Update ABACUS_16_04_2025.md

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@@ -23,7 +23,7 @@ Magnetic anisotropy plays a crucial role in maintaining the long-range magnetic
 
  *Figure 1 : (a) Top and side views of monolayer MnSe₂; (b - c) Side and oblique views of AA-stacked bilayer MnSe₂; (d) Definition of polar angle θ and azimuthal angle φ in the spherical coordinate system; (e - f) Energies of magnetic moments of monolayer (e) and bilayer (f) MnSe₂ along different directions.*
 
-The calculation results of the interlayer differential charge density (Figure 2a) indicate that MnSe₂ has a strong interlayer coupling. The researchers further decomposed the contribution of the magnetic anisotropy energy (MAE) to atoms (Figure 2b) and orbitals (Figure 2c - d), and found that the interaction between the $`p_y`$ and $`p_z`$ orbitals of interface Se atoms plays a key role in the transformation of the easy magnetization axis.
+The calculation results of the interlayer differential charge density (Figure 2a) indicate that MnSe₂ has a strong interlayer coupling. The researchers further decomposed the contribution of the magnetic anisotropy energy (MAE) to atoms (Figure 2b) and orbitals (Figure 2c - d), and found that the interaction between the p<sub>y</sub> and p<sub>z</sub> orbitals of interface Se atoms plays a key role in the transformation of the easy magnetization axis.
 
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@@ -33,26 +33,29 @@ The calculation results of the interlayer differential charge density (Figure 2a
 
 According to the second-order perturbation theory, the contribution of electron states to MAE can be expressed by the following formula:
 
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+  <img src="https://dp-public.oss-cn-beijing.aliyuncs.com/community/Blog%20Files/ABACUS_16_04_2025/formula1.webp">
+</center>
 
-where o and u represent the occupied and unoccupied states, respectively. Since the energy difference $`E_{o}-E_{u}`$ between the occupied and unoccupied states appears in the denominator, the states closer to the Fermi level have a greater impact on MAE, while the states far from the Fermi level contribute relatively less.
+where o and u represent the occupied and unoccupied states, respectively. Since the energy difference E<sub>o</sub> - E<sub>u</sub> between the occupied and unoccupied states appears in the denominator, the states closer to the Fermi level have a greater impact on MAE, while the states far from the Fermi level contribute relatively less.
 
-Combined with the electronic structure analysis, the researchers found that in monolayer MnSe₂, the p_z orbital of Se atoms is far from the Fermi level (Figure 3a, 3c), so the coupling between $`p_z`$ and $`p_y`$ is weak; in the bilayer structure, the interlayer coupling causes the $`p_z`$ orbitals of interface Se atoms to hybridize, forming bonding and antibonding states (Figure 3d). The antibonding states split and approach the Fermi level, thus enhancing the coupling between the $`p_y`$ and $`p_z`$ orbitals and making the easy magnetization axis of bilayer MnSe₂ out-of-plane.
+Combined with the electronic structure analysis, the researchers found that in monolayer MnSe₂, the p_z orbital of Se atoms is far from the Fermi level (Figure 3a, 3c), so the coupling between p<sub>z</sub> and  p<sub>y</sub> is weak; in the bilayer structure, the interlayer coupling causes the p<sub>z</sub> orbitals of interface Se atoms to hybridize, forming bonding and antibonding states (Figure 3d). The antibonding states split and approach the Fermi level, thus enhancing the coupling between the  p<sub>y</sub> and p<sub>z</sub> orbitals and making the easy magnetization axis of bilayer MnSe₂ out-of-plane.
 
 In addition, MnSe₂ also exhibits topological properties that change with the number of layers, including the evolution of the Chern number and surface states (Figure 3e - f). The layer evolution of the above electronic structure and topological properties was calculated and verified using the domestic first-principles software ABACUS.
 
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-*Figure 3 here: (a - b) Spin-down band structures of monolayer and bilayer MnSe₂; (c) Projected density of states of $`p_y`$ and $`p_z`$ orbitals of (interface) Se at the Gamma point in monolayer and bilayer; (d) Charge densities of the marked states in (a - c); (e - f) Surface states of monolayer and bilayer MnSe₂.*
+*Figure 3 here: (a - b) Spin-down band structures of monolayer and bilayer MnSe₂; (c) Projected density of states of p<sub>y</sub> and p<sub>z</sub> orbitals of (interface) Se at the Gamma point in monolayer and bilayer; (d) Charge densities of the marked states in (a - c); (e - f) Surface states of monolayer and bilayer MnSe₂.*
 
 Some external regulation methods can also affect the occupation state of the p orbitals of Se atoms, and thus are expected to achieve the regulation of the direction of the easy magnetization axis of the material. Based on this, the researchers systematically studied a variety of external regulation methods. The results show that by changing the interlayer stacking mode (Figure 4a - b), applying charge doping (Figure 4c), introducing biaxial strain (Figure 4d), and replacing non-metal atoms, the direction of the easy magnetization axis of MnSe₂ can be effectively regulated, providing new ideas for realizing the controllable regulation of magnetic anisotropy in 2D magnets.
 
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 <img src="https://dp-public.oss-cn-beijing.aliyuncs.com/community/Blog%20Files/ABACUS_16_04_2025/p4.webp">
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-*Figure 4 here: (a) Top and side views of AB-stacked bilayer MnSe₂; (b) Atom-decomposed MAE of AA and AB stackings; (c - d) Contributions of different atoms to MAE in monolayer MnSe₂ and the changes of $`E_{X}-E_{ea}`$ with doping concentration and in-plane biaxial strain.*
+*Figure 4 here: (a) Top and side views of AB-stacked bilayer MnSe₂; (b) Atom-decomposed MAE of AA and AB stackings; (c - d) Contributions of different atoms to MAE in monolayer MnSe₂ and the changes of E<sub>X</sub> - E<sub>ea</sub> with doping concentration and in-plane biaxial strain.*
 
 ## Conclusion