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Hello VASP Community,

I hope this message finds you well. I am working on calculations of optical properties for a 2D material using VASP and the SUMO package. While analyzing the data, I observed certain aspects that I would like to discuss and seek clarification on. Here are the details:

In the output data, I have extracted the real and imaginary parts of the dielectric function tensor for my 2D material. The data appears as follows:

Based on my computed result of a 2-dimensional material system, I have noticed that the real part of the dielectric function in the last four columns (xx, yy, zz, and xy) is not zero, contrary to my expectations. I understand that in a 2D material, the properties calculated in the third dimension should ideally be zero or very large, if any.

I have observed a similar behavior in the imaginary part of the calculated results as well. I'm trying to interpret these findings and would greatly appreciate your insights on this matter. Could you please help me understand why the calculated values in these dimensions are not zero, especially considering the 2D nature of the material?

Thank you very much for your time and assistance. I look forward to your valuable guidance and suggestions.

Attached please find the above-mentioned computed result.

Best regards,
Zhao

Dear Zhao,

Different components of the dielectric tensor describe the response of the system to an incident photon with polarization along x, y, z direction. In 2D materials, the components of the dielectric tensor in the out-of-plane direction are usually very different from the in-plane ones, but not zero. For example, you can look at the optical absorption of graphene sheet in A.G. Marinopoulos, L. Reining, A. Rubio, V. Olevano, Phys. Rev. B 69, 245419 (2004).

Also, one should keep in mind that the convergence of the dielectric function in 2D materials can be very slow with the number of k-points, number of conduction bands and the amount of vacuum in the out-of-plane direction. Unconverged results can be quite misleading.

Dear alexey.tal,

Thank you for your comments and explanations.

Here, I try to give some my understanding on your following statement:

In 2D materials, the components of the dielectric tensor in the out-of-plane direction are usually very different from the in-plane ones, but not zero.

The dielectric tensor components in the out-of-plane direction of a 2D material generally cannot be exactly zero due to the presence of electronic interactions and the fundamental nature of electromagnetic interactions. Even though the material is confined to a 2D layer, it's important to keep in mind that the physics of electric fields and electromagnetic interactions are not limited to a specific dimension.

Here are a few reasons why the out-of-plane components of the dielectric tensor are not zero in 2D materials:

1. Quantum Effects: Quantum mechanics governs the behavior of electrons in materials. Even in a 2D material, electrons have wave-like properties that extend beyond the physical boundary of the material. This means that even though the material is confined to a 2D plane, the electric field can still interact with the electrons in the out-of-plane direction to induce polarization.

2. Electronic Polarizability: In the presence of an external electric field, the electrons in a material can respond by polarizing, which means they redistribute slightly. This polarization creates an induced electric field that opposes the external field. In a 2D material, the electrons can still polarize in response to an out-of-plane electric field, resulting in a non-zero out-of-plane dielectric response.

3. Electromagnetic Theory: Maxwell's equations, which describe the behavior of electromagnetic fields, are not constrained by material dimensionality. The equations are valid for both 2D and 3D systems. When an electric field is applied, the induced polarization can occur in all directions, including the out-of-plane direction.

4. Boundary Effects: While the material itself might be confined to a 2D layer, interactions with the environment, neighboring layers, or substrates can influence the dielectric response in the out-of-plane direction. These interactions can prevent the out-of-plane components of the dielectric tensor from being exactly zero.

In summary, the non-zero out-of-plane components of the dielectric tensor in 2D materials are a consequence of the underlying physics of electromagnetic interactions, quantum mechanical behavior of electrons, and the influence of neighboring layers or the environment. While the dielectric response might be significantly different along the in-plane and out-of-plane directions, it cannot be exactly zero due to these fundamental principles.

Regards,
Zhao

Dear alexey.tal,

Regarding your below description:

Different components of the dielectric tensor describe the response of the system to an incident photon with polarization along the corresponding plane xx, yy, zz etc.

It seems that the wording here is so not so clear, so I try to restate and supplement it as follows for clarity:

The dielectric tensor, also known as the permittivity tensor or susceptibility tensor, is a mathematical representation used to describe the response of a material to an external electric field. It characterizes how the polarization of the material changes in response to the applied electric field. In the context of an incident photon, the dielectric tensor components describe the response of the material to the photon's electric field component along different directions.

The dielectric tensor is usually denoted by a matrix with components ε_ij, where i and j represent the directions of the electric field and polarization, respectively. In a Cartesian coordinate system (x, y, z), the tensor components correspond to various combinations of these directions. Let's break down the components of the dielectric tensor and their response to an incident photon's electric field:

1. ε_xx: This component describes the response of the material when an electric field is applied along the x-direction, and the resulting polarization also occurs along the x-direction.

2. ε_yy: Similarly, this component represents the response of the material when an electric field is applied along the y-direction, and the resulting polarization occurs along the y-direction.

3. ε_zz: This component characterizes the material's response to an electric field applied along the z-direction, resulting in polarization along the z-direction.

4. ε_xy and ε_yx: These components describe the material's response to an electric field applied along the x-direction, resulting in polarization along the y-direction (cross-polarization response), and vice versa.

5. ε_xz and ε_zx: These components describe the material's response to an electric field applied along the x-direction, resulting in polarization along the z-direction, and vice versa.

6. ε_yz and ε_zy: These components describe the material's response to an electric field applied along the y-direction, resulting in polarization along the z-direction, and vice versa.

When an incident photon interacts with a material, its electric field component can be decomposed into these different directions. The dielectric tensor components provide insight into how the material's polarization will change when subjected to these electric field components. In practical applications, such as optics and materials science, understanding these responses is essential for predicting and engineering the behavior of materials in various electromagnetic contexts.

Keep in mind that the dielectric tensor's components can be complex numbers, indicating phase differences between the electric field and the resulting polarization. This is particularly important when considering the material's frequency-dependent response, such as in optics and photonics.

Regards,
Zhao

Correct. Thank you for clarifying this.