Strain-controllable electronic, magnetic properties, and magnetic anisotropy energy in a 2D ferromagnetic half-metallic MGT monolayer

In recent years, two-dimensional (2D) materials with strong magnetic anisotropy and high Curie temperature $( T c)$ have received much attention.^1–3 These materials, possessing ultrathin thickness and the absence of dangling surface bonds, exhibit heightened sensitivity to external modulation, facilitating the fine-tuning of their magnetic properties.⁴ However, the Mermin–Wagner theory says that 2D magnetic materials cannot exist in the isotropic Heisenberg model at finite temperature.⁵ Hence, 2D ferromagnetic (FM) materials are rare to find.^2,6–8 Researchers have used many methods to find more FM materials, increase $T c$^9, and enhance magnetic stability by the processes of modulating^10–12 and synthesizing.^12–15 In recent years, CrI₃,^12,16–20 Fe₃GeTe₂,^2,13,21–25 VS₂,^14,26–29 and CrGeTe₃ (CGT) with intrinsic ferromagnetism have been successfully synthesized in the experiments.^15,30–34 Using the density functional theory (DFT), Zhang et al. demonstrated that chromium trihalide (SLCT) (CrX₃, X = F, Cl, Br, and I) monolayers (MLs) constitute a series of stable 2D semiconductors with an intrinsic FM order.¹⁷ Further exploration into magnetic anisotropy energy (MAE) was conducted by Webster et al. They investigated the strain dependence of the MAE in 2D chromium trihalides CrX₃ (X = Cl, Br, and I) MLs.¹⁹ In a pioneering study, Zhuang et al. predicted that a mechanically exfoliated Fe₃GeTe₂ ML had strong perpendicular magnetic anisotropy (PMA) with a MAE of 0.92 meV/f.u.²² Furthermore, Kim et al. found antiferromagnetic (AFM) coupling induced by oxide formation in the Fe₃GeTe₂ layer,²⁵ further enriching interfacial effects in these systems. Ma et al. found that isotropic strain enhances the magnetic properties of a pristine 2D VX₂ (X = S, Se) ML, with both magnetic moment (MM) and coupling strength increasing accordingly.²⁸ Chittari et al. investigated the electronic and magnetic properties of 2D transition-metal chromium-based phosphates MPX₃ (M = V, Cr, Mn, Fe, Co, Ni, Cu, Zn, and X = S, Se, Te).¹⁴ However, transition metal dichalcogenides (TMDCs) are predominantly nonmagnetic. Researchers have identified CGT as a ferromagnet and demonstrated that strain is an effective method of tuning magnetic properties.³¹ Previous studies of CGT were based on defect and compositional engineering or proximity effects, which introduced magnetic response only locally or externally. Therefore, Gong et al. investigated the intrinsic long-range FM order in pristine Cr₂Ge₂Te₆ atomic layers and achieved unprecedented control over the transition temperature.¹⁵ On the basis of successfully synthesized 2D CrI₃ and CGT, Huang et al. made a significant enhancement of ferromagnetism by lowering the virtual exchange gap through heterovalent alloying.³² 

There are relatively few 2D FM materials that have been synthesized, and therefore, in this study, we perform calculations to explore novel materials and uncover their properties. Ideal 2D magnetic materials are expected to have a high $T c$⁠,³⁵ a large MAE,^35,36 controllable electromagnetic properties,³⁶ and so on. For a half metal (HM),³⁷ one spin channel is insulating or semiconducting in nature, while the other channel is conducting.³⁸ As a result, HMs could get 100% spin polarized current.³⁰ Moreover, HMs are expected to have a higher $T c$⁠, and the gap should be large enough to prevent the thermally agitated spin-flip transition and preserve half-metallicity at room temperature.³⁹ In addition, a large MAE is essential to generate a potential well, to stabilize the process of magnetization in a certain direction (easy magnetization direction) against thermal fluctuations.⁴⁰ Therefore, magnetization will not be affected by thermal fluctuation.^1,41 Consequently, a larger MAE is needed.^15,41 In the synthesis of 2D materials, substrates are necessary, but a lattice mismatch between substrates and 2D materials often poses a challenge. Accordingly, strain should be considered.^3,35 Additionally, biaxial strain could effectively control both magnetic and electronic properties, as the states near the Fermi level are predominantly supplied by in-plane atomic orbitals.^42–44 Therefore, we use strain to regulate the magnetic properties in this work.

In this article, we performed a study on the properties of Mn₂Ge₂Te₆ (MGT), using the DFT. Mn₂Ge₂Te₆ shows an intrinsic FM order, which originates from the super-exchange interaction of the Mn and Ge atoms. Mn₂Ge₂Te₆ is a HM, whose spin-β electron is a semiconductor with an indirect gap of 1.462 eV. The magnetic easy axis (EA) prefers to be an in-plane magnetic anisotropy (IMA), originating from indirect spin–orbital coupling (SOC). Mn₂Ge₂Te₆ shows good dynamical and thermal stability. Our results indicate that Mn atoms ferromagnetically couple with each other under tensile and compressive strains $( ε < 10 %)$⁠. The energy difference between FM and AFM orders $( Δ E)$ initially decreases, which is attributed to the change in exchange interaction between Mn and Te atoms when the tensile strain increases. The magnetic exchange parameter (J) also changes as the strains change the direct and super-exchange interactions. Additionally, Mn₂Ge₂Te₆ could be transferred from a HM to a spin-polarized metal as the compressive strain $( ε < 2 .1 %)$ is increased. However, Mn₂Ge₂Te₆ is still a HM with a FM order under tensile strains. The MM changes under strains for the different charge transfers. The MM initially increases as the tensile strain increases. Mn₂Ge₂Te₆ tends to be an IMA, with a MAE of −13.2 meV/f.u. When the larger compressive strain $( ε > 11 %)$ is applied, the EA could be switched from the [100] to the [001] direction. The EA remains in the plane under tensile strains and compressive strains $( ε < 11 %)$⁠. The MAE monotonously increases as the tensile strain increases. However, the MAE monotonously decreases as the compressive strains increase. MAE's monotonous decrease origins that the contribution of hybridization between Te's p_y and p_z oribitals to the MAE is changed when the strain changes. Moreover, Mn₂Ge₂Te₆ shows good dynamic stability under strains. Our results provide controllable magnetoelectric properties of Mn₂Ge₂Te₆, which are useful for new magnetoelectric devices.

In this study, we performed first-principles calculations within the framework of a spin-polarized DFT using the Vienna Ab initio Simulation Package (VASP).⁴⁵ Electron exchange interactions were described by the generalized gradient approximation (GGA)⁴⁶ parameterized Perdew–Burke–Ernzerhof (PBE) method.^47,48 The Mn's 3d electron was studied with hybrid-functional HSE06^48,49 and the LDA + U method. The energies with different orders, band structures, and density of states (DOS) were calculated by using the HSE06 functional, while the MAE, phonon spectra, and molecular dynamics were examined by the LDA + U method. Chittari et al. investigated carrier- and strain-tunable intrinsic magnetism in two-dimensional MAX₃ transition metal chalcogenides and found that it was appropriate to take the value of U_eff as 4 meV.⁵⁰ The on-site effective Coulomb interaction parameter (U) was set to 4.60 eV, and the exchange interaction parameter (J₀) was set to 0.60 eV. Accordingly, the effective $U eff$ $( U eff = U − J 0)$ was 4.00 eV,^30,38 and the corresponding magnetic and electronic properties were consistent with the HSE06 functional. A vacuum space of 16 Å in the z direction was implemented to prevent virtual interaction. The kinetic energy cutoff was set as 300 eV for optimizing the geometry and calculating the energy. The geometries were fully relaxed until energy and force converged to 10⁻⁶ eV and 1 meV/Å, respectively. Also, $9×9×1$⁠, $16×16×1$ Monkhorst−Pack grids were used for geometry optimization and energy calculation,⁵¹ respectively.

For the calculation of the MAE, we employ a nonlinear mode with the SOC effect. In the MAE calculation, the total energy converges to $1× 10 − 8 eV$⁠. The MAE is usually small, due to the impact of SOC. Because the calculation of the MAE is sensitive to the parameters, the k-mesh test is performed. A $19×19×1$ k-mesh is adopted without any symmetric constriction, as shown in Fig. S1 in the supplementary material. The MAE is calculated with an energy cutoff of 400 eV and a total energy convergence of $1× 10 − 8 eV$⁠. The phonon spectra and DOS are calculated using the finite displacement method as implemented in Phonon Package.⁵² A $4×4×1$ cell is adopted in the calculation. The total energy and the Hellmann–Feynman force converges to $10 − 8 eV$ and 1 meV/Å in the phonon spectra calculation, respectively. Six thousand uniform k-points along high-symmetry lines are utilized to obtain phonon spectra.

We performed GGA calculations to investigate the influence of strain effects on the electromagnetic properties of the Mn₂Ge₂Te₆ ML.⁵¹ Before studying the Mn₂Ge₂Te₆ ML system, we optimized the lattice parameter of Mn₂Ge₂Te₆. The optimized geometries of the Mn₂Ge₂Te₆ ML are depicted in top, side 1 (along the x axis) and side 2 (along the y axis) views, as shown in Figs. 1(a), 1(b), and 1(c), respectively. The corresponding optimized lattice parameter is a = b = 6.968 Å, which is larger than 5.989 Å of CGT,^15,53 and 6.881 Å of Co₂Ge₂Te₆.³⁴ The difference in lattice parameter values origins that the radius of the Mn atom (137 pm) is larger than that of the Cr (125 pm) and Co atoms (125 pm). The bond length between Mn and Te atoms is 2.915 Å, while the bond length between Ge and Te atoms is 2.617 Å. The bond length between Ge and Ge atoms is 2.477 Å. The distance between the Mn layer and the Ge layer is 1.239 Å, while the distance between the Te layer and the Ge layer is 3.769 Å. From the optimized geometry, we can find that the Mn₂Ge₂Te₆ ML presents the D_3d point group, which is the same with CGT¹⁵ and Co₂Ge₂Te₆.⁵⁴ Also, we can see that the Mn atom is at the center of the octahedron.

We used a $2×2×1$ supercell, each containing two Mn atoms. Thus, there are two magnetic orders: FM and AFM orders. The total MM of the FM order is $8.00 μ B$⁠, while the AFM order is $0.00 μ B$⁠. The spin charge densities of the FM and AFM orders are shown in Figs. S2(a)–S2(d) in the supplementary material, respectively. From the spin charge density, we can see that $4.365 μ B$ MM is mainly localized in the Mn atom with a high-spin octahedral d⁶ configuration, while the MM of the Ge and Te atoms are −0.020 $( 0.01 × 2)$ and −0.876 $( 0.146 × 6)$ $μ B$⁠, respectively. To describe magnetic stability, the $ΔE$ is defined as $ΔE= E AFM− E FM$⁠. The $ΔE$ is 0.123 eV, indicating that the Mn₂Ge₂Te₆ ML shows the FM order. The ground state is determined by the competition between the direct exchange and the super-exchange interactions, as shown in Figs. 1(d) and 1(e), respectively. In Mn₂Ge₂Te₆, the super-exchange interaction is stronger than the direct exchange interaction, and the Mn-d and Te-p orbitals' super-exchange interaction is dominant in determining the order of Mn₂Ge₂Te₆, which is similar to CrI₃ and CGT.¹⁵ The super-exchange interaction arising from the hybridization of the Mn-d and Te-p orbitals dominates, resulting in a FM order of the Mn₂Ge₂Te₆ ML. The states near the Fermi level are mainly contributed by the Te'p orbitals and partially provided by Mn's $d x y$ and $d y z$ orbitals, shown in Figs. 1(f)–1(i). The energy of the FM and AFM orders changes at different strains, which are shown in Fig. S3 in the supplementary material.

Here, we calculate the Te-p_y, Te-p_x, Mn-d_xz, and Mn-d_yz projected band structures of the Mn₂Ge₂Te₆ ML, as shown in Figs. 1(f)–1(i), respectively. The partial density of states (PDOS) of Mn atoms' d orbitals is also calculated with the HSE06 functional, as shown in Fig. S4 in the supplementary material. The charge densities of the VBM contributed by the spin-β electrons at the Γ point are also calculated with the HSE06 functional, as shown in Fig. S5 in the supplementary material. It can be concluded that the spin-α electrons partially occupy the Fermi level, which implies that its channel is conductive. In contrast, the spin-β electron channel is insulating in nature. Therefore, Mn₂Ge₂Te₆ is a HM. There are eight Mn atoms in a $2×2×1$ cell, and the total MM for the FM order amounts to $32.0 μ B$⁠. Moreover, three different AFM orders are considered, including AFM-zigzag (AFM-Z), AFM-stripy (AFM-S), and AFM-Néel (AFM-N) orders. For the AFM order, four Mn atoms have $16.0 μ B$⁠, while the other four Mn atoms have $−16.0 μ B$⁠. Consequently, the total MM of the AFM order equals $0.0 μ B$⁠. The spin charge density difference is shown in Fig. S2 in the supplementary material. The highest energy of the AFM-Z order is 0.637 eV higher than the FM order, and the second highest energy of the AFM-N order is 0.614 eV, as shown in Figs. S2(b) and S2(d) in the supplementary material, respectively.

The AFM-stripy (AFM-S) order exhibits an energy of 0.457 eV, which is higher than that of the FM order, as illustrated in Fig. S2(c) in the supplementary material. The corresponding $J 1$⁠, $J 2$⁠, and $J 3$ are 13.6, 7.5, and 12.0 meV for the Mn₂Ge₂Te₆ ML, as shown in Fig. S2(e) in the supplementary material. Both the nearest- and the next-nearest-neighbor Mn atoms exhibit a FM order. It can be concluded that $J 3$ can be compared with $J 1$⁠. Similar phenomena occur in other materials such as NiCl₂, NiBr₂, and NiI₂.¹⁶ Notably, both the nearest-neighbor and the next nearest-neighbor Mn atoms show FM coupling.

Ma et al. reported two intrinsically ferromagnetic vdW materials with $T c$ higher than room temperature, including the $T c$ of MnGeTe₃ up to 349 K. We also calculated that the $T c$ is about 376 K, as shown in Fig. S6 in the supplementary material, which is in general agreement with this result.⁵⁵ The mechanical properties of Mn₂Ge₂Te₆ are also investigated. Mn₂Ge₂Te₆ is applied with a strain ranging from −10% to 10%. When installing low-dimensional materials on a brittle substrate, bending and stretching strains can be applied experimentally.^56,57 The corresponding strains can be as high up to 10%. In addition, strains can be applied by nanoindentation under atomic force microscopy.⁵⁶ Supersaturation strains can be applied using a diamond anvil cell.⁵⁸ 

In-plane strains are often used to modulate magnetism,⁵⁹ including the MMs of atoms, the energies of different magnetic orders, and the exchange interaction between magnetic atoms.

In Mn₂Ge₂Te₆, the strain can change the atomic MM and enhance or attenuate the FM order. The MMs of various elements constantly change with the direction and magnitude of the applied strains. As shown in Fig. 2(a), the MMs of the Mn and Ge atoms monotonously increase with the strains $( − 2 % < ε < 10 %)$⁠. In contrast, the MM of the Te atom monotonously decreases as the compressive strain decreases, and it also decreases as the tensile strain increases. The MMs of the Mn, Ge, and Te atoms are (4.389, 0.016, −0.15), (4.418, 0.022, −0.155), and (4.45, 0.028, −0.162) $μ B$⁠, respectively, applying compressive strains of −3%, −6%, and −9% to the Mn₂Ge₂Te₆ ML. When the tensile strains of 3%, 6%, and 9% are applied to the Mn₂Ge₂Te₆ ML, the MMs of the Mn, Ge, and Te atoms are (4.356, 0.007, −0.14), (4.373, 0.008, −0.103), and (4.361, −0.005, −0.093) $μ B$⁠, respectively. This suggests that the occupation of each atom near the Fermi level varies with strains, as shown in the DOS plot in Fig. S7 in the supplementary material. Simultaneously, the $Δ E$ varies with in-plane strains, as shown in Fig. 2(b). $J 1$⁠, $J 2$⁠, and $J 3$ also vary with the strains, as shown in Fig. 2(c). Mn₂Ge₂Te₆ maintains the FM order upon strains, while AFM-stripy still has the second lowest energy. The minimum energy difference between the AFM-stripy and the FM orders $Δ E AFM - Stripy$ $( Δ E AFM - Stripy = E AFM - Stripy − E FM)$ also varies with the strain. The $Δ E AFM - Stripy$ of the Mn₂Ge₂Te₆ ML are 0.372, 0.071, and 0.231 eV, respectively, applying compressive strains of −3%, −6%, and −9%. The $Δ E AFM - N e ´ el$ $( Δ E AFM - N e ´ el = Δ E AFM - N e ´ el − E FM)$ are 0.420, 0.266, and 0.388 eV, respectively, and the $Δ E AFM - Zigzag$ $( Δ E AFM - Zigzag = E AFM - Zigzag − E FM)$ are 0.393, 0.123, and 0.257 eV, respectively. The corresponding $J 1$⁠, $J 2$⁠, and $J 3$ are (12.5, 5.4, 5.0), (6.7, −1.1, 4.4), and (11.3, 1.6, 4.8) meV. When the tensile strains of 3%, 6%, and 9% are applied to the Mn₂Ge₂Te₆ ML, the $Δ E AFM - Stripy$ of the Mn₂Ge₂Te₆ ML are 0.359, 0.334, and 0.225 eV, respectively. The $Δ E AFM - N e ´ el$ are 0.645, 0.664, and 0.463 eV. Also, the $Δ E AFM - Zigzag$ are 0.642, 0.652, and 0.536 eV. The corresponding $J 1$⁠, $J 2$⁠, and $J 3$ are (11.3, 5.6, 15.6), (10.8, 5.0, 16.8), and (4.7, 4.7, 14.6) meV. These findings suggest that in-plane strains can effectively modulate direct exchange and super-exchange interactions.

This is most likely caused by a change in electron occupation in the d orbitals. As shown in Fig. S4 in the supplementary material, we plotted the PDOS of the Mn atom and found that the electron occupancy of the Mn atom's d orbitals changed. We speculate that the strain modulates the MM, direct exchange interaction, and super-exchange interaction.

Biaxial strains are also used to control the electronic properties of 2D materials.⁵⁵  $ΔE$ monotonously increases as the tensile strain increases, and Mn₂Ge₂Te₆ is a HM under tensile strains. When the tensile stains of 2%, 4%, 6%, 8%, and 10% are applied, the corresponding $d Mn - Mn$ is 4.130, 4.184, 4.264, 4.345, and 4.425 Å. Under 12% enlarged strain, the corresponding lattice parameter a and $d Mn - Mn$ are 7.804 and 4.506 Å. As the lattice parameter increases, the corresponding $d Mn - Mn$ also increases. It means that the direct exchange and super-exchange interactions weaken, causing a decrease of $ΔE$⁠.

Mn₂Ge₂Te₆ maintains the FM order, as $ΔE$ is always negative under strains. This underscores the superior stability of the FM order. $ΔE$ monotonously increases with the application of tensile strains, as shown in Fig. 3(a). $ΔE$ is −133.55 meV without strain, and $ΔE$ is −149.58 meV under 2% tensile strain. When the tensile strain increases to 6%, $ΔE$ is −195.18 meV. $ΔE$ further increases to −214.08 meV under 8% tensile strain. This indicates that the FM order becomes more stable with the application of tensile strains. However, the effects of compressive strains are complex. $ΔE$ is −86.17 meV under −2% compressive strain, while $ΔE$ is −85.25 meV under −4% strain. $ΔE$ is −101.4 meV under −8% compressive strain. $ΔE$ monotonously increases under the strains $( − 2.7 % < ε < 12 %)$⁠. In contrast, $ΔE$ monotonously decreases under the strains $( − 7 % < ε < − 2.7 %)$⁠. When applying larger compressive strains $( − 10 % < ε < − 7 %)$⁠, $ΔE$ monotonously increases again.

Moreover, strain affects the electronic structure of the Mn₂Ge₂Te₆ ML. The Mn₂Ge₂Te₆ ML is a HM with $E g−β$ of 1.462 eV without strain. The band structures change with the biaxial strains, as shown in Figs. 3(b)–3(g). $E g−β$ changes with tensile strain, while Mn₂Ge₂Te₆ is always a HM. $E g−β$ is 1.244 eV under 2% tensile strain, as shown in Fig. 3(e). $E g−β$ increases to 1.342 eV when 6% tensile strain is applied, as shown in Fig. 3(f). $E g−β$ follows the relationship $E g−β=1.417+10 .41 ε (−0.02≤ε≤0 .06 )$⁠. $E g−β$ is 1.359 eV, under 8% tensile strain, as shown in Fig. 3(g). When −2% compressive strain is applied, $E g−β$ is 1.185 eV, and Mn₂Ge₂Te₆ remains a HM, as shown in Fig. 3(d). When −2.7% compressive strain is applied, $E g−β$ becomes 0 eV and the Mn₂Ge₂Te₆ ML transforms into metal. $E g−β$ remains 0 eV, when compressive strains $( − 10 % < ε < − 2.7 %)$ are applied, as shown in Figs. 3(b) and 3(c), respectively. This suggests that the occupation of each atom near the Fermi level varies with strains, as shown in Fig. S7 in the supplementary material.

The shift of the MM from a soft axis (EA) to a hard axis necessitates energy expenditure to overcome the “energy barrier.”³⁶ This required energy is referred to as MAE.⁴¹ Materials with high MAE demonstrate enhanced magnetic stability.⁶⁰ 

When −10% and −6% strains are applied, the $Δ E 0$ could be calculated by using the following equations: $Δ E 0 ( meV )$ $=−0.603 cos 2⁡θ+0.027 cos 4⁡θ$ and $Δ E 0 ( meV )=−7.17 cos 2⁡θ$ $+1.50 cos 4⁡θ$⁠. The corresponding MAEs are −0.546 and −5.63 meV/f.u. When the tensile strains of 6% and 10% are applied, the $Δ E 0$ could be obtained from the following equations: $Δ E 0 ( meV )=−14.4 cos 2⁡θ−1.14 cos 4⁡θ$ and $Δ E 0 ( meV )=−3.3 cos 2⁡θ$ $−0.359 cos 4⁡θ$⁠. The corresponding MAEs are −15.71 and −18.49 meV/f.u. Also, we calculate that the EA direction of MnGeTe₃ is along the [100] direction, and the MAE of MnGeTe₃ is about 13 204 μV, which is the same as the calculations of Chittari et al.⁵⁰ 

To further explain the variation of the MAE with strains, we also calculated the atomic orbital decomposition MAE, as shown in Figs. 5(a)–5(i). It can be concluded that the MAE partly comes from Mn [Figs. 5(a)–5(c)] and Ge atom contributions [Figs. 5(d)–5(f)] but mainly from Te atom contributions [Figs. 5(g)–5(i)]. Wang. et al. investigated the ferroelectric control of magnetic anisotropy in a multiferroic heterostructure EuSn₂As₂/In₂Se₃ and found that the f(p) orbitals of Eu (Sn and As) atoms were the primary contributors to the SOC-MAE.⁶⁴ They found that the contributions from the As and the Sn atoms stemmed from the interorbital couplings between $p y$ and $p z$ orbitals, as well as between $p x$ and $p y$ orbitals, which are similar with our results. The orbital-resolved MAE of intrinsic Mn₂Ge₂Te₆ without strain is shown in Figs. 5(a), 5(d) and 5(g). The total MAE is −13.40 meV/f.u. Te atoms provide −11.95 meV/f.u. In addition, the Te atoms of Mn₂Ge₂Te₆ contribute −11.76 (2.94 × 4) meV/f.u. to the total MAE, while Mn and Ge atoms' contribution could be negligible. However, Cr atoms provide 0.06 (0.03 × 2) meV/f.u. The Te atoms supply 0.12 (0.03 × 4) and Ge atoms provide −0.050 (−0.025 × 2) meV/f.u. to the total MAE of the CGT ML.⁶⁵ Thus, the MAE of Mn₂Ge₂Te₆ is larger than CGT. When 6% strain is applied, the hybridizations between Mn's d orbital and Ge's p orbital are similar, with neutral Mn₂Ge₂Te₆, as shown in Figs. 5(b) and 5(e). However, the hybridization between Te's p orbitals changes, totally providing −15. 01 meV/f.u. The contribution of hybridization between Te's $p y$ and $p z$ decreases to −18.36 meV/f.u., as shown in Fig. 5(h). Correspondingly, the MAE increases to −15.71 meV/f.u., as shown in Fig. 4(a). When −6% strain is applied to Mn₂Ge₂Te₆, the orbital-resolved MAE of Mn₂Ge₂Te₆ is shown in Figs. 5(c), 5(f) and 5(i). Specifically, the hybridizations between Te's $p x$ and $p z$ orbitals and $p y$ and $p z$ orbitals get weakened, as shown in Fig. 5(i). Consequently, their contribution to the total MAE reduces to −5.28 meV/f.u. Eventually, the MAE decreases to −5.63 meV/f.u., as shown in Fig. 4(a).

Meanwhile, matrix differences for the p orbitals are calculated to clarify the Ge and Te atoms' contribution to the MAE. The matrix differences for the p orbitals, including the $p y$⁠, $p z$⁠, and $p x$ orbitals between the EA along the [100] and [001] directions are calculated, as shown in Table II.

As a result, we can see that the hybridization of the Mn $d y z$ and $d z 2$ orbitals makes positive contributions to the MAE, resulting in a matrix difference of 3 for the d orbitals, as shown in Table I. The hybridization of the $d x y$ and $d x 2 − y 2$ orbitals makes a negative contribution to the MAE, corresponding to a matrix difference of −4 for the d orbital. Compared with the Co₂Ge₂Te₆ ML, the hybridization between Co's $d x y$ and $d x 2 − y 2$ orbitals makes a negative contribution to the MAE (−0.22 and −0.15 meV), which corresponds to the matrix differences of −3 and −4 for the d orbitals, respectively.³⁴ The Ge atoms' contribution to the MAE is negligible compared with that of the Te atoms. The hybridization between the spin-β-occupying $p y$ and the spin-β-occupying p_z orbital of Te is beneficial for the IMA (negative value), corresponding to the matrix difference of −1 for the p orbital. However, the hybridization between the spin-β-occupying $p z$ and the spin-β $p x$ orbital is beneficial for the PMA (positive value), corresponding to the matrix 1 for the p orbital, as shown in Table II. In general, the total MAE is almost dominated by the hybridization between Te's $p y$ and $p z$ orbitals.

The phonon band structure and DOS of the Mn₂Ge₂Ge₆ ML is also calculated with the LDA + U method, as shown in Fig. S8 in the supplementary material. The dynamic stability of Mn₂Ge₂Te₆ after applied strain is confirmed by the phonon dispersion curves and phonon DOS,³⁹ with no obvious virtual phonon modes, as shown in Figs. 6(a)–6(h). For instance, considering the application of 6% tensile strain and −6% compressive strain, the highest vibrational frequencies are 7.389 and 8.135 THz, respectively, as shown in Figs. 6(a) and 6(e). From Figs. 6(b), 6(d), 6(f), and 6(h), we can see that at low frequencies, the contribution mainly comes from Te atoms, while Ge atoms make a higher contribution to the high frequency. The main contribution of the Mn atoms is to the intermediate frequency. This result corresponds to that of the phonon band. We find that the Mn₂Ge₂Te₆ under different strains are thermodynamically stable, resulting from their lowest frequencies being above 0. The highest frequency of intrinsic Mn₂Ge₂Te₆ is 7.39 THz, while these values are 7.36 (9%) and 7.39 THz (6%) under tensile strains, respectively. In contrast, the highest frequencies of Mn₂Ge₂Te₆ are 8.60 (−9%) and 8.14 THz (−6%) under compressive strains, respectively. The highest phonon frequency corresponds to the frequency of the telescopic vibration, which is related to the bonding level.

Thus, the highest phonon frequency is related to its thermal conductivity. A higher telescopic vibration frequency indicates a stronger bond level and enhanced thermal conductivity. Therefore, we find that compressive strain improves the thermal conductivity of the Mn₂Ge₂Te₆ ML.

In summary, our investigation delved into the strain modulation of the electromagnetic state and its impact on the magnetic anisotropy of the Mn₂Ge₂Te₆ ML under in-plane compressive and tensile strains, employing the DFT. We found the intrinsic ferromagnetism of the Mn₂Ge₂Te₆ ML, which arises from super-exchange interactions between Mn and Te atoms. We explored the electronic and magnetic properties of Mn₂Ge₂Te₆ across in-plane strain rates ranging from 10% to −10%. The Mn₂Ge₂Te₆ ML shows a HM under tensile strains, while it can be transferred into a spin-polarized metal under compressive strains. Notably, while Mn₂Ge₂Te₆ maintains the FM order, strain induces a notable alteration in its MM, J, and MAE. This phenomenon shows the sensitivity of MAE to variations of strain, as the contribution of hybridization between Te's $p y$ and $p z$ orbitals to the MAE changes. Meanwhile, Mn₂Ge₂Te₆ always shows good dynamic stability, under both compressive and tensile strains. This effective manipulation of Mn₂Ge₂Te₆ ML magnetism through strain application holds promise for broadening its applications in spintronics.

See the supplementary material for the convergence of MAE is tested with different k-meshes, spin charge density differences and exchange interactions, energy with FM and AFM orders calculated with the HSE06 functional, the Mn d orbitals’ PDOS of the Mn2Ge2Te6 ML, the charge densities of the VBM of spin-β electrons at the Γ point calculated with the HSE06 functional, the Mn orbitals’ PDOS under strains of −8%, −4%, −2% 3%, 6%, and 9%, the phonon band structure and DOS of the Mn2Ge2Te6 ML calculated with the LDA + U method, and the phonon band structure and DOS of the CGT ML calculated with the LDA + U method.

This work received financial support from the Natural Science Foundation of China (Grant No. 11904203), the Fundamental Research Funds of Shandong University (Grant No. 2019GN065), and the Natural Science Foundation of Shandong Province (ZR2023MA019). Dr. Weiyi Wang acknowledges the Postdoctoral Fellowship Program of China Postdoctoral Science Foundation (No. GZC20232540) and is grateful for the computational resources available at the Shanghai Supercomputer Center. The scientific calculations in this paper have been performed on the HPC Cloud Platform of Shandong University. The authors are grateful to Beijing Beilong Super Cloud Computing Co., Ltd for providing the computation resource at the Beijing Super Cloud Computing Center. The numerical calculations in this paper have been done at the Hefei Advanced Computing Center.

The authors have no conflicts to disclose.

Linhui Lv: Conceptualization (lead); Data curation (lead); Investigation (lead); Software (equal); Writing – original draft (lead). Fangyu Zhang: Data curation (supporting); Formal analysis (supporting). Diancong Qi: Data curation (supporting); Formal analysis (supporting); Software (supporting). Zihao Xu: Data curation (supporting); Software (supporting). Weiyi Wang: Data curation (supporting); Formal analysis (supporting). Ya Su: Data curation (supporting); Formal analysis (supporting). Yanyan Jiang: Validation (supporting). Zhaoyong Guan: Project administration (lead); Supervision (lead); Validation (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.