The k-points arise from Bloch's theorem, which say that for a periodic solid, the wave function is given by psi_k = u_k e^(ikr). Quantities like the electron densities are integrals over k,
n(r) = \int |psi_k(r)|^2 dk,
and the selection of an appropriate k-point grid ensures that these Brillouin zone integrals are converged.
There is no general rule about how many k-points to use, it usually has to be found out from a convergence analysis. But note very carefully that k-point sampling is use for at least 2 separate things in ATK:
1) In the selfconsistent loop, the electron density in Fourier space is expanded - these are the "usual" k-points that appear in any DFT code etc. A rule of thumb is given by experts from the
Technical University of Denmark (DTU):
The product, k*a, between the number of k-points, k, in any direction, and the length of the basis vector in this direction, a, should be
k*a ~ 30 Å, for d band metals
k*a ~ 25 Å, for simple metals
k*a ~ 20 Å, for semiconductors
k*a ~ 15 Å, for insulators
Note that there can also be a strong difference between odd and even meshes, since you only include the Gamma point in odd meshes. However, then you have special cases like graphene, where the important part of the Brillouin zone is not at G but at K, and you should have a multiple of 3 k-points; in consequence, 14 is much less accurate than 12.
2) For the computation of the transmission spectrum, you are essentially sampling the tail of the wave function, i.e. how much of an electron is propagating through your system. Such transmission amplitudes can be very sharply varying due to resonant tunneling. For example, to accurately model the minority transmission for the parallel spin case in a FeMgO MTJ you need 300x300 k-points for T(E), but just 9x9 for the electron density. For simpler cases, you can get good results with 6x6 or 7x7.
The main point is, don't just pick a number for either case 1 or 2 blindly, but check it! Vary it a bit to see if the results change - plot the transmission k-point resolved, etc.
Of course, in quasi-1D and 2D structures you just set the sampling in the finite direction to 1.
