the path in the Brillouin zone
Dear all, I used kwant to calculate the band structure of zigzag and armchair graphene nanoribbons. For zigzag graphene nanoribbons, we have sym= kwant.TranslationalSymmetry(lat.a.vec((-1, 0))). For armchair graphene nanoribbons, we have sym= kwant.TranslationalSymmetry(lat.a.vec((-1, 2))). My question is: which is the path in the Brillouin zone that we use and label the x-axis accordingly? My figure is attached. Thanks in advance! Weiyuan
Dear Weiyuan,
Since there's only a single translationally invariant direction, the
Brillouin zone is one-dimensional, and therefore you are viewing the
complete 1D Brillouin zone. If there was a finite order translation
invariance in the transverse direction like in a nanotube, each band
would correspond to a different discrete eigenvalue of momentum, as
described in a lot of educational materials, e.g. [1]. However in a
nanoribbon the momentum in a transverse direction is not conserved,
and therefore the 1D bands cannot in general be mapped on a 1D path in
a 2D Brillouin zone.
Best,
Anton
[1]: https://nanoelectronics.unibas.ch/education/Nanotubes/LCAO-NT.pdf
On Tue, Jan 17, 2017 at 5:59 AM, Weiyuan Tong
Dear all, I used kwant to calculate the band structure of zigzag and armchair graphene nanoribbons. For zigzag graphene nanoribbons, we have sym= kwant.TranslationalSymmetry(lat.a.vec((-1, 0))). For armchair graphene nanoribbons, we have sym= kwant.TranslationalSymmetry(lat.a.vec((-1, 2))). My question is: which is the path in the Brillouin zone that we use and label the x-axis accordingly? My figure is attached. Thanks in advance! Weiyuan
Dear Anton,
Thanks for your help!
Best,
Weiyuan
On Tue, Jan 17, 2017 at 5:43 PM, Anton Akhmerov wrote: Dear Weiyuan, Since there's only a single translationally invariant direction, the
Brillouin zone is one-dimensional, and therefore you are viewing the
complete 1D Brillouin zone. If there was a finite order translation
invariance in the transverse direction like in a nanotube, each band
would correspond to a different discrete eigenvalue of momentum, as
described in a lot of educational materials, e.g. [1]. However in a
nanoribbon the momentum in a transverse direction is not conserved,
and therefore the 1D bands cannot in general be mapped on a 1D path in
a 2D Brillouin zone. Best,
Anton [1]: https://nanoelectronics.unibas.ch/education/Nanotubes/LCAO-NT.pdf On Tue, Jan 17, 2017 at 5:59 AM, Weiyuan Tong
Dear all,
I used kwant to calculate the band structure of zigzag and armchair
graphene
nanoribbons. For zigzag graphene nanoribbons, we have sym=
kwant.TranslationalSymmetry(lat.a.vec((-1, 0))). For armchair graphene
nanoribbons, we have sym= kwant.TranslationalSymmetry(lat.a.vec((-1,
2))).
My question is: which is the path in the Brillouin zone that we use and
label the x-axis accordingly? My figure is attached. Thanks in advance!
Weiyuan
participants (2)
-
Anton Akhmerov
-
Weiyuan Tong