[Pw_forum] Questions on supercell calculation of MgB2

[Gerardo Ballabio]

On 09/02/2005 10:07:10 AM, 张 洪彬 wrote:
> Dear Paolo:
>  Thank you very much for your reply. However, I am not familiar  
> with supercell calculation, especially how "several k-points of the  
> original cell" refold into the Gamma point of the supercell.

This is standard solid state theory. Basically, a Bravais lattice in  
direct space is identified by its three generating vectors, let's  
call them a_1, a_2, a_3. The reciprocal lattice in k-space is then  
generated by the three vectors b_1, b_2, b_3 such that:

   a_i * b_j = 2 pi delta_ij

where "*" means the dot product (a*b = ax bx + ay by + az bz) and  
delta_ij is Kronecker delta, that is, 1 if i=j and 0 otherwise.
You can easily see that as the "a" grow larger, the "b" become  
smaller, and in particular if you take a supercell, the reciprocal  
space cell is a SUBcell of the original one.

Now, all k-points that only differ by a reciprocal lattice vector  
(that is, a linear combination of b_1, b_2, b_3 with integer  
coefficients) are equivalent. This means that you can "refold"  
k-points, that is, replace each of them with its equivalent point  
*that lies in the unit cell*. If the supercell is N times larger than  
the original cell, there is N-to-1 correspondence, that is, N  
k-points that were distinct in the original reciprocal cell refold  
into THE SAME k-point in the reciprocal of the supercell. And since  
each of those N points had different energy levels, every band in the  
original cell splits into N bands in the supercell.

Gerardo