Gamma and X point electrons in semiconductors

[Stu777]

Hi,

I am trying to get a feel for electrons in semiconductros (say GaAs) with regards to the bandstructure of the material. I understand that the velocity of the electron can be written as ~ d(Energy)/d(wavevector) so at band extrema such as Gamma-point or X-point this goes to zero.

However, what I would like to know is what is the difference between a Gamma-point electron in say GaAs compared to an X-point electron? From the band diagram it is clear that the X-point electron has a higher energy but after that I start to struggle. Does the electron at the X-point have a certain momentum but no velocity for example? And, how could a Gamma-point electron become an X-point electron (would it require phonons for the momentum)?

Finally, if an electron is between the Gamma and X-points in the bandstructure does this say something about the direction or position that the electron has in the real lattice?

If anyone can shed some light that would be very helpful

Cheers

Stu

 

[ZapperZ]

Hi,

I am trying to get a feel for electrons in semiconductros (say GaAs) with regards to the bandstructure of the material. I understand that the velocity of the electron can be written as ~ d(Energy)/d(wavevector) so at band extrema such as Gamma-point or X-point this goes to zero.

However, what I would like to know is what is the difference between a Gamma-point electron in say GaAs compared to an X-point electron? From the band diagram it is clear that the X-point electron has a higher energy but after that I start to struggle. Does the electron at the X-point have a certain momentum but no velocity for example? And, how could a Gamma-point electron become an X-point electron (would it require phonons for the momentum)?

I haven't looked at the band structure of GaAs in quite a while, so I'm going on with this based on memory. So someone else, please correct me here if I get this wrong.

The Gamma point is the center of the band, compared with the X-point which is the edge of the Brillouin zone in one of the symmetry direction. If you plot the band dispersion, you'll see another major difference : the curvature of the band in those two places. The 2nd derivative of E vs k is positive at the Gamma point, while it is negative at the X-point. This also means that if you have a filled band up to the X-point, the electrons will have negative mass (shock rings through the crowd who are involved in two different "negative mass" threads on PF).

Now, the only way for the electrons at those two locations to "switch places" or make a transition is via the SAME wavevector symmetry operation. Since these two are at the high symmetry locations, they are separated by one reciprocal wave vector, and so only "something" that can carry such wave vector can cause the transition. If GaAs has that phonon wavevector in its spectrum, then it could conceivably do that, but I'm not so such about this.

Finally, if an electron is between the Gamma and X-points in the bandstructure does this say something about the direction or position that the electron has in the real lattice?

Not position, but definitely momentum, because you are, after all, in momentum space. "Position" is rather meaningless at this point because you have mostly plane wave bloch function, and this implies that the electron is spread out all over the crystal.

Zz.

Edit: P.S. I'm assuming that the X-point is not along the zone diagonal. If it is, you could have a saddle-point there, and so the electronic mass could have different signs depending on the momentum vector.

 

[Stu777]

Thanks Zz,