Help for supercell, Bloch function

[jhzhou2010]

I recently have a puzlle that what is the relation between the periodic part of Bloch function in primitive cell u_nk(r) and that in supercell, e.g., three primitive cells in one of the directions. Are they the same one?



pls give me some hints or explanations!


thanks~~

 

[DrDu]

Well, the unit cell in the reciprocal lattice is smaller in the case of supercells than in the case of normal cells in direct lattice. That is the vector k for the normal lattice will become k'+K in case of the supercell lattice where K may take on 3 values in the case of a supercell which is 3 times longer than a normal cell.
Hence
u_{nk}(x)\exp(ikx)=u'_{NKk}\exp(ik'x)
or
u'_{nKk}(x)=u_{nk}(x)\exp(iKx)
If the length of the original unit cell is d and the length of the supercell is 3d (at least for a space group with orthogonal axes)
then K \in \{0,2\pi/3d,4\pi/3d\}.

 

[jhzhou2010]

Thanks very much for your useful reply!