Fermi velocity in zigzag graphene nanoribbon

[narin]

I tried to calculate the Fermi velocity of the 6 atoms wide graphene nanoribbon structure using the calculateVelocity function as given in the attached file. The velocity is calculated as 136361 m/s as given in the log file. I selected the 10th band with k=[0.475, 0.475, 0.475] looking at at the bandstructure given in the figure attached. I just wanted to know if these band and k-point selection is correct or would you recommend to use another band and k-point?

Thanks a lot.
Narin

[Petr Khomyakov]

The answer really depends on what you want. In this ultra narrow ribbon the Fermi level crosses several bands, unlike pristine graphene with its 6 conical (equivalent) points.

[sonal AGRAWAL]

Dear sir, how can we identify which point should be used for calculating the Fermi velocity. In the pristine graphene case as given in the tutorial K point is selected because of Dirac cone available at that point. In other structure which having bandgap or having more than one band crossed the Fermi level in that case which point should we consider for calculating the Fermi velocity.

Please help me out, sir.
 Thanks & Regards
sonal

[Petr Khomyakov]

The choice totally depends on what you want to do with these velocities, i.e., it is your choice that probably depends on the research problem you are trying to solve.

[sonal AGRAWAL]

Thanks for your reply
Sir, I want to calculate the quantum capacitance of the system for which I need fermi velocity. How can we calculate the total fermi velocity of the system using the calculate velocity function?


Thanks & Regards
sonal

[Petr Khomyakov]

Please clarify what is the total velocity.  I believe the problem is not on the usage of velocity function, but on the problem formulation and choice of the actual approach for quantum capacitance calculations. It would be helpful to see a reference to the theory behind what you are trying to do.

[sonal AGRAWAL]

Thanks for your response
Here I am attaching the reference paper
https://doi.org/10.1016/j.commatsci.2017.09.050
I want to calculate quantum capacitance using the equation given in this reference paper. for which I required fermi velocity.


Thanks & Regards
Sonal

[Petr Khomyakov]

Quantum capacitance is in general defined in terms of density of states: https://en.wikipedia.org/wiki/Quantum_capacitance. For multiple-band case as yours, you would have to sum up DOS defined for each band separately. So, I guess if defined in terms of group velocities, it would be something like sum over inverse velocities corresponding to the bands that are crossed by the Fermi level. For each band, one should also account for state/band degeneracy essentially given by N_ch in the capacitance formula in the "reference" paper.

[sonal AGRAWAL]

Thanks for the help.


Is the value of Nch depends on the band crossing the Fermi level, I have calculated this by calculating the no. of transmission modes.
As you told I have to calculate this for each band that crossing the Fermi level. But in my case, this is the same for all the bands.
Is am I doing something wrong?
please clarify this.
 

[Petr Khomyakov]

It can be the same if degeneracy of all the bands is the same.

[sonal AGRAWAL]

Dear sir,

 As given in the manual of calculate velocity function fermi velocity of graphene is calculated at the valance band maximum band (In the case of graphene band indices 3). In my case as I said multiple bands are crossing the Fermi level than I should calculate the velocity only those valance bands which crosses the fermi level or both conduction and valance band crossing the Fermi level.

2. If bandgap is present in the system in that case which bands should be considered for calculating velocity.


Thanks in advance
sonal

[Petr Khomyakov]

I guess you should calculate velocities for any band that crosses the Fermi level, whether one calls it valence or conduction band. For a system with a gap, you would perhaps need to consider the entire DOS, taking a more general definition of capacitance based on DOS as function of energy.