# Group velocity dependency on unit cell size

Hi Prof. Julian,

I am writing this queries in order to get your guidance with regard to the group velocity (vg) dependency on the unit cell size of single-layer graphene. (Sorry for a long query )

The following group velocity is obtained from lattice dynamics based software GULP. The average vg is coming from the allowed wave vectors in the first BZ.

No. of
basis atoms Avg. vg(km/s)
4 6.8197
12 6.23368
20 5.87847
28 5.6254
36 6.0097
52 4.5093
100 4.4921

Seeing the above data it seems like the average group velocity obtained for the same supercell size of 60X30 Angs^2 does change and that too with a decreasing trend.

Based on the above observations I have few doubts:
(1) Does the average group velocity depend on the no. of basis atoms?
(2) Is the decrease in the vg coming from the BZ-folding? If yes, even then the vg should be same as the folding is the mirror image which should not be affecting the slope of the dispersion curve (i.e. |vg|)
(3) Or are these obseravtions correct but the reason is something different?

I have one more data where I did compare the vg dependency on the size of the supercell.

basis-atoms X-length Y-length vg(km/s)

4 60 30 6.8197

4 90 30 6.6148

4 120 30 6.6418

4 60 60 6.6102

4 60 90 6.8682

4 60 120 6.7505

The above data shows that the average vg for same no. of basis atoms but different supercell size shows somewhat similar group velocity.

Another data set is the following:

basis-atoms X-length Y-length vg(km/s)

100 120 30 4.4921

100 60 30 4.8016

Which shows that the supercell size do not have much effect for same basis atoms.

Seeing the above two data sets, I guess the no. of basis atoms do have effect on the vg but for the same basis atoms the the supercell size do not have much effect on vg. Does it establish the fact of zone-folding? And if this is true then it becomes difficult to chose a particular number of basis atoms while calculating the vg in order to get ultimately get thermal conductivity?

Also, just to cross-verify, dispersion curve for any set of basis atoms can be obtained, isn’t it?

With kind regards.

Kunwar Abhikeern

Research Scholar, Mechanical Engineering Department

Indian Institute of Technology,Bombay(400076),India

Hi Kunwar,
I think I can answer most of the points with some general information. The important thing to understand is that averaging values of anything related to phonons within a cell is just another way of saying “Brillouin zone integration” & there is a vast literature on this topic for phonons and electronic structure. Provided you converge the values with respect to the k point sampling using a Monkhorst-Pack mesh then the answer should be independent of the cell choice. Of course, as the cell gets larger then the number of k points decreases inversely to the supercell size. Once you appreciate this then all should become clear.
Regards,
Julian

1 Like