Drastic change of magnetic anisotropy in Fe3GeTe2 and Fe4GeTe2 monolayers under electric field studied by density functional theory

Magnetic anisotropy energy (MAE) is one of the most important properties in two-dimensional magnetism since the magnetization in two dimension is vulnerable to the spin rotational fluctuations. Using density functional theory calculation, we show that perpendicular electric field dramatically enhances the in-plane and out-of-plane magnetic anisotropies in Fe3GeTe2 and Fe4GeTe2 monolayers, respectively, allowing the change of easy axis in both systems. The changes of the MAE under the electric field are understood as the result of charge redistribution inside the layer, which is available due to the three-dimensional (3D) network of Fe atoms in the monolayers. As a result, we suggest that due to the unique structure of FenGeTe2 compounds composed by peculiar 3D networks of metal atoms, the MAE can be dramatically changed by the external perpendicular electric field.

www.nature.com/scientificreports/ the planar magnetic anisotropy disables magnetic ordering since the spin O(2) symmetry gives the Goldstone mode from the free planar rotation. Thus, the easiest way to tune the magnetism of monolayer magnetic system is to control the MAE. There have been studies on the MAE change of Fe 3 GeTe 2 , with the strain effect on the monolayer 21 or the hole doping on the bulk system 22 . In this work, by using first principles calculations, the MAE changes by the perpendicular electric fields in the Fe 3 GeTe 2 and Fe 4 GeTe 2 monolayers are investigated. Since there are 3D networks of Fe atoms in FGT monolayers as shown in Fig. 1a,b, the electric field can move the electrons from one side of the layer to the other side. Thus, the electric field effect can be schematically understood as doping oppositely on Fe atoms in each sides of the layer. As a single unit of the FGT monolayer, the MAE is drastically changed by the external electric field, so we suggest that this system can be applied to the efficient in-situ magnetism control in spintronic devices.

Methods
For this study, density functional theory based on projector augmented wave (PAW) method implemented in Vienna ab initio simulation package (VASP) was used for entire work [30][31][32][33] . For the exchange correlation potential, the generalized gradient approximation (GGA) by Perdew-Burke-Ernzerhof (PBE) was used 34 . The cut-off energy for plane wave basis set of 700 eV was used. For k-mesh, 21 × 21 × 1 centered at Ŵ point was used. Crystal structures were obtained by the structure relaxation with the same options described above. The MAE, defined as E(001)-E(100) were determined by the comparison between the total energy of self-consistently calculated results with spin direction along z (001) and x (100) axis.

Figure 1.
Crystal structures of (a) Fe 3 GeTe 2 and (b) Fe 4 GeTe 2 monolayers. Space group numbers are 187 and 164 for Fe 3 GeTe 2 and Fe 4 GeTe 2 respectively without electric field, but as electric field is induced, both changes to 156. Both compounds have 3D networks of Fe atoms embracing Ge atoms at the center of the network and sealed by Te atoms. The total and Fe partial DOSs of (c) Fe 3

Results and discussions
Intralayer electron redistribution by the electric field. Crystal structures of Fe 3 GeTe 2 and Fe 4 GeTe 2 monolayers are shown in Fig. 1a,b, respectively. Both of these materials have the 3D networks of Fe atoms allowing stable itinerant ferromagnetism. Moreover, the 3D networks allow the electrostatic potential bias between Fe atoms by the perpendicular electric field. Since these systems are metallic, the continuous change of the induced electric field can result in charge transfer from one side to another side of the monolayer. Densities of states (DOS) near the Fermi level are dominated by Fe-d orbitals, as shown in Fig. 1c,d for Fe 3 GeTe 2 and Fe 4 GeTe 2 , respectively. Thus, the charge redistribution by the external electric field or carrier doping are expected to influence mostly on Fe-d orbital occupation. However, owing to the Fe-Te hybridizations, the charge redistribution by the electric field can indirectly influence on Te atoms which provide strong spin-orbit coupling. Thus MAE can be changed by the charge redistribution.
As the perpendicular electric field is induced, the electrostatic potential biases enter in these systems and move the electrons from one side of the layer to the other, as schematically shown in Fig. 2a. Since these systems have z-inversion symmetry in the structures, we can consider the charge redistribution as charge transfers between z-inversion paired atoms. Among the Fe atoms, Fe OUT , which are located far from the center, get large charge redistribution by the electric field due to the large electrostatic potential bias between Fe OUT (1) and Fe OUT (2) .
The changes of the magnetic moments of each Fe atoms show clear evidence of such charge transfer. As depicted in Fig. 2b, the Fe atoms that gain electrons lose the net spin moment and the Fe atoms that lose electrons  Fig. 3a,b, respectively. In case of Fe 3 GeTe 2 , the MAE is about -1.25 meV per Fe atom at zero electric field, indicating uniaxial anisotropy. As the electric field is induced, the MAE increases after plateau of 0.1 V/Å and finally the sign gets changed to positive near 0.25 V/Å, flipping the preferred spin direction from uniaxial to planar. In case of Fe 4 GeTe 2 , the MAE is about 1.0 meV per Fe atom at zero electric field, indicating planar anisotropy. As electric field is induced, the MAE decreases after plateau of 0.05 V/Å and finally sign gets changed to negative near 0.2 V/Å, flipping the preferred spin direction from planar to uniaxial. To summarize, the increase of the perpendicular electric field induces inplane and out-of-plane spin directions in Fe 3 GeTe 2 and Fe 4 GeTe 2 respectively. Since Curie temperature is strongly dependent to the MAE in 2-dimensional systems, the stabilization of uniaxial magnetic anisotropy especially in the monolayer or few-layer limit can strongly increase the Curie temperature. Thus, our results show that the induction of perpendicular electric field is expected to increase the Curie temperature of monolayer Fe 4 GeTe 2 and decrease the Curie temperature of monolayer Fe 3 GeTe 2 .
Since the effect of the electric field in our work is expected to originate directly from the electron redistribution, the MAE change under charge doping should show consistent result to the MAE change under electric field. For the analysis of the MAE change, first, we consider the MAE of the system as a function of the occupancies of Fe atoms, for instance, as E MAE n OUT(1) , n OUT(2) , n IN in case of Fe 3 GeTe 2 , where n OUT(1) , n OUT (2) and n IN denotes the electron occupancies of Fe OUT(1) , Fe OUT (2) and Fe IN , respectively. Secondly, we can consider that the total MAE coupled to each Fe atoms are divisible, so that where ǫ OUT and ǫ IN denotes the MAE contributions from Fe OUT and Fe IN atoms, respectively. Here, we assume that the MAE depends only on the charge occupancies of each Fe atoms, based on the previous analysis on Fig. 2. Note that the changes in Te atoms are indirectly covered through the charge occupancies of Fe atoms as discussed in Fig. 2.
Based on the considerations above, the change of the MAE during the charge redistribution among Fe OUT atoms by the electric field can be expressed as where n is the initial occupancy of Fe OUT atoms and n is the amount of charge transfer from one Fe OUT to the other Fe OUT atom by the electric field. Thus, if ǫ OUT (n) shows convex curve around original occupancy ( ǫ OUT (n + �n) + ǫ OUT (n − �n) > 2ǫ OUT (n) ), the MAE increases under electric field. Since the contribution www.nature.com/scientificreports/ to E MAE from ǫ IN should be much smaller than the contribution from ǫ OUT due to the reason discussed in the previous analysis on Fig. 2., the convexity of ǫ OUT (n) curve can be considered as equivalent to the convexity of E MAE curve. Figure 4a shows the MAE change under charge doping for Fe 3 GeTe 2 .

MAE change by the charge doping.
First of all, the result is consistent to the previous work on hole doped bulk Fe 3 GeTe 2 showing the weakened uniaxial anisotropy as hole doping, although we dealt the monolayers 22 . Furthermore, the convex MAE behavior of Fe 3 GeTe 2 shows consistency to the increasing MAE under electric field shown in Fig. 3a. Thus, the MAE changes under the charge doping can explain the electric field effect on the MAE. Figure 4b shows the MAE change under charge doping in Fe 4 GeTe 2 . This curve is concave ( ǫ OUT (n + �n) + ǫ OUT (n − �n) < 2ǫ OUT (n) ), thus the electric field should decrease the MAE of Fe 4 GeTe 2 , which is consistent to the result in Fig. 3b. Since the key of electric field dependence of the MAE is the convexity of the MAE-doping curve, we also figured out the origin of the convexity to keep track of the MAE changes under the electric field. The different convexity between Fe 3 GeTe 2 and Fe 4 GeTe 2 can be analyzed by the shifts of the electronic bands contributing to the MAE as charge doping 22 . Here, the states contributing to MAE can be identified better by the band structures instead of the localized orbitals due to its itinerant characteristics. Figure 5 shows the band structure change under charge doping in Fe 3 GeTe 2 and Fe 4 GeTe 2 respectively. In all cases, bands obtained with perpendicular spin direction were denoted as black line and those with planar spin direction were denoted as red line. The bands that the black lines are lower than the red lines contribute to the perpendicular magnetic anisotropy (PMA) and the bands that the black lines are upper than the red lines contribute to the planar magnetic anisotropy. Thus, the change of the occupation of the bands with split black and red lines determine the MAE change by the charge doping 22 . The representative bands contributing to the PMA are denoted as blue regions, which move near the Fermi level by the charge doping.
The same negative sign of the gradient in Fig. 4a,b can be explained by the fact that the bands near the Fermi level mostly to favour the PMA. However the convexity of the curve in Fig. 4a,b were the opposite due to the fact that in Fe 3 GeTe 2 , the blue region is already filled in non-doped case and gets unfilled as hole doping, whereas in Fe 4 GeTe 2 the blue region is not filled in non-doped case and gets filled as electron doping. Thus in Fe 3 GeTe 2 hole doping influences stronger than electron doping on MAE and in Fe 4 GeTe 2 electron doping influences stronger than hole doping.
From these results, we suggest that the convexity of the MAE curve under doping is relevant to the direction of magnetic anisotropy change under electric field. To ensure this idea more certainly, we need to verify that the charge redistributions only in Fe OUT atoms actually affects in the same way. Figure 4c  www.nature.com/scientificreports/ by controlling the atomic valence numbers of Fe OUT (1) and Fe OUT (2) individually, mimicking the situation of charge transfer between Fe OUT atoms by the electric field. The results show that the charge redistribution on Fe OUT atoms increase the MAE in Fe 3 GeTe 2 and decrease the MAE in Fe 4 GeTe 2 after plateau, being consistent both to the electric field effect and the convexity arguments above. Thus, the idea that the electric field effect on the MAE in these systems is explainable by charge redistributions and that the convexity of the MAE curve under doping determines the direction of the change of the MAE is confirmed.
Since the electron doping is expected to strengthen the uniaxial MAE both in Fe 3 GeTe 2 and Fe 4 GeTe 2 , Curie temperatures are also expected to increase. To relate this result to experimental situations, substituting Fe to Mn corresponds to hole doping and Fe to Co corresponds to electron doping. Also, the formation of vacancies also can be understood as charge doping. For instance, Te vacancies can effectively dope electrons to Fe atoms and Fe vacancies can effectively dope holes to Fe atoms as discussed in previous study on the MAE change under hole doping 22 . However, Fe vacancies in the bulk FGT systems are expected to influence on Curie temperature through the changes of ferromagnetic exchange or disorder rather than charge doping as pointed out in previous experimental study on Fe 3-x GeTe 2 35 .

Conclusion
The possibility of the electric field control of the MAE of Fe 3 GeTe 2 and Fe 4 GeTe 2 monolayers has been investigated by using first principles calculation. Our results showed that the induced electric field along perpendicular direction stabilizes planar magnetic order in case of Fe 3 GeTe 2 and uniaxial magnetic order in case of Fe 4 GeTe 2 . Moreover, the amounts of the change of MAE were dramatic, allowing the change of easy axis in both systems easily. The origin of the electric field effects on the MAE are well explained through the charge redistribution scheme inside of the layer. While the uniform charge dopings both in Fe 3 GeTe 2 and Fe 4 GeTe 2 showed that the electron doping stabilizes the uniaxial magnetic ordering, the charge redistributions among Fe atoms in Fe 3 GeTe 2 and Fe 4 GeTe 2 stabilized the planar and uniaxial magnetic ordering, respectively. Since MAE in two-dimensional systems strongly influence on the Curie temperature, our result also implies that the Curie temperature can be controlled by the electric field in qualitative manner. Quantitative prediction on the Curie temperature change, which is beyond our scope, can be performed by Monte Carlo simulation on anisotropic Heisenberg model 36 .
The MAE control by electric field can be utilized for various practical usages. One certain case is the voltagetransfer torque magnetoresistive random-access memory (VTT-MRAM), suggested for fast and energy efficient memory device for nonvolatile computation without ohmic dissipation, which might support the big-data science and applications in hardware level 37 . Voltage controlled magnetic anisotropy (VCMA) has been studied for thinfilm made magnetoresistive tunnel junctions (MTJ) with surface perpendicular magnetic anisotropy (PMA) [37][38][39][40][41] . The value of VCMA coefficient, defined as the MAE change from out-of-plane to in-plane per area divided by the electric field, are about few hundreds fJV −1 m −1 in current status, while it needs to be few pJV −1 m −1 for practical VTT-MRAM application 37 . Our calculations suggest that the VCMA coefficients of Fe 3 GeTe 2 and Fe 4 GeTe 2 are about 2 and − 2 pJV −1 m −1 respectively, even in the monolayer with the thickness of nearly 0.5 nm. Alongside with the large absolute value of VCMA coefficient, the FGT materials have one more advantage that it is much www.nature.com/scientificreports/ free to obtain and manipulate clean layers due to the vdW nature. Although the scale of the electric field in this study is a lot larger than the usual scale of electric field in practical applications, since the same scale of the potential bias is easily reachable with small electric field as the number of the layers increase, the possibility for practical application based on electric field controlled MAE is still open. These advantageous properties for applications arise from the vdW structure while keeping Fe 3D networks inside of layer. While 3D networks allow intralayer charge redistribution, Te atoms allow large absolute value of VCMA coefficient by offering strong spin-orbit coupling with the hybridization with Fe atoms, working similar to heavy metal doping at the interface. Since the ligand sealed structure is common for the vdW magnetic materials, any vdW materials with 3D metal atom networks are expected to be practically applicable for VTT-MRAM.
Furthermore, various vdW magnetic materials are expected to show drastic MAE change under the electric field in finite multilayers by the mechanism based on the interlayer charge redistribution might work. However, since the other interlayer transitions such as ferromagnetic to antiferromagnetic transitions frequently occur, the study on spin dynamics in such cases might open a new possibility of practical applications.