### 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.

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

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

(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)

-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

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

### Task 3

Check the lecture notes for the free electron model and:

### 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

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