What is inversion symmetry in solids and why does it break at the surface?

[amjad-sh]

Hello
Can somebody explain for me what is the meaning of inversion symmetry in solids?
and why does it breaks at the surface?
and also why this inversion symmetry breaking leads to SOC(spin orbit coupling)?
If somebody also know a document that explain this in full details(from A to Z) please send it here.
Thanks

 

[DeathbyGreen]

Inversion symmetry is simply flipping a system and seeing that it looks the same after flipping. So if you have something, then you turn it upside down, and it looks the same, you would say that object has inversion symmetry. For example, a layered system ABC -> CBA, or $z \rightarrow -z$, $E(k,\uparrow)\rightarrow E(-k,\uparrow)$. Breaking inversion symmetry means that the system does not look the same after flipping it. This leads to a non-centrosymmetric arrangement of lattice sites and creates an electric field gradient throughout the system, and since the electric field can couple to the crystal momentum, creates spin orbit coupling. Does that help? This textbook might be helpful: https://link.springer.com/chapter/10.1007/978-3-540-36616-4_5