10.2 Molecular Crystals as Tunable Model System
205
10.2.2 Unified Description of Structural Phase Transitions
We have learned that crystalline phases may undergo a transition into other crystalline
phases by either a structural instability described in Sect. 5.5.3 or the ordering from
a disordered state typically exemplified by the Ising model in Sect. 6.2.1. Indeed,
both mechanisms have been proposed and identified for the emergence of the ferroelectricity. The classification is usually into two groups depending on the primary
mechanism that brings the ferroelectricity. The order-disorder ferroelectrics is evident from its name, while the term “displacive” is for those derived from the lattice
instability. Although the classification implies that the division is concrete because
of a fundamental difference between two mechanisms, many experiments have indicated the presence of intermediate cases, which exhibits characteristics of either both
or none of the two types. Such a situation prompted theorists to build a model that
should describe phase transitions of both types in a unified manner.
A representative of such models is as follows [32, 33]: Suppose interacting particles, the mass of which is m, trapped in a single-particle potential V (x),
V (x) = Ax
4
+ Bx
2
,
(10.2)
where x is the coordinate, and A (> 0) and B are parameters characterizing the potential. Depending on the sign of B, the potential changes its form between the single
minimum form (B > 0) and double minima one with the energy hump V 0 = B
2
/4 A
at x = 0 (B < 0), as shown in Fig. 10.4. The interaction has a bilinear form, −γx i x j .
The exact calculation except for the mean-field treatment on the interaction (i.e.,
−γx i x j ≈ −γxx) indicates the occurrence of a phase transition, even B > 0 [32].
When the thermal energy at the transition temperature T c is well below V 0 for B < 0,
the transition is qualitatively characterized as an order-disorder transition. The different sites where the particle resides correspond to the states of spin. On the other hand,
the transition with B > 0 or k B T c V 0 with B < 0 is of the displacive type accompanying a soft mode, which softens (decreases in frequency) upon approaching the
transition temperature from the high-temperature side. This simple model reproduces
characteristic features of not only such dynamical behaviors but also thermodynamic
characteristics [33]. Although the model is not quantitative but qualitative, it convinces us of the possibility of artificial control of the transition mechanism through
tuning material parameter(s).
It is noteworthy that the single-particle potential is not easy to identify in general
because the dynamics of a “particle” in real systems reflect the entire effect of its environment. The division into the single-particle potential and the relevant interaction
is nontrivial. If we overcome this difficulty, we can potentially tune material parameter(s) through designing molecules. Resultant systems are neat and clean under the
ambient pressure. These properties are significantly distinct from the tuning by doping and/or mixing, which unavoidably suffer from disorders. The availability under
the ambient pressure is crucially preferable for detailed studies, thought the possibility of continuous tuning by compression (change in pressure) might be beneficial
unless we mind experimental difficulties.
205
10.2.2 Unified Description of Structural Phase Transitions
We have learned that crystalline phases may undergo a transition into other crystalline
phases by either a structural instability described in Sect. 5.5.3 or the ordering from
a disordered state typically exemplified by the Ising model in Sect. 6.2.1. Indeed,
both mechanisms have been proposed and identified for the emergence of the ferroelectricity. The classification is usually into two groups depending on the primary
mechanism that brings the ferroelectricity. The order-disorder ferroelectrics is evident from its name, while the term “displacive” is for those derived from the lattice
instability. Although the classification implies that the division is concrete because
of a fundamental difference between two mechanisms, many experiments have indicated the presence of intermediate cases, which exhibits characteristics of either both
or none of the two types. Such a situation prompted theorists to build a model that
should describe phase transitions of both types in a unified manner.
A representative of such models is as follows [32, 33]: Suppose interacting particles, the mass of which is m, trapped in a single-particle potential V (x),
V (x) = Ax
4
+ Bx
2
,
(10.2)
where x is the coordinate, and A (> 0) and B are parameters characterizing the potential. Depending on the sign of B, the potential changes its form between the single
minimum form (B > 0) and double minima one with the energy hump V 0 = B
2
/4 A
at x = 0 (B < 0), as shown in Fig. 10.4. The interaction has a bilinear form, −γx i x j .
The exact calculation except for the mean-field treatment on the interaction (i.e.,
−γx i x j ≈ −γxx) indicates the occurrence of a phase transition, even B > 0 [32].
When the thermal energy at the transition temperature T c is well below V 0 for B < 0,
the transition is qualitatively characterized as an order-disorder transition. The different sites where the particle resides correspond to the states of spin. On the other hand,
the transition with B > 0 or k B T c V 0 with B < 0 is of the displacive type accompanying a soft mode, which softens (decreases in frequency) upon approaching the
transition temperature from the high-temperature side. This simple model reproduces
characteristic features of not only such dynamical behaviors but also thermodynamic
characteristics [33]. Although the model is not quantitative but qualitative, it convinces us of the possibility of artificial control of the transition mechanism through
tuning material parameter(s).
It is noteworthy that the single-particle potential is not easy to identify in general
because the dynamics of a “particle” in real systems reflect the entire effect of its environment. The division into the single-particle potential and the relevant interaction
is nontrivial. If we overcome this difficulty, we can potentially tune material parameter(s) through designing molecules. Resultant systems are neat and clean under the
ambient pressure. These properties are significantly distinct from the tuning by doping and/or mixing, which unavoidably suffer from disorders. The availability under
the ambient pressure is crucially preferable for detailed studies, thought the possibility of continuous tuning by compression (change in pressure) might be beneficial
unless we mind experimental difficulties.