How to calculate the exchange coupling parameters of the TM oxides

 

 In order to extract the exchange interactions from first-principles calculations, one should first map the electronic Hamiltonian onto the Heisenberg model. This mappingis by itself not uniquely defined. In the conventional picture, the Heisenberg model fortypical AFM TM oxides such as NiO and MnO  involves TM sites only, whereas the oxygen atoms mediate the magnetic interaction via superexchange mechanism.

 

Summarizing, two basic options exist to construct the Heisenberg model for TM oxides. The first one is based on the Anderson principle and assumes that oxygen and ligand effects are a property of the TM, whereas induced spin polarization should beignored. [29] In this approach, only exchange interactions between the TM sites are considered, while oxygen plays the role of a mediator of indirect (superexchange) interactions. In the second option, oxygen is also considered as a magnetic center provided it is sufficiently polarized. As a result, additional exchange interactions between TM and oxygen sites may come into play. In turn, the extended model that includes oxygen sites can be mapped onto an effective model involving renormalized interactions between the TM sites.

 In this work, we compare the performance of both approaches for three prototypeTM oxides: NiO and MnO. We calculate magnetic moments and exchange interactions starting from different magnetic configurations (FM and AFM) for the two materials. We derive an effective model to consider spin polarization on oxygen and show how the exchange interactions between TM sites are primarily affected. Finally, we perform a general comparison between the two approaches. We find that explicit treatment of oxygen spin polarization reduces the non-Heisenberg effects in NiO. That is, the exchange interactions depend on the magnetic state considered. In contrast, the inclusion of spin polarization in MnO makes the non-Heisenberg effects more pronounced.

 Calculation of exchange coupling parameters

In this approach, we calculate the energies of multiple magnetic configurations using DFT and fit them to the Heisenberg model. For NiO and MnO,we used three magnetic states: (i) FM; (ii) AFI: AFM order in the [001] direction; and(iii) AFII: AFM order in the [111] direction, being the magnetic groundstate.

 The 

corresponding total energies can be expressed as follows,

 EFM = E0−12J1 (FM coupling) −6J2 (FM coupling)

EAFI = E0+ 8J1(AFM coupling) − 4J1 (FM coupling) + 6J2 (AFM coupling)= E0+ 4J1− 6J2 

EAFII = E0+ 6J1 (AFM coupling)− 6J1 (FM coupling)+ 6J2 (FM coupling)= E0+ 6J2

 

 

https://www.researchgate.net/publication/231101608_Spin_Hamiltonian_effective_parameters_from_periodic_electronic_structure_calculations

 where J1 and J2 can be identified using figure 1 and E0 corresponds to the non-magnetic part of the energy. Using these energies, the exchange interactions per TM-oxygen paircan be calculated as

 J1=16(EAFI−EFM)

 J2=48(4EAFII−EFM−3EAFI).

 

 

 

Results of NiO:

 The magnetic ground state of NiO is formed by AFM ordering along the [111] crystallographic axis. Using (2), we calculate the magnetic moments in the AFM ground state and in the FM configuration for both the Ni and O sites. The Nimagnetic moments are 1.63μBand 1.69μBin the AFM and FM configuration, respectively. Whereas the Ni magnetic moments are independent on the magnetic state, the O sites show a pronounced dependence with 0.00μBand 0.30μBfor the AFM and FM states, respectively. The calculated magnetic moments of the Ni sites agree very well with previous calculations using the local self- interaction correction (LSIC) [41]: 1.68μBand LDA+DMFT [42]: 1.85μB., The experimental magnetic moment of Ni is larger:1.90μB, but it comprises both spin and orbital contributions to the magnetization. [43]

 

https://arxiv.org/pdf/1611.07288.pdf