# Confusion regarding on whether to include conduction charges(intrinsic to the material) as a part of bound charges in the calculation of polarization

Consider a semi-conductor in which conduction is mainly due to free electrons(valence electrons that are free to roam throughout the material) , and since it's a semi-conductor say not all atoms contribute these free electrons(some do,some don't ) and under the application of an external electric field these free electrons start moving in the material and the other non-contributing atoms(atoms that do not contribute any electrons for conduction) get stretched as in the case of a dielectric .I have the following questions :

1 )The neutral non-contributing atoms get stretched as in the case of a pure dielectric,but do the contributing ionised atoms also stretch a bit? I do think they stretch but i wonder if they contribute anything to the polarization vector $$\vec P$$ .

2 ) In calculating $$\vec P$$ should we also consider the free conduction electrons ,because they are also from the atoms of the material and we want the total bound charge to be zero ,i.e when we write Div.$$\vec P =-\rho_b$$ , is the charge due to these conduction electrons included in $$\rho_b$$? I think it should be included because we wish that the continuity equation hold for these bound charges and that requires total bound charge to be constant,if these conduction charges are not included then continuity equation won't hold since an atom which is non-contributing might release an electron if the electric field let's say is increased ,then this electron now no longer is bound,hence for this reason i think all the charges belonging to the material be counted as bound.

3)If the charge due to conduction electrons is also included in $$\rho_b$$,then is the conduction current also somehow included in the polarization current $$\frac{d\vec P}{dt}$$?