Abstract
Schematics of different spin order and the total energy difference of the phases are compared with ferromagnetic groud state energy. The first, third, and fifth rows of the basis functions related to spin ordering represent signs of spin of the top, middle, and bottom irons, respectively. Therefore, the new order parameters can represent in-plane AFM spin ordering as well as out-of-plane AFM spin ordering.
Our expectation was that symmetry breaking of the spin order can induce structural polar displacement in the low-dimensional magnetic materials. Next, A-type ferrimagnetic spin ordering, meaning the top and bottom iron sublattices have A-type AFM spin structure and the middle iron sublattice has ferromagnetic spin ordering, breaks σh. We confirmed our prediction of the emergence of polar displacement through symmetry breaking induced by spin ordering.
The first, third, and fifth rows of the base functions relating to spin order represent the top, middle, and bottom iron spin signs. Therefore, the new order parameters can represent the in-plane AFM spin order as well as the out-of-plane AFM spin order.
Computational method
Density Functional Theory
- The Hohenberg-Kohn Theorems
- The Kohn-Sham equation
We relaxed the FGT structure with A-fri spin ordering and found that approximately 0.18 angstroms of z-directional polar displacement occurred. We expected the occurrence of in-plane and out-of-plane polar displacements as the G-AFM spin ordering breaks both two σv (vertical mirror symmetry) and σh. For example, in the case of fri-centro spin ordering which has a ferromagnetic relationship between Top iron and Bottom iron and an antiferromagnetic relationship between Top iron and Middle iron, it also shows atomic displacement upon relaxation of the structure.
Here we will confirm the idea that symmetry breaking of the spin order can cause structural polar displacement in the low-dimensional magnetic materials more clearly by using group theoretical analysis. By using this equation, we can finally obtain the power difference between ferromagnetic spin ordering and ferrimagnetic spin ordering. For the following we will apply group theory to the more complicated spin order G-AFM.
Since atomic displacements revive the effect, the energy differences between ferromagnetic structure and other spin-ordered structures are reduced. As figure 2 shows, G-AFM structure has the lowest energy close to ferromagnetic structure energy under the spin order causing polar displacement.
Design of low dimensional multiferroic material
Introduction
After Maxwell established the relationship between the electric field and the magnetic field in 18657, it was first demonstrated experimentally with Cr2O3 that the electric field can control the magnetic property8. Among them, the most well-known mechanism is that the polarization is caused by the inverse interaction of DM in a non-collinear spin structure by magnetic frustration1, 9. The polar distortion caused by the collinear spin structure is rare, most of which is induced by the exchange double in the complex. Antiferromagnetic order of type E10, 11.
However, we have found that polar distortion can only arise from symmetry breaking of a very simple collinear spin order in a low-dimensional system that lacks translation. Top and bottom irons have the same Wyckoff position and oxidation number, while the middle iron has a different Wyckoff position and different oxidation number12. Fe3GeTe2 possesses unusually strong magnetic interaction evidenced by one of the highest ferromagnetic critical temperatures (> 200 K)13, 14 among van der Waals materials.
In addition, FGT consists of three iron layers with other ligand ions with a lot of space between trilayers. Therefore, we first speculated that if bonds between Fe ions could be broken (recovered) by antiparallel (parallel) spin configuration, Fe ions would be displaced largely perpendicular to the layers. The formation of a large ferroelectric by simple collinear spin ordering in a low dimensional system will open a new chapter in the discovery and application of new states of low dimensional materials as a general method applicable to all two dimensional materials to apply.
Computational details
Result and Discussion
- Spin induced ferroelectricity in FGT monolayer
- Origin of the polar displacements
- Microscopic origin
- Macroscopic origin
- Achieving ferroelectric phase by strain engineering
In addition to the above three, we also tried many other spin arrangements on the FGT system and confirmed that the symmetries of the atomic movements depend on the symmetries of the spin arrangement. The upper figures show the FM and A-fri exchange interaction, and the lower figure shows the simplified density of states of Fe3GeTe2, Mn3GeTe2 and Cr3GeTe2. In Figure 3, the A-fri spin order is used as the simplest spin order that induces a polar shift to explain the origin.
The interaction between the top iron and the middle iron and between the bottom iron and the middle iron, indicated by thick red lines. To enable the electron hopping from the d6 state of the middle iron to the d5 state of the top and bottom iron, the double exchange interaction strongly promotes the ferromagnetic relationship between them. In the case of A-fri spin ordering, the double exchange path between the middle iron and the bottom iron is broken, since electron hopping is forbidden by the Pauli exclusion principle.
But it cannot explain the large force that arises from changing spin order without lattice relaxation. These results show us that the d states close above and below the fermi level and the small gap between them contribute to the strong interaction between the middle iron and the top or bottom iron and this makes J2 very strong. The calculated J5 value is -2.2 meV and means that the sum of the effects of all exchange interactions between the top iron and the top iron favors antiferromagnetic coupling very weakly.
As Table 3 shows, the f and a order parameters of ferromagnetic spin order are 1 and 0, and that of A free-spin order is 0 and 1. The result clearly shows that the appearance of dipolar force is only related to the coupling of antiferromagnetic relation between top iron and bottom iron and ferromagnetic middle iron. The second, fourth, and sixth rows also represent signs of spin of the top, middle, and bottom irons, respectively, but they are different from the previous ones.
Although we have found that spin ordering can induce polar displacement, there is still a problem in using a ferroelectric phase of FGT. Red, blue, green and magenta colors represent top iron, bottom iron, middle iron and germanium of G-AFM structure, respectively. If we only look at z-direction displacement, upper iron and lower iron go up and down respectively and they can cancel most of each other's impact because they have the same sign of charge.
Conclusion
If we calculate the polarization with bon effective charge, the polarization value will be much smaller. Because FGT is metal and most of the charge is shielded by free electrons, the effective charge is very small.
Design of low dimensional multi state material
- Introduction
- Computational details
- Result and Discussion
- Finding ground state structure of GeSnTe 2
- Two different polarized phases of GeSnTe 2
- Energy barrier for polarization switching (NEB)
- Conclusion
But for all the previous research, the polarized state is only one as x-direction and y-direction polarization are not distinguishable. This simple phase does not show in-plane anisotropy, but other phase gives the anisotropy and splits a direction and b direction polarized state. First principles calculations were performed using density functional theory (DFT) within the general gradient approximation GGA method with Perdew-Becke-Erzernhof parametrization as implemented in the Vienna ab initio simulation package (VASP 5.4.4).
The K-point sampling uses the Monkhorst-Pack scheme22 and employs a 6×6×1 and a 3×6×1 mesh for structure optimization of the primitive cell and the 2×1×1 supercell of the GeSnTe2 monolayer. The activation barrier for the polarization switch of GeSnTe2 was calculated using the nudged elastic band (NEB) method and 3×6×1 k-point grid used for the NEB calculation34. To define the ground phase of GeSnTe2, we made three candidate structures and compared the total energy of each structure.
Fortunately, the ground state structure was constructed in such a way that SnTe and GeTe are connected alternately along one direction and therefore have anisotropy. From this result we can conclude that mixing while retaining the original structure of SnTe and GeTe is energetically more stable. In the previous section we showed that the ground state phase of GeSnTe2 has in-plane anisotropy.
Therefore, we can expect that a-direction polarization and b-direction polarization of GeSnTe2 are no longer the same. The distance between the minimum of the red line and the minimum of the black line comes from the total energy difference between the Pa phase and the Pb phase. Since Pb phase has lower energy, we need to apply a directional electric field to stabilize the Pa phase.
The rotational path from -Pb phase to Pb phase stopping through Pa phase reduces energy barrier to 25meV/f.u. From the calculation, we figured out that SnTe and GeTe are alternately connected along a-direction in the mixture structure which has the lowest energy since mixture structure tends to keep the original structure of SnTe and GeTe. For this anisotropic mixture structure, a-direction polarization and b-direction polarization of GeSnTe2 are no longer the same.
The calculated polarizations of the mixed structure are 12.434 μC/cm for the a-direction and 13.344 μC/cm2 for the b-direction. These two different polarization values and barrier energies of a-directional and b-directional polarized phase enable its multi-state application.
