On 09/02/2005 10:07:10 AM, ? ?? 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
