Perovskite Framework Transformations

Basics

Aristotype - Cubic $Pm\bar3m$ (221)

An aristotype is a high-symmetry structure type that can be viewed as an idealized version of a lower-symmetry structure.

The lower-symmetry structure is called hettotype.

Aristotypes are also known as basic structures and hettotypes as derivative structures.

• Corner-connected $BX_6$ octahedral framework

• Face-sharing $AX_{12}$ cuboctahedra

$ABX_3$ Perovskite Framework

Topological transformation mechanism: Anti-parallel $<111>_\text{Cubic}$ Rotation

Quantitative Analysis of $BX_6$ Tilting

Glazer Notation:

Untilted - $a^0/b^0/c^0$

Tilted in the same direction - $a^+/b^+/c^+$

Tilted in opposite directions - $a^-/b^-/c^-$

Ref:

The classification of tilted octahedra in perovskites - Glazer - 1972 - Acta Crystallographica Section B - Wiley Online Library

(IUCr) Some structures topologically related to cubic perovskite (E21), ReO3 (D09) and Cu3Au (L12)

Cubic $Pm\bar3m$ (221) Aristotype Framework

Putative Composition: $ABX_3$

Unit Cell Parameter: a = 2d, where d is the octahedral B-X bond length

Unit Cell Volume: V(0) = $(2d)^3$

Unit Cell Contents:

Ideal Kernel: $AX_{12}$ cuboctahedra - 2 $\times$ hemi-cuboctahedron ($AX_9$)

Rotating the upper part of a twin-cuboctahedron with a mirror plane generates a hemi-cuboctahedron, in which case the mirror plane is destroyed.

O’Keeffe & Hyde Tilt System Approach

• Rotation axes are specified for the eight octahedra with centers at the corners of the perovskite unit cell, i.e. for a 2 $\times$ 2 $\times$ 2 ‘supercell’ in which eight octahedra surrounding an $AX_{12}$ coordination sphere.
• Rotation is clockwise when viewed in the direction of the axis, e.g. [111] and [$\bar1\bar1\bar1$] represent rotations about the same axis, but in opposite senses.
• The size of the octahedra is maintained constant, the B-X distance being d and the octahedron edge length $\sqrt2d$

Four rotation systems are considered:

1. Trigonal $R\bar3c$ (167)
2. Cubic $Im\bar3$ (204)
3. Tetragonal $I4/mmm$ (123)
4. Orthorhombic $Pnma$ (62)

Each of these different tilt systems gives rise to different $AX_{12}$ coordination spheres, which are considered to be the kernels of the perovskites.

Perovskite Hettotypes

1. Trigonal $R\bar3c$ (167)

Putative Composition: $ABX_3$

Rotation System: $<111>$

Unit Cell Parameter (hexagonal metric):

a = $\sqrt8d\cos\phi$

c = $\sqrt{48}d$

where d is the octahedral edge length

Unit Cell Volume: V($\phi$) = V(0)$\cos^2\phi$

Unit Cell Contents:

Ideal Kernel: $AX_{12}$ triangular bifrustum - 2 $\times$ triangular frustum

Topological Transformation: $Pm\bar3m$ (221) $\rightarrow$ $R\bar3c$ (167)

Anti-parallel rotation - $a^-$

At $\phi=30\degree$ conversion from CCP to HCP complete with

2. Cubic $Im\bar3$ (204)

Putative Composition: $A^\prime A^{\prime\prime}3B_4X{12}$

Rotation System: 4 $\times$ $<111>$ directions

Unit Cell Parameter:

a = $\frac{8\cos\phi+4}3d$, where d is the octahedral edge length

Unit Cell Volume: V($\phi$) = $\frac{(2\cos\phi+1)^3}{27}$V(0)

Unit Cell Contents:

Ideal Kernel: $AX_{12}$ icosahedron

Topological Transformation: $Pm\bar3m$ (221) $\rightarrow$ $Im\bar3$ (204)

Parallel rotation - $a^+$

At $\phi=22.24\degree$ there is perfect $AX_{12}$ icosahedron with

3. Tetragonal $I4/mmm$ (123)

Putative Composition: $A^\prime A^{\prime\prime} A^{\prime\prime\prime}2B_4X{12}$

Rotation System: $<110>$

Unit Cell Parameter:

a = $2d(1+\cos\phi)$

c = $4d\cos\phi$

where d is the octahedral edge length

Unit Cell Volume: V($\phi$) = $\frac{(1+\cos^2\phi)}4\cos\phi$V(0)

Unit Cell Contents:

Ideal Kernel:

• $AX_{12}$ face-sharing square antiprism: 2 $\times$ square antiprism ($A^{\prime\prime}X_8$)
• $AX_{12}$ tetracapped cube: cube ($A^\prime X_8$) + 4 $\times$ tetrahedron

Topological Transformation: $Pm\bar3m$ (221) $\rightarrow$ $I4/mmm$ (123)

Parallel rotation - $a^+/b^+$

At $\phi=19.47\degree$ there is near perfect $AX_{12}$ face-sharing square antiprisms and tetracapped cubes with

4. Orthorhombic $Pnma$ (62)

Putative Composition: $A_4B_4X^\prime_4X^{\prime\prime}_8$

Rotation System: $<0\bar11>$

Unit Cell Parameter:

a = $d[\frac{8(2+\cos^2\phi)}3]^{\frac12}$

b = $d[\frac{48}{1+\sec^2\phi}]^{\frac12}$

c = $d\sqrt8\cos\phi$

where d is the octahedral edge length

Unit Cell Volume: V($\phi$) = $\cos^2\phi$V(0)

Unit Cell Contents:

Ideal Kernel: $AX_{12}$ augmented tetracapped trigonal prism (approx)

$AX_6$ trigonal prism $\rightarrow$ $AX_9$ tricapped trigonal prism $\rightarrow$ $AX_{10}$ tetracapped trigonal prism $\rightarrow$ $AX_{12}$ augmented tetracapped trigonal prism

Topological Transformation: $Pm\bar3m$ (221) $\rightarrow$ $Pnma$ (62)

Anti-parallel rotation - $a^-/b^+$

For $\phi=30\degree$ the $BX_6$ octahedra are perfectly regular and can be described as twinned HCP. Trigonal prisms ($AX_6$) are created between the twin planes.

Anti-perovskite

In cementite $Fe_3C$ only the $CFe_6$ trigonal prisms are observed at the twin planes and the octahedral sites are unoccupied.

Tolerance Factor

A tolerance factor ($t_p$) for perovskite defines the size constraints for A cations to fit exactly inside the cavities in the $BX_6$ framework, i.e. $AX_{12}$ unit. \((r_A+r_X) = \sqrt2(r_B+r_X)\\ t_p = \frac{r_A+r_X}{\sqrt2(r_B+r_X)}\) For a perfect fit $t_p=1.0$ but perovskites exist from ~0.8 - 1.0

Complications include:

• Coordination number and effective bond valence
• Size as function of temperature and pressure
• Molecular dynamics of organic component
• Hydrogen bonding

Ref:

An extended Tolerance Factor approach for organic–inorganic perovskites - Chemical Science (RSC Publishing)

Summary

• The tilting of the octahedra leads to a densification of the structure.
• The patterns of $BX_6$ octahedral tilting in 3D $ABX_3$ perovskites will follow regular hierarchical sequences.
• While in some cases the $BX_6$ octahedra are regular (remain as rigid bodies), there is sometimes distortion required or a small secondary tilt used to maintain the corner-connected topology.
• The same rules apply to the slabs (even single octahedral sheets) of ‘intercalated’ perovskite that are 2D structures.

Key Text Book

Perovskites Modern and Ancient. By Roger H. Mitchell. Thunder Bay, Ontario: Almaz Press , 2002. Price USD 70.00. ISBN 0-9689411-0-9

Perovskites and High Tc Superconductors. Francis S. Galasso. ISBN 978288124391

Crystal Structures: A Working Approach. Helen D. Megaw: 9780721662602