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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.
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 co construct the monoclinic phase of HfO2
Execution of the python script will open 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)
Have a look at the atomic coordinates, for example in the file hfo2.xyz (where you also find teh 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
Check the lecture notes for the free electron model and:
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