What kx/ky Values Are Used for Graphene Band Structure Plots?

In summary, the band structures plotted in the link provided are for graphene nanoribbons, not graphene itself. The left graph shows the band structure for a zigzag graphene nanoribbon, which is a 1D structure. The right graph shows the band structure for an armchair graphene nanoribbon. The wave vector for these structures is referred to as "k" since they are 1D systems. The values of k_x and k_y used to plot these band structures depend on the width of the system.
  • #1
Boccard
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TL;DR Summary
Band structure of graphene and its energy dispersion.
Hello,

https://www.researchgate.net/figure/Top-Graphene-nanoribbons-with-zigzag-left-and-armchair-right-edges-with-disordered_fig1_258105786

With what kx/ky values are these band structures plotted?

The energy dispersion of graphene is given by this
1611510931504.png

, yet I am confused to what kx/ky value choices I make to plot the graphs depicted in the link.
 
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  • #2
The band structure of graphene is not the same as the band structure of graphene nanoribbons. Graphene is a two-dimensional system, whereas GNRs are quasi-one dimensional, so ##k_x## and ##k_y## don’t make sense for GNRs.

Also the equation you provide is the tight binding energy of the valence and conduction bands of graphene, whereas the link gives a plot of the complete band structure for nanoribbons.
 
  • #3
This does not make sense to me.

Using that equation I can reproduce the left plot if I plot kx for some values of ky (with the exception of the lowest energy mode; where does this come from?).
 
  • #4
I’m not sure what you mean. There is no two dimensional band structure in a one dimensional system.
 
  • #5
The band structure plotted on the left graph is simply the 2D energy dispersion function of the graphene, but for specific k_y values that depend on the width of the system. The discretized bands of the system are different k_y values.

My question is, what are these values``?
 
  • #6
Boccard said:
The band structure plotted on the left graph is simply the 2D energy dispersion function of the graphene, but for specific k_y values that depend on the width of the system. The discretized bands of the system are different k_y values.

My question is, what are these values``?
Maybe we aren’t talking about the same thing. In the link you provided, the image at the left is the band structure for a zigzag graphene nanoribbon. This is a 1D structure. Hence the wave vector being referred to simply as “k.” A zigzag graphene nanoribbon is not graphene. Draw the unit cell and this will be obvious.
 
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  • #7
Yes, I was indeed confusing the structures. Thank you for the help.
 

1. What is a band structure calculation?

A band structure calculation is a computational method used in condensed matter physics and materials science to study the electronic structure of solids. It involves solving the Schrödinger equation for a periodic lattice, taking into account the interactions between electrons and the crystal lattice.

2. Why are band structure calculations important?

Band structure calculations are important because they provide insights into the electronic properties of materials, such as their electrical conductivity, optical properties, and magnetic behavior. This information is crucial for designing new materials with desired properties for various applications.

3. What factors affect the accuracy of band structure calculations?

Several factors can affect the accuracy of band structure calculations, including the choice of the computational method, the size of the crystal lattice, and the inclusion of electron-electron interactions. The accuracy can also be influenced by the quality of the input parameters and the convergence criteria used in the calculations.

4. How are band structure calculations performed?

Band structure calculations are typically performed using quantum mechanical methods, such as density functional theory (DFT) or tight-binding approximation. These methods involve solving the Schrödinger equation for a periodic lattice and calculating the energy levels and wavefunctions of the electrons in the material.

5. What are some limitations of band structure calculations?

While band structure calculations are a powerful tool for studying the electronic properties of materials, they also have some limitations. These include the computational cost of accurately modeling large systems, the reliance on input parameters that may not be experimentally determined, and the inability to accurately capture some complex phenomena, such as electron-electron interactions and disorder in the material.

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