How to calculate the exchange matrix of a bilayer magentic system?

Question

I have been using the Vampire atomistic simulation software for a while and it has a useful simulation for predicting the curie temperature. But in order to do that I need the exchange matrix of Co-CoO system which should include the constant which represents the interaction between spins in the Co part, spins in the CoO part and a constant representing the interaction of spins in the Co-CoO interface. So I have been able to obtain the values for the Co and CoO part by couldn't find any resources to calculate the term representing the interaction at the interface. Should I use the average of the two values I've obtained. How should I approach this problem?

An excerpt from the VAMPIRE manual regarding this is given below:

material:exchange-matrix[index] = float [default 0.0 J/link] Defines the pair-wise exchange energy between atoms of type index and neighbour-index. The pairwise exchange energy is independent of the coordination number, and so the total exchange integral will depend on the number of nearest neighbours for the crystal lattice. The exchange energy must be defined between all material pairs in the simulation, with positive values representing ferromagnetic coupling, and negative values representing antiferromagnetic coupling. For a ferromagnet with a nearest neighbour exchange, the pairwise exchange energy can be found from the Curie temperature by the mean-field expression:

$$J_{ij} = \frac{(3k_{b}T_{c})}{\epsilon z}$$

where $J_{ij}$ is the exchange energy, $k_{b}$ is the Boltzmann constant, $T_{c}$ is the Curie temperature, z is the coordination number (number of nearest neighbours) and $\epsilon$ is a correction factor to account for spin-wave fluctuations in different crystal lattices. If a custom unit cell file is used the exchange values defined here are ignored.

I have included the manual here in this link. VAMPIRE Manual

Answer 1

But in order to do that I need the exchange matrix of Co-CoO system which should include the constant which represents the interaction between spins in the Co part, spins in the CoO part and a constant representing the interaction of spins in the Co-CoO interface.

The Co-CoO (FM- AFM) interfacial exchange energy can be taken to be equal to the exchange coefficient of the Bulk Ferromagnetic layer. This is based on what is being specified on VAMPIRE's tutorial on bilayered systems.

Information from VAMPIRE

The exchange between the two materials (the interfacial exchange) is defined as a ferromagnetic interaction with the same strength as the bulk. Note that the interaction strength from material 1 to 2 must be the same as from material 2 to 1.

This is for a bilayered system. But for a core-shell system, the following input file and material file can be used.

The input file

create:crystal-structure = fcc 
create:sphere 
dimensions:unit-cell-size = 3.524 !A 
dimensions:system-size-x = 11.0 !nm 
dimensions:system-size-y = 11.0 !nm 
dimensions:system-size-z = 11.0 !nm 
dimensions:particle-size = 11.0.0 !nm 
material:file = Co.mat 
sim:temperature = 300.0 
sim:minimum-temperature=0 
sim:maximum-temperature=2000 
sim:temperature-increment=10 
sim:time-steps-increment=1 
sim:equilibration-time-steps:1000 
sim:loop-time-steps=1000 
sim:program = curie-temperature 
sim:integrator =monte-carlo 
output:real-time 
output:temperature 
output:magnetisation 
output:magnetisation-length 
output:mean-magnetisation-length 
screen:temperature 
screen:mean-magnetisation-length 
config:atoms 

Material file

materials:num-materials=2 
# The shell
material[1]:material-name = "AFM" 
material[1]:material-element = "Co"   
material[1]:exchange-matrix[1] = -4.27e-22    
material[1]:exchange-matrix[2] = 5.75e-21 
material[1]:atomic-spin-moment = 2.55 !muB    
material[1]:damping-constant = 1.0    
material[1]:uniaxial-anisotropy-constant = 6.96e-24 
# The core  
material[2]:material-name = "FM"  
material[2]:material-element = "O" 
material[2]:exchange-matrix[1] = 5.75e-21 
material[2]:exchange-matrix[2] = 5.75e-21 
material[2]:atomic-spin-moment = 1.59 !muB    
material[2]:damping-constant = 1.0    
material[2]:uniaxial-anisotropy-constant = 2.98e-24   
material[1]:core-shell-size = 1.0 
material[2]:core-shell-size = 0.90909  

For more files, you can look at this Github repo

Hope this helps :)