Parallel plate capacitor with layers of dielectrics in between

[ShayanJ]

Consider a parallel plate capacitor with layers of dielectric between its plates somehow that the interfaces between them are parallel to the plates of the capacitor. If the surface charge density on the plates of the capacitors be $\sigma$ , gauss's law gives $D=\sigma$ which is the same inside all dielectric layers.
But as boundary conditions for interfaces between dielectrics, we have $| D_i-D_j |=\sigma_b^{ij}$ and the fact that the displacement field is the same inside all dielectrics, gives $\sigma_b^{ij} =0$. But I know that there should be a surface density of bound charges on the interfaces which tells me something is wrong in the above argument.
What is that?
Thanks

 

[Born2bwire]

The divergence of the displacement field gives you the free charge, not the bound charge. So the continuity of the normal displacement field tells you that there is no free charge at the interface, not bound charge.

The bound charge is the negative divergence of the polarization density, the difference between the vacuum displacement field and the displacement field.