What is the effective mass of electrons in graphene?

[FranzDiCoccio]

Hi all,

according to textbook definition the effective mass of a particle in a periodic potential is

$$\frac{\hbar^2}{m*} = \frac{d^2}{d k^2} E(k)$$

where $$E(k)$$ is the energy dispersion.
Is this definition applicable at a generic point of a band, or only at the center and edge of the Brillouin zone, where the band is actually "curved"?
The above definition results in an infinite mass at a point where the band is flat, where by flat I mean zero curvature.

The reason why I'm asking these question is related to the peculiar (tight-binding) band structure of graphene. The two bands of this material touch at two points in (two-dimensional) k-space. Since the local structure of the two bands around these points is conical, under suitable conditions electrons are expected to behave as free massless relativistic particles.

I sort of see this, in view of the similarity between the local dispersion around the special points of graphene and the dispersion of the Dirac equation for free particles.

However, I'm confused by the statement that an electron has an effective mass of zero [Physics Today, Jan 06, p. 21] . This is not the same effective mass $$m*$$ defined above, is it?
Because it seems to me that it should be infinite, since the local dispersion is flat...

I'm aware that one can define different effective masses. Does this mean that in this particular case the electron has both an infinite effective mass (according to one definition) and zero effective mass (from another point of view)?

Or perhaps the definition of $$m*$$ does not apply at the special points of graphene Brillouin cell?This is perhaps related to a https://www.physicsforums.com/showthread.php?t=277782" about silicon.

Thanks a lot for any insight

F

 

[FranzDiCoccio]

I have found an https://www.physicsforums.com/showthread.php?t=99223" about this subject.
Unfortunately, it does not clarify my doubts completely.

I understand that one should be careful in making a parallel between electrons and relativistic particles. But, even with this proviso in mind, it is not clear to me to what extent electrons do behave as free relativistic particles.

I'm in the process of reading the "serious literature", but I have some difficulty in interpreting the experimental findings. Plus, it seems to me that most of them are about quantum Hall effect, which adds more things on top of the relativistic parallel...

What's the experimental signature of the electron being relativistic as opposed to classical?

Is this just a matter of dispersion relation, like lonewolf seems to be suggesting in the last post of the thread I mention?

And what about the "other effective mass"?

 

[Cthugha]

What's the experimental signature of the electron being relativistic as opposed to classical?

Is this just a matter of dispersion relation, like lonewolf seems to be suggesting in the last post of the thread I mention?

And what about the "other effective mass"?

It means, that the linear approximation E=vp of the dispersion relation is valid.
Note that this also means, that the average speed of the particle has a fixed value, which does not depend on the energy of the particle.

 

[FranzDiCoccio]

Hi Cthuga,

and thanks a lot for your reply.

It means, that the linear approximation E=vp of the dispersion relation is valid.
Note that this also means, that the average speed of the particle has a fixed value, which does not depend on the energy of the particle.

Yes, after writing my post it occurred to me that one could think in terms of
speed as well. Perhaps I should actually focus on speed rather than on effective mass.
But to what extent this is "fictitious" and to what extent it is "real"?

Is this spectral feature of graphene actually mirrored in the average speed of
electrons? Can you point me to some experimental result (also a "thought experiment"
would do) which highlights this feature of graphene, and show that the situation with graphene is entirely different from the usual one?

Thanks a lot again

F

 

[Cthugha]

I am afraid I do not know much about graphene. About the only paper I know of, is this classical one:

Two-dimensional gas of massless Dirac fermions in graphene (Nature 438, 197-200 (10 November 2005)) by Novoselov et al.

 

[FranzDiCoccio]

I have that paper, but for some reason I do not find it very helpful.
This "relativistic parallel" is still a bit elusive to me. I understand the similarity
between the dispersion surfaces, but I do not find it completely satisfactory.
So far I have the impression that it is mostly a decoy to capture the reader's
(editor's) attention, but probably this is just due to my illiteracy in
the subject.
I'm trying to get better bearings by reading more literature, but if any of the
forum people has any illuminating comment, it is mostly welcome.

F

 

[edguy99]

but if any of the
forum people has any illuminating comment, it is mostly welcome.

Thank you, I will try one. I hope someone with working knowledge can help.

Usually the carbon atom (6 electrons) is thought of as 2 electons in an inner orbital (less then 53pm from the center - trapped at fairly high energies). The next 2 electrons are caught in the 2s oribital on either side of the molecule at say 77pm. The last 2 electrons are a little farther out in the 2p orbital. Carbon binds into diamond with the 2s and 2p orbitals fairly far apart. Carbon binds into organic molecules (hydrocarbons, etc) with the 2s and 2p orbitals very close together. Carbon in this form generally binds into tetrahedral type structures.

Graphene is quite a bit different. Here, one of the 2p electrons is close in energy to the 2s level and the other is "almost" free. The two 2s electrons and the one 2p electrons make 3 electrons that bind the carbon into sheets with one electron orbital left that sticks above the sheet that binds the sheets into layers. This electron between the sheets is what is readily available to move around.

 

[f95toli]

This "relativistic parallel" is still a bit elusive to me. I understand the similarity

You are not alone. The "connection" to the Dirac equation and relativistic physics is very controversial. I've been to meetings where this has been discussed and also have collegues who work with graphene (for the record: I don't, so I am NOT an expert, my knowledge is mainly tea-room based).
There are plenty of people who work in the field that thinks that this is a bit of a ploy; it looks could when applying for a grant but that does not mean that there is anything of substance.

 

[FranzDiCoccio]

Thank you, I will try one. I hope someone with working knowledge can help.

Hi edguy 99,

and thanks for your input! The filling is surely an important ingredient of the whole thing,
and at some point I was actually wondering why on Earth it should be 1/2 in graphene.
I have found basically the same answer as you are suggesting in a paper dating back to 1947
(a Phys Rev by one Wallace).

But, important as it might be, half filling is just the starting point, in my view.
I mean, half filling (or somewhere in its vicinity) is the condition in which the odd dispersion
of graphene is expected to have some role.

What I was asking is: assuming that we're around half-filling, in what sense the electrons
behave relativistically? What's the relativistic signature in their behaviour, and can I tell it apart from their "usual" behaviour
in an experiment?

 

[FranzDiCoccio]

You are not alone. The "connection" to the Dirac equation and relativistic physics is very controversial. I've been to meetings where this has been discussed and also have collegues who work with graphene (for the record: I don't, so I am NOT an expert, my knowledge is mainly tea-room based).
There are plenty of people who work in the field that thinks that this is a bit of a ploy; it looks could when applying for a grant but that does not mean that there is anything of substance.

Hi f95toli,

and thanks for replying. I have actually the same impression as you. But for some reason I'm willing to resist it.
I mean, are Nature's and Physical Review Letters' referees and editors so gullible to accept
a paper just because the authors do some handwaving and write cool things like "bench-top high-energy physics"?
There must be some meat under all of this smoke.

My (optimistic) impression is that one must be careful in drawing the parallel. I have the
feeling that "relativistic effects" might be actually there, although they might have been
known for ages under different names. I mean, they could be well known effects in
condensed matter physics that could have a "relativistic" interpretation.
I'm not sure whether this new point of view is anywhere near useful, because I still don't get it.
But if it has some real basis, I'd like to understand it.

 

[edguy99]

What I was asking is: assuming that we're around half-filling, in what sense the electrons
behave relativistically? What's the relativistic signature in their behaviour, and can I tell it apart from their "usual" behaviour
in an experiment?

I don't know if this would be relativistic, quantum or just plain wierd. Above a sheet of graphene (or between 2 sheets), a grid type energy is going to form with the dominating feature being the big heavy carbon protons that are bound by the 3 electrons. This grid has high points and low points of energy and that's where the electrons fill in. Adding an electron into this grid, does not just change the distribution of electrons like springs, but causes an electron in that grid spot (or orbital) to jump to a grid spot right beside it. If this spot is occupied, the electron will pop to the next and so on. This seems more like a quantum effect rather then a relativistic effect since the electrons are traveling less then 1% of the speed of light.

Someone has posted:

"If the string of zeroes are electrons in the graphite and electron A kicks in
resulting in electron B being kicked out at the other side then the speed
of A to B would appear to be extremely high.

A ---> 00000000000000000000000000000000000000000000
------ 00000000000000000000000000000000000000000000 ----> B"

 

[FranzDiCoccio]

I don't know if this would be relativistic, quantum or just plain wierd. Above a sheet of graphene (or between 2 sheets), a grid type energy is going to form with the dominating feature being the big heavy carbon protons that are bound by the 3 electrons. This grid has high points and low points of energy and that's where the electrons fill in. Adding an electron into this grid, does not just change the distribution of electrons like springs, but causes an electron in that grid spot (or orbital) to jump to a grid spot right beside it. If this spot is occupied, the electron will pop to the next and so on.

I'm not sure I'm following you... What do you mean by "grid type energy"?
Probably it is a limit of mine, but your referring to protons and orbitals confuses me.
Plus, most of the papers I've seen ultimately deal with a simple "pure-hopping" tight-binding
model, assuming half filling.
I'd like to understand the (allegedly) cool relativistic effect in this framework.

This seems more like a quantum effect rather then a relativistic effect since the electrons are traveling less then 1% of the speed of light.

Well, surely it has a quantum origin. And I'd say that here "relativistic" does not mean that
the speed of electrons is anywhere near c, since it is obviously not true.
I'm not even sure any more that the "relativistic" particle is the electron.
Perhaps it is something "more structured".
As far as I understand (but I am by no means sure) the pseudospin appearing in the Dirac
equations involves both of the sites in a graphene unit cell.

"If the string of zeroes are electrons in the graphite and electron A kicks in
resulting in electron B being kicked out at the other side then the speed
of A to B would appear to be extremely high.

A ---> 00000000000000000000000000000000000000000000
------ 00000000000000000000000000000000000000000000 ----> B"

This might have something to do with what I say above... But perhaps it has not..
I should probably read the whole thread again.

Thanks for your inputs

 

[iamquantized]

T. Ando had a paper on graphene. He performed a k.p expansion about the Dirac point and arrived at an equation which has the exact form as the Dirac equation.

A more theoretical treatment would be this paper "Condensed-Matter Simulation of a Three-Dimensional Anomaly" by GW Semenoff - Physical Review Letters, 1984