Table of Contents

Crystallographic point groups, free electron model

Starting from 2006 Hafnium silicates replaced SiON as gate oxide in MOSFETS. The high dielectric constant of HfO2 and the ability of HfO2 to form silicates played a key role in the industrial transition.

connect to hypatia: 
ssh -X EMPA-USER@jump1.empa.ch
ssh -X hypatia
module load povray
module load python/2.7.12

go in the directory where you want to put the exercise and do: 

cp /home/cpi/exercise_10.tar ./
tar -xvf exercise_10.tar
cd exercise_10

Task1

Follow the instructions contained in the ASE webpage dedicated to spacegroups

https://wiki.fysik.dtu.dk/ase/ase/spacegroup/spacegroup.html

and find in this article (a pdf copy is included in the tar file of the exercise)

https://journals.aps.org/prb/abstract/10.1103/PhysRevB.65.233106

all data necessary to construct the monoclinic phase of HfO2

modify the python script hfo2.py according to your needs.

(replace “…”)

Execution of the python script 

python hfo2.py

will open the ASE visualizer showing you the structure, will produce the file hfo2.xyz, hfo2.png and hfo2.pov

to create a “nice” image of the primitive cell you can render the output file hfo2.pov with the command 

povray +W320 +H320 -I./hfo2.pov -Ohfo2 +P +X +A +FJ +C

or executing the script 

./povray.sc

(that will take care of removing files hfo2.jpg and hfo2.inc)

-how many atoms are contained in the unit cell?

-compute the volume of the unit cell

Task2

Have a look at the atomic coordinates, for example in the file hfo2.xyz (where you also find the cell vectors in cartesian coordinates) and

try to reproduce them (just the 4 Hf atoms), starting from the coordinates that you find in the article and applying the symmetry operations of the space group:

1)x,y,z

2)-x,y+1/2,-z+1/2

3)-x,-y,-z

4)x,-y+1/2,z+1/2

DO NOT FORGET Periodic Boundary Conditions!! but it's simple: x,y,z are given in crystal coordinates so, if for example x=0.276 for “-x” you can use -x+1 = 0.724

usually the coordinates are provided in crystal coordinates so if a1=(a1x,a1y,a1z),a2=(a2x,a2y,a2z),a3=(a3x,a3y,a3z) are the three basis vectors of the crystalin cartesian coordinates and (x1,y1,z1) will be the crystal coordinates of atom 1, the cartessian coordinates of atom 1 will be: x1*a1 + x2*a2 + x3*a3

Task 3

Check the lecture notes for the free electron model and:

Compute the Fermi energy (in eV) and the Fermi wavevector (in cm-1) for Cu,Au,Ag

Task 4

Have a look at this ASE page to compute bandstructures and the symmetry points of the Brillouin zone of a crystal:

https://wiki.fysik.dtu.dk/ase/ase/dft/kpoints.html

Compute the free electron bandstructure of Si and Cu (Have a look at the scripts included in the exercise directory)

Write the CARTESIAN COORDINATES of the Gamma, X, W points of FCC