Crystal Structures: Bulk and Slab Calcula3ons

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Jocelyn Boyd
5 years ago
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1 Crystal Structures: Bulk and Slab Calcula3ons

2 Periodic Boundary Condi3ons So far in lab 2 and 3, we have done calcula3ons for small clusters/nanopar3cles with <39 atoms The small clusters were always placed in a box Today we will define what the box represents and how we can use them to simulate various chemical systems.

3 Periodic Boundary Condi3ons The box represents boundary condi,ons which take into account symmetry of a physical system and are used to simulate an infinitely large system. In lab 2 and 3 you simulated a nanopar3cle in a box

4 Periodic Boundary Condi3ons Here is a visual of what you were actually simula3ng with using periodic boundary condi3ons which is an infinitely large system..

5 Periodic Boundary Condi3ons So in this simula3on two nanopar3cles are interac3ng We selected the distance between these nanopar3cles to be large enough that they are not interac3ng. There is no formal op3miza3on that needs to take place when selec3ng this distance.

6 Periodic Boundary Condi3ons: Another Example 1 2 3

7 Periodic Boundary Condi3ons In lab 1 and 2 we have done calcula3ons for small clusters with <39 atoms. However, most materials are much larger and for our purposes infinitely large In order to model infinitely large materials with a computer, we will need to make some approxima3ons. In order to introduce how we can model infinitely large materials, we will need to introduce some terminology

8 Unit Cells The structure of solid materials have a repea3ng papern The Unit Cell is the simplest repea3ng structure in a solid. In the next few slides I will give many 2D examples of solid structures. Let s look a few examples of unit cells and introduce some common terminology. Note that when you actually simulate these materials they will be three dimensional but the principles remain the same.

9 A 2D Example Below is a 2D example of a simple cubic unit cell

10 A 2D Example Next, we can pull out the smallest repea3ng area (or volume in 3D) papern of this crystal structure..

11 A 2D Example And get the Unit Cell for the 2D simple cubic (the simplest repea3ng structure in a solid) A Θ The la3ce constants are the physical dimensions of the the unit cell The lengths A and B of the box and the angle Θ are the laxce constants for this simple 2D examples B

12 A 2D Example! A C! B We can also describe the unit cell with laxce vectors A and B and laxce point C. A la3ce point is a point in a crystal that is repeated several 3mes in the structure The la3ce vectors are vectors where if you translate the laxce point along these vectors you will be at another laxce point. (i.e. LaXce vectors connect laxce points)

13 A 2D Example A Θ B Cubic unit cells are a special case where the laxce constants are all equal and the laxce vectors are all orthogonal (perpendicular). In this 2D example: A = B Θ = 90 o

14 Bulk Calcula3ons Now we can use the periodic boundary condi3ons to simulate a infinitely large material Infinitely large systems in all three direc3ons are known as bulk systems. In order to op3mize the structure of a bulk system we will need to find the op3mal value of the laxce constants of your unit cell. E 1 E 2

15 Common 3D Unit Cells Body Centered Cubic (BCC) Two laxce points per cubic unit cell Below is an example of a 2D structure with 2 laxce points 2 1

16 Common 3D Unit Cells Face Centered Cubic (FCC) This unit cell has 4 laxce points

17 Slab Calcula3ons In Lab 2, we calculated the binding energy of oxygen on the surface of a nanopar3cle. We would like to calculate the binding energy of oxygen on an infinitely large system. However, for bulk there is no surface where you can add an oxygen

18 Slab calcula3ons In order to simulate a surface for an infinitely large material, we can use periodic boundary condi3ons in 2 dimensions and expose the third dimension to vacuum. Below is a diagram of the 2D case:

19 Slab calcula3ons Next lets describe what the different components of the slab represent: x x x Vacuum 3 atomic layers BoPom layers are typically fixed in place to bulk posi3ons. This done since we assume the bopom layers behave as bulk.

20 Slab calcula3ons Also remember that we are s3ll using PBC in all direc3ons (for this example 2 dimensions) We select the vacuum distance to be long enough to not interact with PBC in the z direc3on Vacuum distance is selected to be long enough that two slabs are not interac3ng

21 Slab calcula3ons: Miller Indices Another decision we will need to make is where we cut or cleave the bulk structure

22 Slab calcula3ons: Miller Indices Let s consider a 2D example and define the nota3on for describing the various planes in a bulk material. 1 y 1 x

23 Slab calcula3ons: Miller Indices Let s consider a 2D example and define the nota3on for describing the various planes in a bulk material. 1 y To describe the plane, we need to first write down the where the plane intersect the laxce vectors. If the plane is parallel to a laxce vector then (1, ) The symbol is for when the plane is parallel to the laxce vector. 1 x

24 Slab calcula3ons: Miller Indices Let s consider a 2D example and define the nota3on for describing the various planes in a bulk material. 1 y Then take the reciprocal of (1, ) To get.. (1/1,1/ ) 1 x which is equal to: (1,0) This is known as the miller index of the blue hyperplane shown on the lea.

25 Bulk to Slab Transforma3on 1. Select a plane. The miller index is (1,0) in the plane below. 2. Cut or slice the bulk material along the plane along. 3. Rotate the material and add vacuum.

26 Slab calcula3ons: Miller Indices Let s consider another 2D example 1 y 1 x First get the intercepts of the laxce vectors (1,1) Take the reciprocal (1/1,1/1) which is equal to: (1,1) This is known as the miller index of the blue hyperplane shown on the lea.

27 Bulk to Slab Transforma3on 1. Select a plane. The miller index is (1,0) in the plane below. 2. Cut or slice the bulk material along the plane along. 3. Rotate the material and add vacuum.

28 Comparison of Surfaces

29 Binding sites 3D Gold FCC 100 Slab 3D Gold FCC 100 Slab Top view- looking through the xy plane to the lea

30 Binding sites 3D Gold FCC 100 Slab There are three general binding sites on a surface. The first of these sites is the hollow site. The absorbent is equidistant to 3 or more atoms.

31 Binding sites 3D Gold FCC 100 Slab The next is the bridge site. The absorbent is equidistant to 2 atoms on the surface

32 Binding sites 3D Gold FCC 100 Slab The next is the top site. The absorbent is equidistant to 1 atoms on the surface

Problem with Kohn-Sham equations

Problem with Kohn-Sham equations (So much time consuming) H s Ψ = E el Ψ ( T + V [ n] + V [ n] + V [ n]) ϕ = Eϕ i = 1, 2,.., N s e e ext XC i i N nr ( ) = ϕi i= 1 2 The one-particle Kohn-Sham equations