Symmetry Breaking in the Perovskite SrTiO3

T/

T

c

0.949

Solid-solid phase transitions result when at least one normal mode of the system softens (i.e. becomes weaker) as the temperature decreases. Once the temperature is decreased below a critical temperature

T

c

, the higher symmetry structure is no longer stable and the lattice distorts to form a new structure with lower symmetry. This results in a second-order phase transition, since the entropy and free-energy vary continuously with temperature while the heat capacity

C

p

=T

(∂S/∂T)

p

is discontinuous at

T

c

.

In the example shown here, the oxygens located in the face-centers of the

SrTiO

3

perovskite unit cell undergo a slight rotation about the icosohedral axis once

T

is less than

T

c

. This distortion is related to an order parameter

η

. We can write the free energy

G

in terms of this order parameter as

G(η)=

G

o

+α(T)

2

η

+γ

4

η

, with

α=a(T-

T

c

)

as a simple approximation. When

T>

T

c

, the system has a single free energy minimum at

η=0

, corresponding to the undistorted, high-symmetry case. When

T<

T

c

the system has two equivalent minima corresponding to the two possible ways the equatorial O's can move. The upper and lower oxygen planes move in opposite directions to preserve the center of mass of the cell.

In the graphics, the rotation angle is exaggerated in order to visualize the distortion.