PbTiO3 (perovskite)

PbTiO₃ has a simple perovskite structure. Many compounds of this type undergo one or more structural phase transitions. PbTiO₃ undergoes a cubic-to-tetragonal ferroelectric phase transition at around 770K.
In terms of the strain in a thin film, equilibrium theories of epitaxy predict that, below a critical thickness, the strain coming from the lattice mismatch will be accommodated by film itself. Above the thickness, the strain will be partially relaxed by forming dislocations. If the film undergoes a strutural phase transition from a high-symmetry phase to a low-symmetry phase during cooling from the growth temperature, the epitaxial strain can be relieved by domain formation as suggested by Roitburd and Bruinsma and Zangwill.
The potential surface was initially mapped out using the Linear Augmented PlaneWave (LAPW) method, and the charge density and electronic struccure were analyzed. PbTiO₃ showed a much deeper well when tetragonal strain was included. Thus the tetragonal strain is responsible for the tetragonal ground state in PhTiO3. In BaTiO3, the Ba is quite spherical in the ferroelectric phase, whereas the Pb in PhTiO3 is not very spherical in the ferroelectric phase, and polarization of the Pb helps stabilize the large strain and the tetragonal ground state in PbTiO3.
In PbTiO3, the O 2p states strongly hybridize with the d₀ Ti⁴⁺ cation, which reduces the short-range repulsions thus allows off-center displacements.

Elastic constants of PbTiO₃:
c₁₁=1.433
c₁₂=0.322
c₁₃=0.241
c₃₃=1.316
c₄₄=0.558
c₆₆=0.556
Y=1.34
μ=0.58
ν=0.16
All terms except ν have been divided by a factor of 10¹¹N/m²

Some constants of MgO:
Y=3.105
μ=1.332
ν=0.161
Ref:B.S.Kwak, Physical Review B, Vol49, 14865, 1994

Lattice constants(Å) and TEC of various materials:
PbTiO₃:
tetragonal(RT), a=3.899(bulk); c=4.153(bulk);
cubic (823K), a=3.986,
TEC=12.6 x 10^-6.
Room temperature crystal structure of tetragonal PbTiO₃ was determined by Shirane et al.(1956) with displacements parallel to the polar axis (relative to the Pb ion at the origin):
dz_Ti=17pm
dz_OI=dz_OII=47pm

Note: Oxygen octahedra suffers no distortion in going to ferroelectric phase.
KTaO₃:
cubic(RT),a=3.989;
cubic(823K),a=4.003;
TEC=6.67 x 10^-6.
STO:
cubic(RT),a=3.905;
cubic(823K),a=3.928;
TEC=11.7 x 10^-6.
ferroelectric phase transition: 35-40K
MgO (NaCl structure):
cubic(RT),a=4.213;
cubic(823K),a=4.239;
TEC=14.8 x 10^-6.
BaTiO₃
In the tetragonal phase, the Ti and O ions move relative to Ba at the origin from their cubic position:
Ti: (1/2,1/2,1/2) to (1/2,1/2,1/2+dz_Ti)
O: (1/2,1/2,0), (1/2,0,1/2) and (0,1/2,1/2) to (1/2,1/2,dz_OI),(1/2,0,1/2+dz_OII) and (0,1/2,1/2-dz_OII
dz_Ti=5pm
dz_OI=-9pm
dz_OII=-6pm
(Harada et al.1970)
prototype cubic perovskite: >120°C;
ferro 4mm tetragonal: 120°C>T>5°C;
ferro mm orthorhombic: 5°C>T>-90°C;
ferro 3m trigonal: <-90°C
PbZrO₃:
Antiferroelectricity
Paraelectric-antiferroelectric: 230°C in zero field
but upon application of an external electric field below T_c will induce a transition to a rhombohedral ferroelectric phase.

Ref: Principles and Applications of Ferroelectrics and Related Materials by M.E. Lines and A.M. Glass