Compute subband structure of graphene

In summary, there are two ways to calculate the subband structure of graphene nanoribbon using the tight-binding model for a zigzag chain. The first method involves diagonalizing the Hamiltonian and obtaining the eigenvalues, while the second method involves diagonalizing the Hamiltonian shown in Untitled-1.jpg. Although the resulting Hamiltonians are slightly different, they have the same graph of the dispersion relation. When the hopping energy is changed to t*exp(i*φ), the graph plotted by the first method shifts horizontally, while the second method remains the same. It is unclear why this occurs and further investigation is needed. Additionally, the attached figures show the dispersion relation for an armchair nanoribbon when φ=pi/4 and the
  • #1
Isaac.Wang88
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0
There’re 2 ways to calculate the subband structure of graphene nanoribbon using tight-binding model, for Zigzag chain:
The first one is to diagonize the Hamiltonian $H = H_0 + H_10*exp(i*k_x) + H_01*exp(-i*k_x)$ and obtains the eigenvalues.
The second method is just diagonize the Hamiltonian in Untitled-1.jpg.
The Hamiltonians derived by the methods above are slightly different, but they have the same graph of the dispersion relation.
The question is, when I change the hopping energy t to t*exp(i*φ), here φ is any value you like, the graph plotted by the first method has been shifted horizontally while the graph of the second method remains the same. This is why it confuses me, I don’t know what’s wrong in here…

P.S. The attached figures are my dispersion relation of armchair nanoribbon when φ=pi/4, number of atoms N=16 using those two methods respectively. I don’t know why the plot has been shifted horizontally by using the first method…

Thank you very much
 

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  • #3
Isaac.Wang88 said:
There’re 2 ways to calculate the subband structure of graphene nanoribbon using tight-binding model, for Zigzag chain:
The first one is to diagonize the Hamiltonian $H = H_0 + H_10*exp(i*k_x) + H_01*exp(-i*k_x)$ and obtains the eigenvalues.
The second method is just diagonize the Hamiltonian in Untitled-1.jpg.
The Hamiltonians derived by the methods above are slightly different, but they have the same graph of the dispersion relation.
The question is, when I change the hopping energy t to t*exp(i*φ), here φ is any value you like, the graph plotted by the first method has been shifted horizontally while the graph of the second method remains the same. This is why it confuses me, I don’t know what’s wrong in here…

P.S. The attached figures are my dispersion relation of armchair nanoribbon when φ=pi/4, number of atoms N=16 using those two methods respectively. I don’t know why the plot has been shifted horizontally by using the first method…

Thank you very much
Isaac.Wang88 said:
There’re 2 ways to calculate the subband structure of graphene nanoribbon using tight-binding model, for Zigzag chain:
The first one is to diagonize the Hamiltonian $H = H_0 + H_10*exp(i*k_x) + H_01*exp(-i*k_x)$ and obtains the eigenvalues.
The second method is just diagonize the Hamiltonian in Untitled-1.jpg.
The Hamiltonians derived by the methods above are slightly different, but they have the same graph of the dispersion relation.
The question is, when I change the hopping energy t to t*exp(i*φ), here φ is any value you like, the graph plotted by the first method has been shifted horizontally while the graph of the second method remains the same. This is why it confuses me, I don’t know what’s wrong in here…

P.S. The attached figures are my dispersion relation of armchair nanoribbon when φ=pi/4, number of atoms N=16 using those two methods respectively. I don’t know why the plot has been shifted horizontally by using the first method…

Thank you very much
thanks for nice answer,can you tell me that when we are changing from armchair to zigzag ,what will happen?
How we can the magnetic term in my Hamiltonian ,which is of the order of n by n matrix in block form.
regards
 

Related to Compute subband structure of graphene

1. What is graphene?

Graphene is a two-dimensional material composed of a single layer of carbon atoms arranged in a hexagonal lattice. It is the basic building block of other carbon-based materials, such as graphite and carbon nanotubes, and has unique physical and chemical properties that make it a promising material for various applications.

2. How is the subband structure of graphene computed?

The subband structure of graphene is typically computed using a combination of theoretical models and experimental techniques. The most common approach is to use first-principles calculations, such as density functional theory, to determine the electronic band structure of graphene. This is then validated and refined through experiments, such as angle-resolved photoemission spectroscopy, which measures the energy and momentum of electrons in graphene.

3. What is the significance of computing the subband structure of graphene?

The subband structure of graphene provides valuable information about its electronic properties, such as the band gap, electron mobility, and conductivity. This information is crucial for understanding the behavior of graphene in different environments and for designing and optimizing graphene-based devices for various applications, such as electronics, energy storage, and sensors.

4. What factors can affect the subband structure of graphene?

The subband structure of graphene can be affected by various factors, such as the number of layers, doping, strain, and defects. These factors can alter the electronic properties of graphene, leading to changes in its subband structure. For example, doping can introduce additional charge carriers, while strain can modify the band gap and electronic band structure of graphene.

5. How does the subband structure of graphene compare to other materials?

The subband structure of graphene is unique due to its two-dimensional nature and the presence of a linear energy-momentum dispersion relation, which gives rise to massless Dirac fermions. This is in contrast to traditional materials, such as metals and semiconductors, which have a parabolic energy-momentum dispersion relation. Additionally, the subband structure of graphene is highly tunable through external factors, making it a promising material for future technological applications.

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