Plotting Graphene in Mathematica

In summary, the conversation discusses an issue encountered while plotting a 3D band structure of graphene in Mathematica. The cones at the K-points are not touching, which is an important electronic property of graphene. The code provided includes a function f and a GrapheneHam function, and the Energies and Plot3D commands are used to plot the band structure. The variable a is set to 2.46 and there is discussion about potential issues with PlotPoints and WorkingPrecision that may affect the accuracy of the plot.
  • #1
nuclearpasta
4
0
I am encountering an issue when I plot attempting to plot 3d band structure of graphene in Mathematica. While the general shape and curvature looks fine, the cones at the K-points are not touching, which is an important qualitative electronic property of graphene. Since further down the line I want to test the effects of extra terms in the Hamiltonian, I need to first ensure that this is working as expected. Code is provided below:

Code:
f[kx_, ky_,
   t_] := -t E^(-I kx a) (1 + 2 E^(I (3 kx a)/2)*Cos[Sqrt[3]/2 ky a]);

GrapheneHam[kx_, ky_, t_] :=
  ComplexExpand[{{0, f[kx, ky, t]}, {Conjugate[f[kx, ky, t]], 0}}];

Energies[kx_, ky_, t_] := Eigenvalues[GrapheneHam[kx,ky,t]];

Plot3D[Energies[kx,ky, 2.8], {kx, -Pi/a, Pi/a},{ky, -Pi/a, Pi/a}]

In this case the variable a is set to 2.46. Would anyone know if there is something in the syntax I am doing incorrectly, or perhaps the dispersion is being calculated wrong? Any help is appreciated. Thanks!
 
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  • #2
It plots fine for me as long as I set a=2.46 at the beginning of the code
 
  • #3
That's strange. When I do it I get that the two surfaces never touch each other, which is what is supposed to happen at the six corners of the hexagon.
 
  • #4
nuclearpasta said:
That's strange. When I do it I get that the two surfaces never touch each other, which is what is supposed to happen at the six corners of the hexagon.

Ok, I only meant that it plotted. Can't say if the plot is what you want though.
 
  • #5
I think it's an issue with PlotPoints, when I set them on 60 (fair bit of warning: this usually takes quite long to render), the peaks at the "hexagon" are much sharper, you could try experimenting with that. Tampering with the WorkingPrecision also seems to refine it even more, I've tried setting it to 5 and the gaps are almost unnoticeable.
 

FAQ: Plotting Graphene in Mathematica

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 thinnest and strongest material known, and has excellent electrical and thermal conductivity.

How can I plot graphene in Mathematica?

To plot graphene in Mathematica, you can use the GrapheneModel function, which generates a hexagonal lattice of carbon atoms. You can then manipulate the lattice parameters to adjust the size and shape of the graphene lattice.

What are some important properties to include when plotting graphene?

Some important properties to include when plotting graphene in Mathematica include the lattice parameters (such as the lattice constant and angle), the number of carbon atoms in the lattice, and the desired boundary conditions (such as periodic or free boundaries).

Can I add defects or impurities to my graphene plot?

Yes, you can add defects or impurities to your graphene plot in Mathematica. You can use the GrapheneModel function to generate a perfect graphene lattice, and then use the ReplacePart function to replace specific carbon atoms with different elements or structures to simulate defects.

How can I visualize the electronic band structure of graphene in Mathematica?

You can visualize the electronic band structure of graphene in Mathematica by using the BandStructurePlot function. This function takes into account the lattice parameters and boundary conditions of your graphene plot, and generates a plot of the energy bands for both the conduction and valence bands.

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