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 <weiyuantongtrans@gmail.com> wrote:
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 <anton.akhmerov+kd@gmail.com
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 <weiyuantongtrans@gmail.com> wrote:
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)
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Anton Akhmerov -
Weiyuan Tong