In this project, the energies and lattice constants of the ferromagnetic and antiferromagnetic states of Co, MnF2, and MnBi were found and compared. To perform these calculations, a suitable energy cutoff and k-space sampling were selected to obtain a 0.01eV tolerance, and research was done in order to determine a proper starting c/a value of the lattice constant. From there, the the materials were placed into a ferromagnetic or antiferromagnetic state by aligning or misaligning their atomic spin polarizations and their energies were found at varying values of lattice constant a using CASTEP plane wave calculations [1].

These calculations were performed in Materials Studio using the GGA Perdew Burke Ernzerhof (PBE) functional and the pseudopotential was solved using the Koelling-Harmon atomic solver. In Materials Studio, spin polarizations are defined by the the total number of up electron spins minus down electron spins, so a spin polarization of +3, for example, implies that the atom has 3 more spin up than spin down electrons.

Cobalt

Fig. 1. Picture of cobalt cell used

For these calculations, cobalt was imported using the default Materials Studio structure, which gives a lattice constant of 2.507Å, a c/a value of 1.623, and forms the P63/MMC symmetry. From there, the spins in the 2 atom unit cell were polarized into a ferromagnetic +3, +3 state and energy calculations with varying energy cutoffs and k-samplings were performed until the results converged to a 0.01eV tolerance. The psuedopotential was calculating using the 1s2 2s2 2p6 3s2 3p6 electrons for the inner shells and 3d7 4s2 as the valence shells. Ultimately, a 420eV cutoff with a total k space sampling of 108 points were used.

Fig. 2. Energy cutoff calculations for Co

Fig. 3. k-space Sampling calculations for Co

From here, the lattice constant a was varied in 0.1Å and 0.005Å increments to find a suitable ferromagnetic lattice constant. These same calculations were repeated for the antiferromagnetic state, which were started with opposite +3 and -3 spin polarizations.

Fig. 4. Lattice constant calculations for ferromagnetic and antiferromagnetic states of Co

In the antiferromagnetic calculations, as the atoms approached one another, the calculations ceased to reach a local minimum in the antiferromagnetic state and began converging to non-spin polarized states. Additionally, some calculations failed to converge within 200 steps. Regardless, Co favors a ferromagnetic state, with a lattice constant of approximately 2.49Å, over the antiferromagnetic state, which appears to have a lattice constant between 2.465Å and 2.45Å (energy decreases past 2.465Å, but calculations converge to a non-spin-polarized state). This is in good agreement with the experimental value of 2.507Å in the ferromagnetic state.

Table 1. Results of the cobalt ferromagnetic and antiferromagnetic calculations

MnF2

Fig. 5. Picture of cell used

For these calculations, MnF2 was constructed using the analogous rutile TiO2 structure, which has the P42/MNM symmetry. The starting value of lattice constants c/a and a, 0.68 and 4.875Å respectively, used in these calculations were found by Dormann et. al [2]. From there, the spins in the atomic unit cell were polarized into a ferromagnetic state and energy calculations with varying energy cutoffs and k-samplings were performed until the results converged to a 0.01eV tolerance. The psuedopotential was calculating using the 1s2 electrons for the inner shells of fluorine and 2s2 2p5 as the valence shells. For manganese, the 1s2 2s2 2p6 electrons were used for the inner shells and 3s2 3p6 3d5 4s2 for the valence shells.

Fig. 6. Energy cutoff calculations for MnF2

Fig. 7. k-space Sampling calculations for MnF2

Calculations were made by shifting the lattice constant a in 0.1Å increments to get a rough idea of the lattice constant and 0.01Å measurements to take a finer sweep around the minimum energies of the two states. Ultimately, MnF2 converged to a lattice constant of approximately 4.930Å in the favored antiferromagnetic state and 4.945Å in the less favorable ferromagnetic state. This is relatively close to the experimental value of 4.875Å in the antiferromagnetic state.

Fig. 8. Graph of energies vs. lattice constants for the MnF2 lattice with ferromagnetic and antiferromagnetic spin polarization.

Table 2. Table of energies vs. lattice constants for MnF2.

MnBi

Fig. 9. Picture of cell used

For these calculations, MnBi was constructed using the analogous NiAs structure, which has the P63/MMC symmetry. The starting value of lattice constants c/a and a, 1.426 and 4.270Å respectively, used in these calculations were found by Koyama et. al [3]. From there, the spins in the atomic unit cell were polarized into a ferromagnetic state and energy calculations with varying energy cutoffs and k-samplings were performed until the results converged to a 0.01eV tolerance. The psuedopotential was calculating using the 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 4f14 5s2 5p6 electrons for the inner shells of bismuth and 5d10 6s2 6p3 as the valence shells. For manganese, the 1s2 2s2 2p6 electrons were used for the inner shells and 3s2 3p6 3d5 4s2 for the valence shells.

Fig. 10. Energy cutoff calculations for MnBi

Fig. 11. k-space Sampling calculations for MnBi

Next, the lattice constant was varied in 0.05Å increments to find an approximate energy minimum for the ferromagnetic and antiferromagnetic states. The lattice constant was then incremented in finer 0.01Å increments around the minimum value to obtain more accurate results. Ultimately, the lattice constant for both states came out to 4.23Å, with the ferromagnetic state being favored over the antiferromagnetic state. This is extremely close to the experimental value of 4.27Å in the ferromagnetic state.

Fig. 12. Graph of energy vs. lattice constant for the two ordered magnetic states of MnBi

Table 3. Table of energy vs. lattice constant for the two ordered states of MnBi

Discussion

These results all agree with the experimentally accepted lattice constant values for the three materials to a good degree of accuracy. In addition, they correctly determined that Co and MnBi are ferromagnetic, while MnF2 is antiferromagnetic. However, the effect of spin polarization in these materials is quite small – on the order of <1eV – suggesting that the energy change is relatively small even for materials considered to be ferromagnets and antiferromagnets.This provides some justification for ignoring spin polarization in our previous calculations, such as those involving copper, which is diamagnetic.

Another interesting note is that the antiferromagnetic calculations required breaking the initial lattice symmetry. This resulted in the antiferromagnetic calculations taking approximately 3-4 times longer than the ferromagnetic calculations; for example, the a = 4.18Å MnBi calculations  took 1701s for the P1 antiferromagnetic state versus 458s for the P63/MMC ferromagnetic state.

Though all the antiferromagnetic calculations in this post were performed with only P1 symmetry, the author realized afterwards that Materials Studio enables one to find a higher symmetry state factoring in the spins. By going into build->Symmetry->Find Symmetry->Options, one can force Materials Studio to consider the formal spins of each atom when searching for symmetries. Using the symmetry finder with these settings, one can reduce the  a = 4.18Å MnBi antiferromagnetic calculation time to 295s, a nearly 6-fold decrease in calculation time.

References