Multi-step spin-state transitions in organometallics and frustrated antiferromagnetic Ising models

In previous posts, I discussed how "spin-crossover" material is a misnomer because many of these materials do not undergo crossovers but phase transitions due to collective effects. Furthermore, they exhibit rich behaviours, including hysteresis, incomplete transitions, and multiple-step transitions. Ising models can capture some of these effects.

Here, I discuss how an antiferromagnetic Ising model with frustrated interactions can give multi-step transitions. This has been studied previously by Paez-Espejo, Sy and Boukheddaden, and my UQ colleagues Jace Cruddas and Ben Powell. In their case, they start with a lattice "balls and spring" model and derive Ising models with an infinite-range ferromagnetic interaction and short-range antiferromagnetic interactions. They show that when the range of these interactions (and thus the frustration) is increased, more and more steps are observed.

Here, I do something simpler to illustrate some key physics and some subtleties and cautions.

fcc lattice

Consider the antiferromagnetic Ising model on the face-centred-cubic lattice in a magnetic field. 

[Historical trivia: the model was studied by William Shockley back in 1938, in the context of understanding alloys of gold and copper.]

The picture below shows a tetrahedron of four nearest neighbours in the fcc lattice.

Even with just nearest-neighbour interactions, the lattice is frustrated. On a tetrahedron, you cannot satisfy all six AFM interactions. Four bonds are satisfied, and two are unsatisfied.

The phase diagram of the model was studied using Monte Carlo by Kammerer et al. in 1996. It is shown above as a function of temperature and field. All the transition lines are (weakly) first-order.

The AB phase has AFM order within the [100] planes. It has an equal number of up and down spins.

The A3B phase has alternating FM and AFM order between neighbouring planes. Thus, 3/4 of the spins have the same direction as the magnetic field.

The stability of these ordered states is subtle. At zero temperature, both the AB and A3B states are massively degenerate. For a system of 4 x L^3 spins, there are 3 x 2^2L AB states, and 6 x 2^L   A3B states. At finite temperature, the system exhibits “order by disorder”.

On the phase diagram, I have shown three straight lines (blue, red, and dashed-black) representing a temperature sweep for three different spin-crossover systems. The "field" is given by h=1/2(Delta H - T Delta S). In the lower panel, I have shown the temperature dependence of the High Spin (HS) population for the three different systems. For clarity, I have not shown the effects of the hysteresis associated with the first-order transitions.

If Delta H is smaller than the values shown in the figure, then at low temperatures, the spin-crossover system will never reach the complete low-spin state.

Main points.

Multiple steps are possible even in a simple model. This is because frustration stabilises new phases in a magnetic field. Similar phenomena occur in other frustrated models, such as the triangular lattice, the J1-J2 model on a chain or a square lattice.

The number of steps may change depending on Delta S. This is because a temperature sweep traverses the field-temperature phase diagram asymmetrically.

Caution.

Fluctuations matter.

The mean-field theory phase diagram was studied by Beath and Ryan. Their phase diagram is below. Clearly, there are significant qualitative differences, particularly in the stability of the A3B phase.

The transition temperature at zero field is 3.5 J, compared to the value of 1.4J from Monte Carlo.

Monte Carlo simulations may be fraught.

Because of the many competing ordered states associated with frustration, Kammerer et al. note that “in a Monte Carlo simulation one needs unusually large systems in order observe the correct asymptotic behaviour, and that the effect gets worse with decreasing temperature because of the proximity of the phase transition to the less ordered phase at T=0”. 

Open questions.

The example above hints at what the essential physics may be how frustrated Ising models may capture it. However, to definitively establish the connection with real materials, several issues need to be resolved.

1. Show definitively how elastic interactions can produce the necessary Ising interactions. In particular, derive a formula for the interactions in terms of elastic properties of the high-spin and low-spin states. How do their structural differences, and the associated bond stretches or compressions, affect the elastic energy? What is the magnitude, range, and direction of the interactions?

[n.b. Different authors have different expressions for the Ising interactions for a range of toy models, using a range of approximations. It also needs to be done for a general atomic "force field".]

2. For specific materials, calculate the Ising interactions from a DFT-based method. Then show that the relevant Ising model does produce the steps and hysteresis observed experimentally.