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

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

The Hamiltonian used is:

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

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?

• It depends what behavior you are interested in. Are you comparing to an experimental system? Another numerical model? Mar 23, 2021 at 17:55
• You would consider S to be the number of unpaired electron on your magnetic atom, typically. For eg, if you have an Fe atom with a moment of 3 bohr magneton, S would just be 3*(1/2). If you can't eyeball 'S', you can simply infer it from your DFT calculation. All other parameters in your question can be estimated through first-principles, through DFT. I just answered a question regarding the calculation of Js, hopefully that points you in the right direction: mattermodeling.stackexchange.com/questions/1548/… Mar 23, 2021 at 20:23
• @Xivi76 As this is one of our longest standing unanswered questions, do you think you'd be able to turn this into an answer at some point (if you have time)? Apr 18, 2021 at 2:59
• @Xivi76 If you do have something more in mind, I think this question would benefit from a full answer.
– Tyberius
Oct 19, 2021 at 20:51
• @taciteloquence why don't you answer it? Since you know what the answer depends on, then you can pick one and answer it based on that! This question has gone unanswered for 16+ months! Nov 30, 2021 at 20:43

## 1 Answer

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.