Temperature-induced phase transitions between ordered phases

Landau symmetry breaking theory describes phase transitions from disordered states (high symmetry) to ordered states (lower symmetry) by a continuous emergence of a symmetry-breaking order parameter. While beyond this scope, there exists a large number of phase transitions where both sides of the transitions are sort of "ordered". No doubt they are interesting but subtle phenomena. How to understand them?

A general notion to understand this class of phenomena can be found in a description of structural phase transition from A (low temperature) to B (high temperature) in Kittel Solid State Physics P467, "The stable structure A at absolute temperature generally has the lowest accessible internal energy of all the possible structures. Some other structure B may have a softer or lower frequency phonon spectrum than A. As the temperature is increased the phonons in B will be more highly excited than the phonons in A. Because entropy increases with the occupancy, the entropy of B will become higher than the entropy of A as the temperature is increased." Thus both terms in the free energy F = U - TS play important roles in a competing manner. On each side of the phase transition, there is an ordered state, the lower temperature is relatively favoured by energy while the higher temperature one is relatively favoured by entropy. Though this statement seems general to all the temperature-induced phase transitions such as solid-to-liquid, liquid-to-gas transition, here we focus on transitions between different ordered states with energies not far from each other.

Here I will try to note the examples I have collected for this kind of phase transitions (in solids) without invoking the details of U and S.


Structural Phase Transitions

Here are many interesting examples of structural phase transitions (in solid phase) induced by temperature (it is surprisingly commonly existing in nature!).

The elemental metal Manganese seems to possess a variety of crystal structures as a function of temperature, as seen below in Fig.1. One way to look at the very complicated structure at low temperatures is in them, each Mn atom has a huge number of neighbours, as to gain cohesive energy. The symmetry of each phase is very different and there are 54 sites in the unit cell of alpha manganese, the low temperature stable phase!

 

Fig1. Alpha, beta, gamma, delta Manganese respectively, which are the stable phase for elemental Mn at < 1100 K, 1100 ~ 1300 K, 1300 ~ 1411 K and 1411 K ~ 1519 K, respectively.
Ref: http://som.web.cmu.edu/structures/S025-alpha-Mn.html

The example in Fig. 2 shows the structural phase transition in a Heusler alloy Mn2NiSn, which changes the symmetry of the crystal, for example from cubic to hexagonal.



Fig.2 Structural phase transition in Mn2NiSn
Ref: http://www.spring8.or.jp/en/news_publications/press_release/2010/100427/



Various Magnetic Orders

Transition between different ordered states can be found in the magnetic degrees of freedom of crystals as well. This does not happen in simple ferromagnets or antiferromagnets, where the ordered state is unique; however, this is commonly observed in frustrated magnetic systems.

In the following phase diagram of TbFeO3, there are at least three different ordered phases labeled LT (low temperature), IT (intermediate temperature), HT (high temperature) phases with slightly different antiferromagnetic order of Fe and Tb, indicating 1) there are competing magnetic interactions in the system therefore 2) there are multiple ordered phases close in energy. More interestingly, at finite field, the temperature could induce 4 phases, LT', IC (incommensurate), IT and HT phases. The neutron scattering pattern of IC phase is shown below, and this phase is anticipated as a periodic array of domain walls, which are stabilised by Yukawa-like interactions. Though clearly the domain wall state is not a ground state at zero temperature, it is the lowest free energy state at finite temperature thus is a well-defined thermodynamical state. Similar is true for Skyrmion lattices found in chiral magnets.

Fig.3 (a) Temperature-field phase diagram of TbFeO3, which has a perovskite-related structure. It shows three different phases (LT, IT, HT) when changing the temperature.  In field, there is an IC (incommensurate phase) whose neutron scattering pattern is shown in (b).  It is conceived as a periodic ordering of domain walls. (Ref: Nature Materials)



Spin Liquids Stabilised by Finite Temperature

Spin liquid, a disordered state with strong antiferromagnetic correlation is long-sought in condensed matter physics. Though finding a material with spin liquid as the zero temperature ground state is tough, spin liquid might exist as a finite temperature state in some frustrated magnets because they gain a lot of entropy compared to ordered states.

Here are two examples Yb2Pt2Pb and Cs2CuCl4, both have prototype frustrated magnetic lattice, the former with a Shastry-Sutherland lattice and the latter with a distorted triangular lattice. Both of them show antiferromagnetic order at lowest temperatures, while a spin liquid state is realized at finite temperature. (Spin liquid phase boundary determined by a bump in the susceptibility as a function of temperature.)

Here the entropy helps a lot! And this seems common because spin liquid systems are intrinsically with many competing states close to energy!

 
Fig.4  (a) Phase diagram of Yb2Pt2Pb with an underlying Shastry-Sutherland lattice. The dome shaped phase is an AFM ordering while between the solid circles and empty circles is the spin liquid state. (b)(c) phase diagram of Cs2CuCl4.