### 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.

**hfo2.py, PhysRevB…pdf, bands_si.py, bands_cu.py**from this link

### Task1

Construct the primitive cell of the monoclinic phase of HfO2, using ASE and teh information contained in the HfO2 manuscript

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

and find in this article (a pdf copy is included in dropbox)

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 “…”)

**(comment the line “hfo2.write(“hfo2.png”)”)**

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)

-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

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 crystal in cartesian coordinates and (x1,y1,z1) the crystal coordinates of atom 1, the cartesian coordinates of atom 1 will be: x1*a1 + x2*a2 + x3*a3

### 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