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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.

please download, PhysRevB…pdf,, from this link


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

Follow the instructions contained in the ASE webpage dedicated to spacegroups

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

all data necessary to construct the monoclinic phase of HfO2

modify the python script according to your needs.

(replace “…”)

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

Execution of the python script


will open the ASE visualizer showing you the structure, will produce the file, (hfo2.png)

-how many atoms are contained in the unit cell?

-compute the volume of the unit cell


Have a look at the atomic coordinates, for example in the file (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:





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

obtain teh HfO2 structure from

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:

Compute the free electron bandstructure of Si and Cu Write the CARTESIAN COORDINATES of the Gamma, X, W points of FCC
Have a look at the scripts included in the exercise directory
How does the Fermi energy that you computed for Cu compares with the one plotted in the bands?
exercises/2018_ethz_mmm/bands_i_2018.txt · Last modified: 2020/08/21 10:15 by