Andrii - do you mean what positions do the two surface layers take up when you relax them?

If I take your question as read, I would start by constructing a slab of Si with vacuum on one side (either pin a couple of layers to bulk positions and allow 2-3 unrestricted monolayers on top, or have at least 5 monolayers so that the internal one *might* represent the bulk) and performing a geometry optimisation (or relaxation). The manual for whichever DFT code you prefer to use should detail how to perform this type of calculation, and there should be an example tutorial file available too.

If you just want to know values that someone else has calculated, I suggest searching the literature for silicon 001 surface dimer... Or look at Dabrowski & Scheffler, App. Surf. Sci. 56-58:15-19 (1991) and the references they cite.

Thank you for the answer Daniel.

The question was how to find the initial positions of atoms in the unit cell (out of symmetry group or any tables?) so to perform a geometry optimization. Suppose,we choose two atomic layers on surface in a certain crystallographic direction say 100 or 111 with vacuum. What would be the positions for those 3 (?) atoms? Because this is not supposed to be a supercell that might contain whole lattice.

And thanks for the refference!

Ah - the basic cubic 8-atom cell has atoms at:

0 0 0

0 0.5 0.5

0.5 0 0.5

0.5 0.5 0

0.25 0.25 0.25

0.25 0.75 0.75

0.75 0.25 0.75

0.75 0.75 0.25

in fractional units. (You can find this on wikipedia, or in any textbook on semiconductor physics, or probably at some US National Lab site where they have XRD data for many crystals, or mentioned in almost any thesis involving diamond or silicon, or in the seminal works on silicon.) The lattice parameter will depend on what code/basis/pseudo you are using - you'll have to test this yourself. Run a few 8-atom test single-point energy calculations, varying the lattice parameter both sides of the experimental value of 5.43A and plot; you should get a parabola, and you'll want to set the lattice parameter to the value at the parabola's minimum.

Once you have that basic cell set up, you can easily multiply the fractional coordinates by the lattice parameter to get the Cartesian coordinates. Then just extend as necessary to get a surface large enough to study the effect you are looking for.

The problem with only taking two layers in your model is that you will effectively have vacuum on both sides of the slab (assuming periodic boundary conditions or unpinned bottom layer) and thus neither monolayer will behave like it does when attached to bulk silicon on one side. The atomic positions will certainly deviate from bulk positions in the outside two monolayers, and possibly more (though increasingly less as you go deeper into the crystal). If you are after a realistic model you should include more layers so that at least one of them represents the bulk physics below the "surface" and you can see the changes up to the dimer rows that will form on the 100 surface.

Doing the 111 surface is a little trickier, but if you have access to some programs like Diamond or VESTA (free) you can happily create a slab and then cleave it along a particular plane to give you the surface you want, saving the output. The DFT code CRYSTAL09 includes a way of building slabs like this in the actual code - you specify the cleave planes etc. in the input file.

Either way, you are certainly going to need more than three atoms in your cell.

It clarifies a lot =)

Thank you for such a helpful answer, Deniel. References on software are really valuable.

P.S. Why I thought of such a small amount of atoms for a surface case... it was a mistaken suggestion only, which goes out of a correct way of calculating the bulk Silicon with 2 atoms per unit cell though.

0.0 0.0 0.0

1/4 1/4 1/4

That gave me a feeling that one might add an atom on top of it to get a "slab" unit cell in 111 direction

I would build a unit cell according to ME Fleet (1981) Acta Crystallographica B, 37, 917 :

Space Group Fd-3m (No. 227)

a=8.3941 Å

a=90.00

Z=8

Atomic Positional Parameters

Fe1 8a 0.1250 0.1250 0.1250

Fe2 16d 0.5000 0.5000 0.5000

O 32e 0.2549 0.2549 0.2549

With a software like Avogadro it is possible to fill the unit cell, multiply it and to build a slab.