31055531, 0.00000000) a2 = (1.91135000, 3.31055531, 0.00000000) a3 = (0.00000000, 0.00000000, 6.26070000) Volume = 79.23078495

S0.33333333 0.66666667 0.37480000 1.91135000 1.10351844 2.34651036

S0.66666667 0.33333333 –0.12520000 1.91135000 –1.10351844 –0.78383964

RMnO3 rare-earth manganites show a wide variety of physical properties. For R3+ cations with large ionic size RMnO3 oxides crystallize in a perovskite-type structure, with orthorhombic symmetry, as was illustrated in Sect. 2.3.4 by the example of LaMnO3 manganite. For R = Y, Sc, Ho–Lu the perovskite structure becomes metastable and a new hexagonal polytype stabilizes. The hexagonal manganites are an interesting group of compounds because of their unusual combination of electrical and magnetic properties: at low temperatures they show coexistence of ferroelectric and magnetic orderings. ScMnO3 plays a prominent role in this series. It has the highest Neel temperature and the smallest distance between magnetic ions Mn3+ along the hexagonal axis direction. At temperatures below 1200 K ScMnO3 structure belongs to space group P63cm with hexagonal lattice and six formula units (30 atoms) per cell. The six Mn atoms occupy Wycko position 6c(x, 0, z;0, x, z;−x, −x, z;−x, 0, z + 1/2;0, −x, z + 1/2;x, x, z + 1/2). The six Mn atoms are distributed in the z = 0 and z = 1/2 planes. Each Mn atom occupies the center of a triangular bipyramid whose vertices are oxygen atoms (see Fig. 2.18). As a result, each Mn atom is coordinated by five oxygen atoms in a bipyramidal configuration. The same Wycko position 6c is occupied by nonequivalent (with the di erent internal parameters x, z) oxygens O1, O2. Sc1 and O3 atoms occupy Wycko position 2a (0, 0, z;0, 0, z +1/2), Sc2 and O4 atoms – Wycko position 4b(1/3, 2/3, z;2/3, 1/3, z + 1/2;1/3, 2/3, z + 1/2;2/3, 1/3, z). One O3 atom and two O4 atoms are in the equatorial plane of the bipyramid, whereas the O1 and O2 atoms are at the apices. Sc atoms occupy two crystallographic positions Sc1 and Sc2, both of them bonded to seven oxygen atoms. Both RO7 polyhedra can be described as monocapped octahedra. The capping oxygens are O3 for Sc1 and O4 for Sc2. Along the axis z , the structure consists of layers of corner-sharing MnO5 bipyramids separated by layers of edge-sharing RO7 polyhedra. The structure is defined by two hexagonal lattice parameters and 10 internal parameters – two for each type of atoms Mn, O1, O2, and one for each type of atoms Sc1, Sc2, O3 and O4. The numerical values of all the structure parameters can be found, for example in [25]. The description of the ScMnO3 shows it as a very complicated one with many atoms in the primitive cell.

46 2 Space Groups and Crystalline Structures

Nevertheless, the first-principles LCAO calculations of ScMnO3 were recently made (see Chap. 9).

Aluminum oxide (α − Al2O3, corundum) has a large number of technological applications. Due to its hardness, its chemical and mechanical stability at high temperatures and its electronic properties as a widegap insulator it is used for the fabrication of abrasives, as a carrier for thin metal films in heterogeneous catalysis and in optical and electronic devices. α − Al2O3 crystallizes in a rhombohedral structure with the space group R3c. The primitive rhombohedral unit cell consists of two Al2O3 units with experimental lattice parameters a = 5.128 ˚Aand α = 55.333 ˚A. Four Al atoms occupy Wycko position 4c: ±(x, x, x;−x+1/2, −x+1/2, −x+1/2). Six oxygen atoms occupy Wycko position 6e: ±(x, −x + 1/2, 1/4;1/4, x, −x + 1/2;−x + 1/2, 1/4, x). Using the transformation of the rhombohedral lattice vectors with the matrix (2.10) one obtains the hexagonal setting of the α − Al2O3 structure. Its unit cell consists of 3 primitive unit cells (the determinant of the transformation matrix equals 3). The lattice parameters in this hexagonal setting are a = 4.763 ˚A, c = 13.003 ˚A. In hexagonal axes the positions occupied by atoms are written in the form 4c ±(0, 0, z;0, 0, −z+1/2) and 6e ±(x, 0, 1/4;0, x, 1/4;−x, −x, 1/4). The two internal parameters of the corundum structure define the Al and O atoms positions; in terms of the hexagonal lattice, xAl(hex) = 0.35228 and xO(hex) = 0.306. On the rhombohedral lattice this translates to z1 = xAl(hex) = 0.35228 and x1 = 1/4 − xO(hex) = −0.0564. In rhombohedral axes the numerical data for corundum structure are given, for example, in [20]:

Primitive rhombohedral vectors

a1 = (0.256984, 3.621398, 3.621398) a2 = (3.621398, 0.256984, 3.621398) a3 = (3.621398, 3.621398, .256984)

Volume = 84.89212148

O0.05640000 –0.55640000 –0.25000000 –2.90580145 –0.84408855 –1.87494500

O 0.55640000 0.25000000 –0.05640000 0.84408855 1.87494500 2.90580145

O–0.55640000 –0.25000000 0.05640000 –0.84408855 –1.87494500 –2.90580145

O0.25000000 –0.05640000 0.55640000 1.87494500 2.90580145 0.84408855

O–0.25000000 0.05640000 –0.55640000 –1.87494500 –2.90580145 –0.84408855

By consideration of the hexagonal and trigonal structures we conclude the discussion of the structure definitions by space groups and Wycko positions. In the next chapter we consider the symmetry of crystalline orbitals, both canonical and localized.