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Hi all,

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

[tex]\frac{\hbar^2}{m*} = \frac{d^2}{d k^2} E(k)[/tex]

where [tex]E(k)[/tex] 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

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 [tex]m*[/tex] 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

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

[tex]\frac{\hbar^2}{m*} = \frac{d^2}{d k^2} E(k)[/tex]

where [tex]E(k)[/tex] 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 [tex]m*[/tex] 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 [tex]m*[/tex] 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

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