Magnetic exchange interactions and Curie temperature

4.3 Results and Discussions

4.3.6 Magnetic exchange interactions and Curie temperature

electronic, and magnetic properties of Mn₂NiGa

0 0.25 0.5 0.75 1

Fe content (x)

400 600 800 1000 1200 1400

Curie Temperature (K)

(Mn_2-xFe_x)NiGa Mn₂(Ni_1-xFe_x)Ga Mn₂Ni(Ga_1-xFe_x)

0 0.25 0.5 0.75 1

Co content (x)

400 600 800 1000 1200

1400 (Mn_2-xCo_x)NiGa Mn₂(Ni_1-xCo_x)Ga Mn₂Ni(Ga_1-xCo_x)

Figure 4.12: Calculated Curie temperatures as a function of substituent (a) Fe concentra- tion and (b) Co concentration for substitutions at different sites in Mn₂NiGa. Calculations are done by the MFA method.

substituent moments, both of which vary slowly across compositions. The linear in- crease in the total moment of the Mn-substituted system, on the other hand, can be attributed to the gradual loss of contributions of MnI due to its decreasing concen- tration which boosts the ferromagnetic component in the system with concentration of the substituent.

4.3.6 Magnetic exchange interactions and Curie tempera-

4.3 Results and Discussions

0 0.25 0.5 0.75 1

Fe content (x)

300 400 500 600 700 800 900

Curie Temperature (K)

(Mn_2-xFe_x)NiGa Mn₂(Ni_1-xFe_x)Ga Mn₂Ni(Ga_1-xFe_x)

0 0.25 0.5 0.75 1

Co content (x)

300 400 500 600 700 800

900 ^(Mn_Mn^2-xCox)NiGa

2(Ni_1-xCo_x)Ga Mn₂Ni(Ga_1-xCo_x)

Figure 4.13: Calculated Curie temperatures as a function of substituent (a) Fe concentra- tion and (b) Co concentration for substitutions at different sites in Mn₂NiGa. Calculations are done by the MCS method. Open symbols represent the experimental results which are adopted from Refs. [239,240,248,257].

0 0.25 0.5 0.75 1 -150

-100 -50 0 50 100 150

J eff (meV)

MnI-Fe MnI-Ni Mn1-MnII

0 0.25 0.5 0.75 1 Fe content (x)

Fe-Ni Fe-MnII Ni-MnII

A A A A

0 0.25 0.5 0.75 1

MnI-MnIII Fe-MnIII Ni-MnIII A AMnII-MNIII

(Mn_2-xFe_x)NiGa Mn₂(Ni_1-xFe_x)Ga Mn₂Ni(Ga_1-xFe_x)

Figure 4.14: Effective exchange coupling constant (J_{ef f}) as a function of Fe content at different sites in Mn2NiGa.

Tc rises sharply with increase in x. For the experiment [240], on the other hand, although a decrease in Tc up to x = 0.24 and a rise after that were obtained, the changes were not this substantial. The experimentally obtained Tc decreased from 538 K (x = 0) to 517 K for x= 0.24 and rose to only 537 K for x = 0.52. It may be noted that their Tc for Mn2NiGa is lower by 50 K in comparison to Tc obtained from other experiments [64, 239]. Such discrepancies could be due to antisite dis- order or off-stoichiometric compositions present in the samples used by the authors of Ref. [240]. In fact, we have reported discrepancies in the magnetic moments calculated by us and obtained from their magnetic measurements in the previous subsection. Thus the origin of these discrepancies could be the same. Nevertheless, the agreement between the MCS results and the experiments for the end compounds, wherever available, is remarkable.

In order to understand the trends in the Curie temperature, we have calculated

electronic, and magnetic properties of Mn₂NiGa

0 0.25 0.5 0.75 1 -150

-100 -50 0 50 100 150 200

J eff (meV)

MnI-Co MnI-Ni Mn1-MnII

0 0.25 0.5 0.75 1 Co content (x)

Co-Ni Co-MnII Ni-MnII

A A A A

0 0.25 0.5 0.75 1

MnI-MnIII Co-MnIII Ni-MnIII A AMnII-MNIII

(Mn_2-xCo_x)NiGa

Mn₂(Ni_1-xCo_x)Ga

Mn₂Ni(Ga_1-xCo_x)

Figure 4.15: Effective exchange coupling constant (J_{ef f}) as a function of Co content at different sites in Mn₂NiGa.

the inter-atomic magnetic effective exchange coupling(Jef f), presented in Figs. 4.14 and 4.15 for Fe and Co substituted Mn2NiGa, respectively. The Jef f are calculated asJ_{ef f}^µν =P

jJ_0j^µν, 0 fixed to sub-lattice µ and the sites j belonging to sublattice ν.

We find that the dominant J_{ef f}s remain either antiferromagnetic or ferromagnetic across compositions. The dominant antiferromagnetic J_{ef f} is due to the MnI-MnII pairs while the other dominant inter-sublattice Jef f s are ferromagnetic. For the (Mn2−xCox)NiGa system, the strength of antiferromagnetic J^{M nI−M nII}_{ef f} decreases withxwhile the strength of the largest ferromagnetic effective exchange interaction, J^{Co−M nII}_{ef f} , increases with x. This is because of the increasing hybridizations between nearest-neighbor Co and MnII which couple parallel resulting in the weakened anti- ferromagnetic nearest-neighbor MnI-MnII exchange interaction as the system grad- ually becomes MnI deficient (Co excess). The increasing ferromagnetic exchange interaction between nearest-neighbor Ni and MnII and next-nearest-neighbor Ni- MnI and Ni-Co gives rise to an increase in the overall ferromagnetic interaction. Up tox= 0.25, there is a competition between the ferromagnetic and antiferromagnetic interactions which brings Tc down. Beyondx= 0.25, the ferromagnetic components overwhelm the antiferromagnetic interactions resulting in the rise of Tc for higher values of x. In contrast, for (Mn2−xFex)NiGa, the J^{M nI−M nII}_{ef f} remains nearly con- stant withx. The strength of the ferromagnetic Fe-MnII, Fe-Ni, and MnI-Ni in these compounds is weaker in comparison to those in (Mn2−xCox)NiGa with the Ni-MnII exchange interaction having nearly the same strength and larger than the Fe-MnII one, exactly opposite to the Co-substituted compound. Such weaker ferromagnetic interactions might be the result of weaker Fe-MnII hybridizations. This weak inter- action of MnII with one of the components in the 4a site keeps the strength of the interaction with the other component, MnI, at the same site intact across composi-

4.3 Results and Discussions tions although the concentration of MnI decreases gradually. This manifests itself in bringing down the Tc considerably in (Mn2−xFex)NiGa for low x and keeping it almost like that as x increases.

For Mn2(Ni1−xCox)Ga and Mn2(Ni1−xFex)Ga, we find the variations in the ex- change interactions quite similar, qualitatively and quantitatively. The antiferro- magnetic MnI-MnII interactions remain largely unaffected across compositions as the substitutions are not done in either of these sites. The inter-sublattice an- tiferromagnetic MnI-Fe and MnI-Co second-neighbor interactions are found to be substantial and identical in strengths. The strongest ferromagnetic interaction in the Co-substituted system is that of Co-MnII pairs while the strength of the Ni- MnII interaction is considerably weaker across the concentration range. In the case of Fe-substituted system, though the strongest ferromagnetic interaction is due to Fe-MnII pairs for smaller x values, the strength of Ni-MnII interactions quickly catches up with it. Thus, the initial decrease in Tc for Mn2(Ni1−xFex)Ga is due to weakening of the overall ferromagnetic interactions, primarily due to weak Fe-MnII hybridizations. For higher x, the J^{N i−M nII}_{ef f} compensates for the J^{F e−M nII}_{ef f} , strength- ening the ferromagnetic interactions in the system leading to an increase in Tc with x. For Mn2(Ni1−xCox)Ga, the strong ferromagnetic exchange interactions for all x values lead to an increase in Tc with x.

In systems with substitutions done at Ga sites, a greater number of interactions compete with each other as Mn atoms are present at three different sites. The an- tiferromagnetic interactions are due to nearest neighbor MnI-MnII and MnI-MnIII and second neighbor MnII-MnIII. The ferromagnetic components in the exchange in- teractions are due to nearest neighbors X-MnIII (X=Co, Fe), X-MnII, Ni-MnII, and Ni-MnIII and second neighbor Ni-MnI. For X=Co, that is, when Ga is replaced with Co, the antiferromagnetic MnI-MnII interaction loses its strength as does the second-neighbor antiferromagnetic MnII-MnIII interaction while the nearest- neighbor MnIII-MnI interaction becomes more antiferromagnetic as concentration of Co increases. Among the ferromagnetic interactions, except for Co-MnIII which decreases with concentration of Co, the other three increase. Strong hybridizations of Co with MnII and MnIII atoms are responsible for the strong ferromagnetic interac- tions in this system. This was predicted earlier [240] without explicit computations of the exchange interactions. Our calculation corroborates this with quantitative estimates. The antiferromagnetic exchange interactions behave the same way in systems with X=Fe. The ferromagnetic components for Fe-substituted systems are much weaker than those for Co-substituted systems. In fact, although Fe-MnIII

electronic, and magnetic properties of Mn₂NiGa

was the strongest ferromagnetic interaction initially, it drops fast giving way to Ni-MnII and Ni-MnIII. As Fe concentration increases, the Fe majority spin states move towards higher energies in comparison with MnII/MnIII states, thus making the hybridizations weaker. The weaker ferromagnetic components in Fe-substituted systems explain their smaller Tc values, in comparison to Co-substituted systems, particularly for higher x values. In both systems, the sharp decrease at x= 0.25 is due to relatively stronger antiferromagnetic interactions. As x increases, the ferro- magnetic interactions build up, effecting an increase in T_c.