456
16 Polarized Semiconductors
(a)
<0
=0
>0
Free energy
(b)
60
50
40
30
20
10
0
P (10 C/m )
S
2
-
2
LiTaO 3
0 100 200 300 400 500 600 700 800 900
Fig. 16.7 a Schematic plot of the free energy versus spontaneous polarization for a second-order phase transition. α > 0
(α < 0) corresponds to the paraelectric (ferroelectric) phase. b Spontaneous polarization of LiTaO 3 as a function of
temperature exhibiting a second-order phase transition. The dashed line is theory with suitable parameters. Adapted
from [1497]
=
C
T − T C
.
(16.8)
16.3.3.2 First-Order Phase Transition
When the P
6 term is included in (16.2) (γ = 0), a first-order phase transition is modeled. However,
in order to obtain something new, compared to the previous consideration, now β < 0 (and γ > 0) is
necessary. The dependence of the free energy on P is schematically shown in Fig. 16.8a for various
values of α. The condition
∂ F
∂ P
= 0 yields
α P + β P
3
+ γ P
5
= 0 ,
(16.9)
with the solutions P = 0 and
P
2
= −
β
2γ
1 +
1 −
4 α γ
β 2
.
(16.10)
For a certain value of α, i.e. at a certain temperature T = T 1 , the free energy is zero for P = 0 and
also for another value P = P 0 (second curve from the top in Fig. 16.8a). From the condition
1
2
α(T 1 ) P
2
0 +
1
4
β P
4
0 +
1
6
γ P
6
0 = 0 ,
(16.11)
the values for P 0 and α at the transition temperature T = T 1 are given by
P
2
0 = = −
3
4
β
γ
(16.12a)
α(T 1 ) =
3
16
β
2
γ
> 0 .
(16.12b)
The schematic dependence of P at the phase transition temperature T 1 is depicted in Fig. 16.8b.