[Pw_forum] Questions on supercell calculation of MgB2
Gerardo Ballabio
g.ballabio at cineca.it
Fri Sep 2 10:38:09 CEST 2005
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
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