What are the in-plane and out-of-plane permittivity of a dielectric?
for example, I have read that the in-plane permittivity of the graphene is equal to
1+j*sigma/(eps0*omega*tg)
(sigma: conductivity of graphene, eps0 the permittivity of the air, tg is the thickness of the graphene) but its out-of-plane permittivity is equal to 2.5!
Dear Mostafa,
Shortly! The electric permittivity, like many other properties of matter, depends on the crystal orientation and generally anisotropic. That's permittivity is a constant, but this constant changes with crystal (material) direction, when this material is not isotropic (anisotropic).
Mathematically speaking, this may be represented by 2nd order tensor (3x3 matrix). When this matrix is diagonalized, the 3 diagonal elements represent the permittivity constant values across the main directions (x,y,z). When two of theses 3 elements are equal (say x and y), the permittivity in x-y plane is therefore called "in-plane" permittivity, while its value in the 3rd direction (z) is called "perpendicular" permittivity. Obviously, all the 3 component are equal if the material is isotropic. The permitivity is therefore a scalar quantity (zero-order tensor).
Good luck.
(Cont. about graphene)
Concerning graphene, it is a 2D (plane) material. Therefore the in-plane permittivity means the permittivity in the graphene 2D plane (say x-y plane), while out of plane permittivity means permittivity in the normal direction (z).
Hi Dr Mostafa Shabani . I hope the following links could help you:
1.https://ieeexplore.ieee.org/iel5/5135/13956/00642338.pdf
2.https://www.google.com/amp/s/www.researchgate.net/figure/In-plane-a-and-out-of-plane-b-components-of-the-permittivity-extracted-from_fig9_273918180/amp
3.Article In-Plane Versus Out-of-Plane Dielectric Response in the Thin...
Behnam Farid
Muhammad Hamza El-Saba
Thank you for your answers
my conclusion is that the permittivity matrix of such a anisotropic material will be look like the attached image. Am I correct?
Brinkman, K., A. Tagantsev, V. Sherman, D. Su, and N. Setter. "In-plane versus out-of-plane dielectric response in the thin-film relaxor Pb (Sc 1∕ 2 Ta 1∕ 2) O 3." Physical Review B 73, no. 21 (2006): 214112.
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