Cakculation of the exchange coupling constants J

To calculate the coupling constants J, we take the example of binary transition metal oxides like NiO.

Introduction

TMO have a rock-salt structure (see Fig.1), so each TM has 12 nearest-neighbor (NN) and 6 next-nearest neighbors (NNN). The NNN are connected through oxygen bridges, and their interaction J2 is dominated by superexchange. On the other hand, NN interact via a typically smaller exchange coupling J1 whose sign may depend on the specific TMO; J1 involves direct TMTM exchange (giving a robust AF contribution) and a 90◦ -oriented TM-O-TM superexchange (expected to be weakly FM). The observed ground state magnetic phase is antiferromagnetic (111) A-type, labeled AF2 hereafter. It can be seen as a stacking of (111) planes of like spin alternating along the [111] direction, as illustrated in Fig. 1. In AF2 each TM has 6 spin-paired intra-(111)-plane NN and 6 spin-antipaired inter-(111)-plane NN; on the other hand, all 6 NNN bonds are inter-planar and antipaired. Thus, this configuration maximizes the energy gain associated to the NNN antiparallel spin alignment. As for beyond-NNN magnetic interactions, there is ample experimental54 and theoretical16 evidence that they can be safely discarded (e.g. according to inelastic neutron scattering54 in NiO they are two order of magnitude smaller than the dominant J2.

                            Fig.1


How to calculate the exchange coupling constants J

In order to evaluate J1 and J2 we need to consider at least two competing high-symmetry magnetic phases beside the observed AF2. Natural choices are the ferromagnetic (FM) order and the AF (110)  order with (110) spin-paired planes compensated along [110] (labeled AF1). AF1 can also be seen as made of FM (001) planes alternating along [001] (see Fig. 1). The AF1 phase has all the 6 NNN spin-paired, 4 of the NN spinpaired, and 8 NN spin-antipaired.

To extract J1 and J2 we fit our calculated total energies to a standard 2-parameter classical Heisenberg Hamiltonian of the form:

This is solved to give:

With this choice of the Hamiltonian, negative and positive J values correspond to energy gain for spinantiparallel and spin-parallel orientations, respectively.



Reference: https://www.researchgate.net/publication/51915278_Exchange_interactions_and_magnetic_phases_of_transition_metal_oxides_Benchmarking_advanced_ab_initio_methods