Abstract
The emergence of ferromagnetism in twodimensional van der Waals materials has aroused broad interest. However, the ferromagnetic instability has been a problem remained. In this work, by using the firstprinciples calculations, we identified the critical ranges of strain and doping for the bilayer Cr_{2}Ge_{2}Te_{6} within which the ferromagnetic stability can be enhanced. Beyond the critical range, the tensile strain can induce the phase transition from the ferromagnetic to the antiferromagnetic, and the direction of magnetic easy axis can be converted from outofplane to inplane due to the increase of compressive strain, or electrostatic doping. We also predicted an electron doping range, within which the ferromagnetism can be enhanced, while the ferromagnetic stability was maintained. Moreover, we found that the compressive strain can reverse the spin polarization of electrons at the conduction band minimum, so that two categories of halfmetal can be induced by controlling electrostatic doping in the bilayer Cr_{2}Ge_{2}Te_{6}. These results should shed a light on achieving ferromagnetic stability for lowdimensional materials.
Introduction
Since 2004 when the graphene was exfoliated by Geim and Novoselov^{1}, researchers have revealed many unique physical properties from various twodimensional (2D) materials, e.g. quantum spin Hall candidate monolayer WTe_{2}^{2}, stanine^{3} with topological band inversion, and highmobility black phosphorus^{4,5}. Nevertheless, the absence of intrinsic ferromagnetism limits their application in spintronic devices. Recently, the intriguing intrinsic ferromagnetism has been proved both theoretically^{6,7,8,9} and experimentally^{10} in Cr_{2}Ge_{2}Te_{6}, one of the layered transition metal trichalcogenide’s family, which broke the longestablished Mermin–Wagner theorem^{11} and greatly enriched the versatility of 2D materials. Some applications have been proposed in newgeneration magnetic memory storage devices^{12} and nanoelectronic devices^{13}.
As the existence of ferromagnetism is one of the most charming features in 2D layered materials, it is important to enhance the stability and realize the tunability of the longrange magnetic ground state. An effective avenue is to increase the magnetic anisotropy energy (MAE), which is based on the energy difference between the inplane and outofplane magnetization direction. Large MAE in van der Waals (vdW) magnets would lift Mermin–Wagner restriction^{11,14}, for that the outofplane magnetic anisotropy would open a spinwave gap and counteract magnetic fluctuations, resulting in the stabilization of the longrange ferromagnetic order^{9,15,16}. The modulation of the magnetic properties based on the band engineering is highly desired in 2D layered ferromagnets, and the applications of external electric field^{17,18,19,20,21}, pressure^{22,23}, electrostatic doping^{24,25,26}, and strain engineering^{7,27,28,29,30,31} offer some valid approaches to tuning electronic structures, as well as the physical properties. Although revealing the transformation of MAE under external factors is imperative, so far, few researches has focused on the tunability of MAE with strain engineering or electrostatic doping for bilayer Cr_{2}Ge_{2}Te_{6}.
In this letter, the firstprinciples calculations were carried out to study the tunability of electronic structures and the magnetism in bilayer Cr_{2}Ge_{2}Te_{6} with biaxial strain or electrostatic doping. We determined a critical range of strain or doping in which the MAE is increased, in other words, the ferromagnetic stability is enhanced. A range of electron doping is also predicted, within which the ferromagnetism and the Curie temperature \(\left(T_{C}\right)\) can be raised. We also showed that two types of halfmetal were induced based on the external regulation. We further explored the possible mechanism involved in the variations of MAE, which would provide deeper understanding of 2D ferromagnetic materials.
Results and discussion
The crystal structures from top and side views of bilayer Cr_{2}Ge_{2}Te_{6} are shown in Fig. 1a,b, respectively. The unit cell of bilayer Cr_{2}Ge_{2}Te_{6} is denoted by the gray shaded region, which contains four Cr atoms, and each of them is bonded to six nearestneighboring Te anions and locates at the center of an octahedron (denoted by light green) formed by these Te atoms. Each layer consists of a honeycomb network of Cr atoms similar to graphene and comprises a Ge–Ge metal bond^{32}, which is a dimer lying perpendicularly at the central position of the CrTe_{6} nets, forming ethanelike groups of Ge_{2}Te_{6}. The bilayer Cr_{2}Ge_{2}Te_{6} contains two layers placed in AB stacking sequence. According to the structure optimization, the vdW gap in the bilayer is 3.437 Å, and the determined lattice parameters is a = b = 6.838(2) Å, as listed in Table 1.
We firstly explore the lattice distortion with the biaxial strain and electrostatic doping. Here, the strain is denoted by \(\eta =\left(a/{a}_{0}1\right)\times 100\%\), where \(a\) and \(a_{0}\) correspond to the strained and pristine lattice constants (without any strain or doping), respectively. So, the positive sign of \(\eta\) represents tensile strain, and the negative sign of it represents compressive strain. The electrostatic doping concentration is tuned by altering the total number of electrons in a unit cell. The positive and negative signs of concentration represent hole and electron doping, respectively. Figure 2a,b exhibit how the Cr–Te–Cr angles \(\alpha\) and Cr–Cr bond \(d\) change versus biaxial strains. According to the previous theoretical predictions^{7,8,22,33,34,35}, since the Cr–Te–Cr angle is close to \({90}^{^\circ }\), the superexchange interaction favors ferromagnetic (FM). The direct exchange interaction favors antiferromagnetic (AFM), which is inversely proportional to \(d\)^{36,37}. The competition between superexchange interaction and direct exchange interaction can be effectively tuned by controlling Cr–Te–Cr angle and the nearestneighbor Cr–Cr bond, which affects the magnetic ground state directly^{22,34}. As shown in Fig. 2b, the \(\alpha\) and \(d\) decrease (increase) with the increase of compressive (tensile) strain. As shown in Fig. 2c, we define the lengths of bonds in Cr–Te as \({l}_{a}\) and \({l}_{b}\) of octahedra a and b, respectively. \({\stackrel{}{l}}_{a}\) and \({\stackrel{}{l}}_{b}\) are the average bonds of six Cr–Te bonds in a and b octahedra, respectively. As shown in Fig. 2d, the \({\stackrel{}{l}}_{a}\) and \({\stackrel{}{l}}_{b}\) decrease (increase) with the increase of compressive (tensile) strain. The increase (decrease) in average bonds \({\stackrel{}{l}}_{a}\) and \({\stackrel{}{l}}_{b}\) means the octahedra expanding (shrinking) with the increase in the tensile (compressive) strain which is closely related to the magnetic and electronic structure, and the relations will be discussed below.
To explore the electronic structures of the bilayer Cr_{2}Ge_{2}Te_{6}, the electron localization function (ELF) is simulated. Figure 3a shows the isosurface demonstration of pristine bilayer Cr_{2}Ge_{2}Te_{6}. In order to explain the electron localization distribution more clearly, the twodimensional contour map in the direction of [001] is shown in Fig. 3b, from which we can see the electron distribution around Te and Ge atoms is highly localized. The ELF value around Cr atom is 0, indicating the highly delocalization of electrons around Cr. The boundary between the localized and delocalized electron distribution is green, which means that the ELF value is about 0.5 and the ionic bond exists between Cr and Te atoms. We also studied the effects on the electron localization distribution around different kinds of atoms under biaxial strains or electrostatic doping concentrations, and found no obvious difference in the electron localization distribution.
The MAE is defined as \(MAE={E}_{[100]}{E}_{[001]}\) for each unit cell (four Cr atoms), where \({E}_{[100]}\) and \({E}_{\left[001\right]}\) denote the total energy for the magnetic moments oriented along inplane and outofplane, respectively. From this definition, we can determine that increased \(\tt MAE\) means the enhanced ferromagnetic stability^{14}. The detailed results are listed in Table 1, from which we can see the positive MAE for pristine bilayer Cr_{2}Ge_{2}Te_{6}, illustrating that the outofplane (\([001]\)) direction is the easy axis for the magnetization. Also, we can see that the spin magnetic moment of Cr increases as the electron doping concentration increases from 0 to 0.2 e/u.c. Furthermore, from our results, the negative spin magnetic moment of Te relative to Cr is obtained, which is crucial for the stability of ferromagnetic ordering of Cr ions^{38}.
We then investigated the spinpolarized band structures of pristine bilayer Cr_{2}Ge_{2}Te_{6} just for comparing, as shown in Fig. 4. From Fig. 4a, we can see that both conduction band minimum (CBM) and valence band maximum (VBM) are of purely spinup character, possessing indirect band gap. The spinpolarized band structures of bilayer Cr_{2}Ge_{2}Te_{6} with \(\eta =2\%\) are plotted in Fig. 4b. Compared that of the pristine one [Fig. 4a], the band gap is significantly reduced due to the increased bandwidth. Attractively, the spinpolarized character at the CBM has even changed from spinup [Fig. 4a] to spindown [Fig. 4b], indicating that the electrons at the CBM and VBM are with opposite spin. As shown in Fig. 4c, the larger compressive strain (\(\eta =5\%\)) further increases the bandwidth due to the stronger interatomic coupling in the fewlayer Cr_{2}Ge_{2}Te_{6}, inducing a semiconductor–metal phase transition. On the contrary, the tensile strain (1%) increases the band gap, and the spin polarization at the CBM is enhanced compared with the pristine Cr_{2}Ge_{2}Te_{6}, as shown in Fig. 4d. Interestingly, in the vicinity of the Fermi level, the compressive strain induces a degeneracy of the spindown energy band along the KM direction, which is marked in the dashed blue circle. The halfmetallic state (conduction electrons being spinup) is induced with electron doping (− 0.1 e/u.c.) due to the obvious spin polarization, as shown in Fig. 4e. While with the hole doping (0.1 e/u.c.), the Fermi level shift down into the valence band, which makes the material metallic, as shown in Fig. 4f.
We further studied the band structure of bilayer Cr_{2}Ge_{2}Te_{6} under dual regulations, demonstrating that the halfmetallic state (conduction electrons being spindown) can also be induced in bilayer Cr_{2}Ge_{2}Te_{6} by combining electron doping (− 0.1 e/u.c.) and compressive strain (\(\eta =2\mathrm{\%}\)), as shown in Fig. 5a. These results indicate the potential usage of fewlayer Cr_{2}Ge_{2}Te_{6} in spintronic devices. In Fig. 5b, the schematic of the electronic structures restructuring illustrates the regulation mechanism more visually.
Figure 6 shows the atomic projected density of states (PDOS) of the bilayer Cr_{2}Ge_{2}Te_{6} under specific strains or electrostatic doping concentrations, as well as the schematic electron configurations for pristine or doping of − 0.1 e/u.c. Here, only the PDOS of Cr d orbitals are presented, which makes a major contribution to MAE based on the perturbation theory analysis. As shown in Fig. 6b, the spin polarization at the CBM of bilayer Cr_{2}Ge_{2}Te_{6} is weakened under the strain of \(\eta =2\%\). Moreover, the states at CBM is transformed into spindown, which originates mainly from Cr \({d}_{yz}{/d}_{xz}\) orbitals. As the compressive strain further increases (\(\eta =5\%\)), the band gap disappears and the orbital overlaps. With tensile strain of \(\eta =1\%\), the contribution of Cr \({d}_{xy}{/d}_{{x}^{2}{y}^{2}}\) orbitals near the Fermi level is reduced compared to the pristine one, as shown in Fig. 6d. As seen in Fig. 6e, the halfmetallic state is induced due to the obvious spin polarization at the doping concentration of − 0.1 e/u.c in the bilayer Cr_{2}Ge_{2}Te_{6}, and the spinup states near the Fermi level are mainly from Cr \({d}_{yz}/{d}_{xz}\) orbitals. To understand the increased magnetic moment for Cr atoms with electron doping (listed in Table 1), the diagrammatic electronic configurations are shown in Fig. 6g for the pristine or doped (− 0.1 e/u.c.) bilayer Cr_{2}Ge_{2}Te_{6}. Compared with the pristine one, the Cr\({d}_{yz}\) state is occupied by doped electrons, so that the net magnetic moment increases at the doping concentration of − 0.1 e/u.c., indicating the enhanced ferromagnetism of the system. It is worth mentioning that Cr \({d}_{yz}\) and \({d}_{xz}\), as well as \({d}_{xy}\) and \({d}_{{x}^{2}{y}^{2}}\) orbitals are degenerate because of the crystal symmetry. The shift of the Fermi level due to the strain or electrostatic doping changes the \(d\) projected orbitals near the Fermi level, which further changes the MAE of bilayer Cr_{2}Ge_{2}Te_{6} according to the following discussion.
The MAE plays a crucial role in the stability of the longrange magnetic ground states^{11,39}. We therefore investigated the variations of MAE and magnetic ground state with various strains or electrostatic doping concentrations in the bilayer Cr_{2}Ge_{2}Te_{6}. As shown in Fig. 7a, the ferromagnetic to antiferromagnetic transition can be induced by applying tensile strain more than 1%. Although larger \(\alpha\) than 90° and longer CrCr bond caused by the tensile strain indicating a weaker superexchange interaction and a weaker direct interaction, combining the results in Fig. 3 and the ferromagnetic to antiferromagnetic transition shown in Fig. 7a, we can deduce that the direct exchange interaction dominates for tensile strain more than 1%. The ground state of the bilayer Cr_{2}Ge_{2}Te_{6} is ferromagnetic when the applied strain is in the range of 4% to 1% while the magnetization direction remains outofplane. More meaningfully, we determine the critical strain range of − 3% to 1%, within which MAE can be effectively enhanced comparing to the pristine one, and the ferromagnetic ground state maintains as well. From Fig. 7a, we can also see that large compressive strain (5%) makes the direction of the easy axis change from outofplane to inplane, which reduces the magnetic stability of the material. In the same way, we determined the critical doping range about 0 to 0.2 e/u.c, as shown in Fig. 7b. Beyond this range, the outofplane magnetization transformed into inplane magnetization as electrostatic doping concentrations increase. Similarly, a range of electron doping from − 0.25 e/u.c. to 0 is also predicted, within which the ferromagnetic stability can be maintained and the \({T}_{C}\) should be increased as the electron doping concentration increases due to the enlarged absolute value of \({\Delta E}_{FMAFM}\) according to the mean field theory^{40}. While the hole doping concentration has little effect on the magnetic ground state.
In order to see the insight of physical mechanisms beneath the variations of MAE, the secondorder perturbation theory is engaged, which indicates that only the occupied and unoccupied Cr d states near the Fermi level make major contributions to MAE in 2D magnetic systems^{41}. Depending on the different spin channels, the contributions to MAE can be divided into two parts^{41,42}, including the same spin polarization and different spin polarizations, namely, \(MAE={E}_{\pm ,\pm }+{E}_{\pm ,\mp }\)^{40}, which are expressed by
where \(o\) and \(u\) denote the occupied and unoccupied states, respectively. The magnetization directions are denoted by \(x\) and \(z\). Positive sign represents spinup states, and reversely, negative sign represents spindown states. \({\varepsilon }_{u}\) (\({\varepsilon }_{o}\)) stands for the energy of unoccupied (occupied) states, and the spin–orbit coupling constant is represented by \(\xi\). Herein, five angular momentum matrix elements between two Cr \(d\) orbitals are nonvanishing^{41}: \(<{d}_{xy}{L}_{x}{d}_{xz}>\), \(<{d}_{{z}^{2}}{L}_{x}{d}_{yz}>\), \(<{d}_{{x}^{2}{y}^{2}}\left{L}_{x}\right{d}_{yz}>\), \(<{d}_{xz}{L}_{z}{d}_{yz}>\), and \(<{d}_{{x}^{2}{y}^{2}}{L}_{z}{d}_{xy}>\). It is seen from Eq. (1) that the outofplane spin polarization is favored for the occupied and unoccupied degenerate states due to the nonvanishing \({{<L}_{z}>}^{2}\) and vanishing \({{<L}_{x}>}^{2}\), while the inplane spin polarization is favored with the nonvanishing \({{<L}_{x}>}^{2}\) for nondegenerate states. In contrary to Eq. (1), for Eq. (2), the outofplane spin polarization is favored for the occupied and unoccupied nondegenerate states due to the nonvanishing \({{<L}_{x}>}^{2}\) and vanishing \({{<L}_{z}>}^{2}\) , while the inplane spin polarization is favored with the vanishing \({{<L}_{x}>}^{2}\) and nonvanishing \({{<L}_{z}>}^{2}\) for degenerate states.
Based on above analyses, we can further understand the variations of MAE under different strains or electrostatic doping concentrations. For pristine bilayer Cr_{2}Ge_{2}Te_{6}, the spinup Cr \({d}_{xy}/{d}_{{x}^{2}{y}^{2}}\) orbitals contribute peaks in both VBM and CBM as shown in Fig. 6a, so the \(<{d}_{{x}^{2}{y}^{2}}{L}_{z}{d}_{xy}>\) in Eq. (1) remained, leading to the positive MAE, which means the outofplane anisotropy is favored. With strain engineering (\(\eta =2\%\)), we can see from the states around the Fermi level in Fig. 6b that MAE is mainly contributed from \({d}_{yz}/{d}_{xz}/{d}_{xy}{/d}_{{x}^{2}{y}^{2}}\) orbitals with different spin states, and the \(<{d}_{xy}{L}_{x}{d}_{xz}>\) and \(<{d}_{{x}^{2}{y}^{2}}{L}_{x}{d}_{yz}>\) remained in Eq. (2), so that we can obtain positive value of MAE, or in another words, the outofplane anisotropy is favored. Also, the reduced energy bandgap in Fig. 6b offers the decreased energy difference between the unoccupied and occupied states in Eq. (2), which results in the enhanced ferromagnetic stability, identical with our results in Fig. 7a. With \(\eta =5\%\) shown in Fig. 6c, MAE are mainly derived from the inverse spinpolarized \({d}_{xz}/{d}_{yz}\), the \(<{d}_{xz}{L}_{z}{d}_{yz}>\) in Eq. (2) maintains, implying the inplane anisotropy is favored, also identical with the results shown in Fig. 7a. From the results in Fig. 6d, the complex competition between orbital interactions may be the reason for maintaining outofplane anisotropy. At small electron doping concentrations (not larger than − 0.1 e/u.c.), the Fermi level can only cross the spinup states due to the obvious spin polarization in bilayer Cr_{2}Ge_{2}Te_{6}, the spinup Cr \({d}_{yz}/{d}_{xz}/{d}_{xy}{/d}_{{x}^{2}{y}^{2}}\) orbitals contribute peaks near the Fermi level in Fig. 6e, the outofplane anisotropy is favored because of the nonvanishing \(<{d}_{xz}{L}_{z}{d}_{yz}>\) and \(<{d}_{{x}^{2}{y}^{2}}{L}_{z}{d}_{xy}>\) in Eq. (2) [see Fig. 7b]. In Fig. 6f, the spinup Cr \({d}_{xy}{/d}_{{x}^{2}{y}^{2}}\) orbitals and spindown Cr \({d}_{xz}{/d}_{yz}\) orbitals contribute peaks near the Fermi level, the \(<{d}_{{x}^{2}{y}^{2}}{L}_{z}{d}_{xy}>\) and \(<{d}_{xz}{L}_{z}{d}_{yz}>\) in Eq. (1) results in the positive MAE, i.e. the outofplane anisotropy [see Fig. 7b].
Conclusion
In summary, we have studied the variations of electronic structures and magnetic properties of bilayer Cr_{2}Ge_{2}Te_{6} with different strains or electrostatic doping concentrations. We proposed a critical strain ranges of − 3% ~ 1% for bilayer Cr_{2}Ge_{2}Te_{6}, within which the ferromagnetic stability can be enhanced, as well as the critical doping range of 0–0.2 e/u.c. While beyond the critical range, the tensile strain induces a phase transition from the ferromagnetic to the antiferromagnetic, which is attributed to the competition between exchange interactions. Moreover, beyond the critical range, the compressive strain or electrostatic doping induced the magnetization direction to change from outofplane to inplane. We also identified a range of electron doping from − 0.25 e/u.c. to 0, within which the magnetic moment and \({T}_{C}\) can be increased, while the ferromagnetic stability was maintained. We have shown two ways for inducing halfmetal in the bilayer Cr_{2}Ge_{2}Te_{6}. The compressive strain induced the reversed electron spin state at the conduction band minimum and the transition from semiconductor to metallic state. The secondorder perturbation theory was applied to explain these variations of MAE. These results illustrated the tunability of electronic structures and magnetic properties by strain and electrostatic doping in the bilayer Cr_{2}Ge_{2}Te_{6} and hopefully shed a light on achieving ferromagnetic stability for lowdimensional materials.
Methods
Ab initio calculations were performed based on density functional theory. The exchange–correlation interaction was treated with the scheme of generalized gradient approximation (GGA) parametrized by the PerdewBurkeErnzerhof revised for solids (PBEsol)^{43} as implemented in the Vienna ab initio Simulation Package (VASP)^{44,45}. The accurate projector augmented wave method (PAW)^{46} was employed for the following electronic configurations: 2p^{6}3d^{5}4s^{1} (Cr), 4s^{2}4p^{2} (Ge), and 5s^{2}5p^{4} (Te). A 500 eV kinetic energy cutoff of the planewave basis set was used for all calculations. The GGA + U was adopted for improving the description of onsite Coulomb interactions to the Cr d orbital^{47}. Different effective onsite Coulomb energy value U_{eff} = UJ were conducted for magnetism, optimized lattice constants, and electronic structures with bilayer Cr_{2}Ge_{2}Te_{6} (as shown in Supplementary Figs. S1, S2, and S3), which indicated that the results of U_{eff} = 1.7 eV were consistent with previous experiments and theoretical calculations^{9,17,48}. The Kmesh of 7 × 7 × 1 was used for structural optimization and others were evaluated with a refined mesh of 20 × 20 × 1 subdivision in the full Brillouin zone. The maximum convergence force of all atoms was optimized until less than 0.01 eV/Å, and the convergence criterion for the energy differences was set as 1 × 10^{–6} eV. To avoid the interaction between adjacent periodic layers, the vacuum space was included larger than 15 Å. To calculate the MAE, the spin–orbit coupling was taken into consideration. The crystal structures in this paper were drawn by VESTA package^{49}.
References
Novoselov, K. S. et al. Electric feld efect in atomically thin carbon flms. Science 306, 666–669 (2004).
Song, Y. H. et al. Observation of Coulomb gap in the quantum spin Hall candidate singlelayer 1T’WTe_{2}. Nat. Commun. 9, 4071 (2018).
Deng, J. L. et al. Epitaxial growth of ultraflat stanene with topological band inversion. Nat. Mater. 17, 1081 (2018).
Li, L. et al. Black phosphorus feldefect transistors. Nat. Nanotechnol. 9, 372 (2014).
Qiao, J., Kong, X., Hu, Z.X., Yang, F. & Ji, W. Highmobility transport anisotropy and linear dichroism in fewlayer black phosphorus. Nat. Commun. 5, 4475 (2014).
Song, C. S. et al. Tunable band gap and enhanced ferromagnetism by surface adsorption in monolayer Cr_{2}Ge_{2}Te_{6}. Phys. Rev. B 99, 214435 (2019).
Sivadas, N., Daniels, M. W., Swendsen, R. H., Okamoto, S. & Xiao, D. Magnetic ground state of semiconducting transitionmetal trichalcogenide monolayers. Phys. Rev. B 91, 235425 (2015).
Li, X. X. & Yang, J. L. CrXTe_{3} (X = Si, Ge) nanosheets: two dimensional intrinsic ferromagnetic semiconductors. J. Mater. Chem. C 2, 7071 (2014).
Fang, Y. M., Wu, S. Q., Zhu, Z.Z. & Guo, G.Y. Large magnetooptical effects and magnetic anisotropy energy in twodimensional Cr_{2}Ge_{2}Te_{6}. Phys. Rev. B 98, 125416 (2018).
Gong, C. et al. Discovery of intrinsic ferromagnetism in twodimensional van der Waals crystals. Nature 546, 265 (2017).
Mermin, N. D. & Wagner, H. Absence of ferromagnetism or antiferromagnetism in one or twodimensional isotropic heisenberg models. Phys. Rev. Lett. 17, 1133 (1966).
Hatayama, S. et al. Inverse resistance change Cr_{2}Ge_{2}Te_{6}based PCRAM enabling ultralowenergy amorphization. ACS Appl. Mater. Interfaces 10, 2725 (2018).
Ji, H. W. et al. A ferromagnetic insulating substrate for the epitaxial growth of topological insulators. J. Appl. Phys. 114, 114907 (2013).
Liu, Y. & Petrovic, C. Anisotropic magnetic entropy change in Cr_{2}X_{2}Te_{6} (X = Si and Ge). Phys. Rev. Mater. 3, 014001 (2019).
Pajda, M., Kudrnovsky, J., Turek, I., Drchal, V. & Bruno, P. Oscillatory Curie temperature of twodimensional ferromagnets. Phys. Rev. Lett. 85, 5424–5427 (2000).
Irkhin, V. Y., Katanin, A. A. & Katsnelson, M. I. Selfconsistent spinwave theory of layered Heisenberg magnets. Phys. Rev. B 60, 1082–1099 (1999).
Sun, Y. Y. et al. Electric manipulation of magnetism in bilayer van der Waals magnets. J. Phys. Condens. Matter 31, 205501 (2019).
Wang, Z. et al. Electricfield control of magnetism in a fewlayered van der Waals ferromagnetic semiconductor. Nat. Nanotechnol. 13, 554–559 (2018).
Xing, W. et al. Electric field effect in multilayer Cr2Ge2Te6: a ferromagnetic 2D material. 2D Materials 4, 024009 (2017).
Castro, E. V. et al. Biased bilayer graphene: semiconductor with a gap tunable by the electric field effect. Phys. Rev. Lett. 99, 216802 (2007).
Xia, F., Farmer, D. B., Lin, Y.M. & Avouris, P. Graphene fieldeffect transistors with high on/off current ratio and large transport band gap at room temperature. Nano Lett. 10, 715–718 (2010).
Sun, Y. et al. Effects of hydrostatic pressure on spinlattice coupling in twodimensional ferromagnetic Cr2Ge2Te6. Appl. Phys. Lett. 112, 072409 (2018).
Lin, Z. et al. Pressureinduced spin reorientation transition in layered ferromagnetic insulator Cr_{2}Ge_{2}Te_{6}. Phys. Rev. Mater. 2, 051004 (2018).
Ma, C., He, X. & Jin, K.J. Polar instability under electrostatic doping in tetragonal SnTiO_{3}. Phys. Rev. B 96, 035140 (2017).
Cao, T., Li, Z. & Louie, S. G. Tunable magnetism and halfmetallicity in holedoped monolayer GaSe. Phys. Rev. Lett. 114, 236602 (2015).
Ma, C., Jin, K.J., Ge, C. & Yang, G.Z. Strainengineering stabilization of BaTiO_{3}based polar metals. Phys. Rev. B 97, 115103 (2018).
Wang, K. et al. Magnetic and electronic properties of Cr_{2}Ge_{2}Te_{6} monolayer by strain and electricfield engineering. Appl. Phys. Lett. 114, 092405 (2019).
Pustogow, A., McLeod, A. S., Saito, Y., Basov, D. N. & Dressel, M. Internal strain tunes electronic correlations on the nanoscale. Sci. Adv. 4, 12 (2018).
Shukla, V., Grigoriev, A., Jena, N. K. & Ahuja, R. Strain controlled electronic and transport anisotropies in twodimensional borophene sheets. Phys. Chem. Chem. Phys. 20, 22952–22960 (2018).
Fang, S., Carr, S., Cazalilla, M. A. & Kaxiras, E. Electronic structure theory of strained twodimensional materials with hexagonal symmetry. Phys. Rev. B 98, 075106 (2018).
Jiang, L.T. et al. Biaxial strain engineering of charge ordering and orbital ordering in HoNiO_{3}. Phys. Rev. B 97, 195132 (2018).
Yang, D. et al. Cr_{2}Ge_{2}Te_{6}: high thermoelectric performance from layered structure with high symmetry. Chem. Mater. 28, 1611–1615 (2016).
Tian, Y., Gray, M. J., Ji, H., Cava, R. J. & Burch, K. S. Magnetoelastic coupling in a potential ferromagnetic 2D atomic crystal. 2D Materials 3, 025035 (2016).
Chen, X., Qi, J. & Shi, D. Strainengineering of magnetic coupling in twodimensional magnetic semiconductor CrSiTe_{3}: competition of direct exchange interaction and superexchange interaction. Phys. Lett. A 379, 60–63 (2015).
Goodenough, J. B. Theory of the role of covalence in the perovskitetype manganites[La, M(II)]MnO_{3}. Phys. Rev. 100, 564–573 (1955).
Goodenough, J. B. An interpretation of the magnetic properties of the perovskitetype mixed crystals La_{1}_{−}_{x}Sr_{x}CoO_{3}_{−}_{λ}. J. Phys. Chem. Solids 6, 287 (1958).
Kanamori, J. Superexchange interaction and symmetry properties of electron orbitals. J. Phys. Chem. Solids 10, 87 (1959).
Kang, S., Kang, S. & Yu, J. Effect of Coulomb interactions on the electronic and magnetic properties of twodimensional CrSiTe_{3} and CrGeTe_{3} materials. J. Electron. Mater. 48, 1441–1445 (2018).
Xu, C., Feng, J., Xiang, H. & Bellaiche, L. Interplay between Kitaev interaction and single ion anisotropy in ferromagnetic CrI_{3} and CrGeTe_{3} monolayers. NPJ Comput. Mater. 4, 57 (2018).
Kittel, C. Introduction to Solid State Physics (Wiley, New York, 2004).
Wang, D., Wu, R. & Freeman, A. J. Firstprinciples theory of surface magnetocrystalline anisotropy and the diatomicpair model. Phys. Rev. B Condens. Matter 47, 14932–14947 (1993).
Lee, S.C. et al. Effect of Fe–O distance on magnetocrystalline anisotropy energy at the Fe/MgO (001) interface. J. Appl. Phys. 113, 023914 (2013).
Perdew, J. P. et al. Restoring the densitygradient expansion for exchange in solids and surfaces. Phys. Rev. Lett. 100, 136406 (2008).
Kresse, G. & Furthmuller, J. Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set. Phys. Rev. B 54, 11169–11186 (1996).
Kresse, G. & Furthmuller, J. Efficiency of abinitio total energy calculations for metals and semiconductors using a planewave basis set. Comput. Mater. Sci. 6, 15–50 (1996).
Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmentedwave method. Phys. Rev. B 59, 1758–1775 (1999).
Dudarev, S. L., Botton, G. A., Savrasov, S. Y., Humphreys, C. J. & Sutton, A. P. Electronenergyloss spectra and the structural stability of nickel oxide: an LSDA+U study. Phys. Rev. B 57, 1505–1509 (1998).
Carteaux, V., Brunet, D., Ouvrard, G. & Andre, G. Crystallographic, magnetic and electronicstructures of a new layered ferromagnetic compound Cr_{2}Ge_{2}Te_{6}. J. Phys. Condens. Matter 7, 69–87 (1995).
Momma, K. & Izumi, F. VESTA 3 for threedimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr. 44, 1272–1276 (2011).
Acknowledgements
This work was supported by the National Key Basic Research Program of China (Grant 2019YFA0308500), the National Natural Science Foundation of China (Grant Nos. 11721404, 51761145104, 11974390 and 11674385), the Key Research Program of Frontier Sciences of the Chinese Academy of Sciences (Grant No. QYZDJSSWSLH020), the Youth Innovation Promotion Association of CAS (Grant No. 2018008). We acknowledge Zhicheng Zhong and Peiheng Jiang for the helpful discussions.
Author information
Authors and Affiliations
Contributions
W.N.R. and K.J.J. contributed the whole idea and designed the research. W.N.R., K.J.J., and J.S.W. wrote up the paper. W.N.R. performed the theoretical calculations. C. M. prepared Fig. 5. K.J.J. supervised the overall project. All authors discussed the results and commented on the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher's note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Ren, Wn., Jin, Kj., Wang, Js. et al. Tunable electronic structure and magnetic anisotropy in bilayer ferromagnetic semiconductor Cr_{2}Ge_{2}Te_{6}. Sci Rep 11, 2744 (2021). https://doi.org/10.1038/s4159802182394y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s4159802182394y
This article is cited by

Research progress of twodimensional magnetic materials
Science China Materials (2023)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.