How to choose the values of J and spin parameters in a heterogeneous spin system?

Question

In this work, graphene-based systems that are described by mixed spin-3/2 and spin-5/2 are studied using the ising-model. A diagram of the structure is shown bellow:

The Hamiltonian used is:

\begin{equation} \tag{1} {\small {H_I} = - J\sum\limits_{\left\langle {i,j} \right\rangle } {{S_i}{\sigma _j}} - {J_\sigma }\sum\limits_{\left\langle {i,j} \right\rangle } {{\sigma _i}{\sigma _j}} - {J_S}\sum\limits_{\left\langle {i,j} \right\rangle } {{S_i}{S_j}} - {K_v}\left( {\sum\limits_i {S_i^2} + \sum\limits_j {\sigma _j^2} } \right). } \end{equation}

Here $\left\langle {i,j} \right\rangle$ refers to the sum over the nearest neighbors pairs, $\left[ i,j \right]$ means sum over the next-nearest neighbors pairs, $J$, $J_\sigma$ and $J_S$ are the exchange interaction constants between sites $\sigma−S$, $\sigma−\sigma$ and $S−S$, respectively (see diagram above), and $K_v$ is the crystal field anisotropy constant. The spins moments can take values $\sigma = ±3/2,±1/2$ and $S=±5/2,±3/2,±1/2$.

How are $J$, $J_\sigma$, $J_S$, $\sigma$ and $S$ chosen?

Answer 1

The problem of defining the parameters comes fundamentally from using the Ising model. The Ising model is extremely simplistic, and is the most basic model of a phase transition. Magnetic materials are generally much better modelled using a 3D Heisenberg spin model (where spins can point in any direction), but the lattice of spins can be 1D,2D or 3D. The advantages over the Ising model are that spin waves are allowed, the Curie temperature is much more accurate when using DFT calculated parameters, and Ms(T) is not overly flat for low temperatures (this requires quantum corrections but not as brutal as spin-up and spin-down in the Ising model). In this case S is a unit vector (representing the local quantisation axis of the spin) and factored into the J coefficients automatically (this is one of the problems with defining J with S as different lengths since different moments give different coupling but this is not necessarily directly proportional to |S||\sigma|). One can approximate the J coefficients by comparing with experimental data, but in general you need an electronic structure calculation (such as DFT or sKKR) to determine nearest and next nearest exchange constants.