Plotting Band Structure in Graphene for Scientists

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Discussion Overview

The discussion revolves around plotting the band structure of graphene using tight-binding models, specifically focusing on the implementation in MATLAB. Participants share their experiences, code snippets, and seek assistance with various aspects of the plotting process, including specific paths in k-space and transitioning from 2D to 3D plots.

Discussion Character

  • Technical explanation
  • Homework-related
  • Debate/contested

Main Points Raised

  • Some participants express difficulty in plotting the tight-binding dispersion for graphene along the path Γ→K→M→Γ and seek guidance.
  • One participant suggests setting kx or ky to zero to simplify the plotting process along specific axes.
  • Another participant provides MATLAB code snippets for calculating and plotting the band structure, indicating how to iterate through k points.
  • There are discussions about the correct selection of k points and symmetry points in the Brillouin zone, with varying opinions on the approach.
  • Some participants share successful code for 2D plots and inquire about extending this to 3D band structure plots.
  • A participant reflects on past work with graphite intercalation compounds, mentioning the importance of including second nearest neighbor interactions in tight-binding models for accurate results.

Areas of Agreement / Disagreement

Participants generally agree on the methods for plotting band structures but express differing views on the specifics of k point selection and the implementation in MATLAB. The discussion remains unresolved regarding the best approach for certain aspects of the plotting process.

Contextual Notes

Some participants mention limitations in their understanding of the physics behind tight-binding calculations versus the implementation in MATLAB, indicating a potential gap in knowledge that could affect their plotting results.

Who May Find This Useful

Researchers and students interested in computational materials science, particularly those working with graphene and tight-binding models, may find this discussion beneficial.

barana
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I just learned how to get a tight-binding dispersion for graphene.
But i can not figure out how can I plot that. I want to plot that in the following path: Γ→K→M→Γ
I would be appreciated if could some one give me a hand on my problem.
thanks for a lot.
 
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barana said:
I just learned how to get a tight-binding dispersion for graphene.
But i can not figure out how can I plot that. I want to plot that in the following path: Γ→K→M→Γ
I would be appreciated if could some one give me a hand on my problem.
thanks for a lot.

What do you mean that you can't figure out how to plot the band?

If you are going along one particular axis, set one of the kx, let's say, set ky=0 and plug that into your dispersion equation. If you are plotting along the zone diagonal, set kx=ky and off you go. Do the same for other high-symmetry directions.

Zz.
 
Thank you very much. Your answer was useful.
 
ZapperZ said:
What do you mean that you can't figure out how to plot the band?

If you are going along one particular axis, set one of the kx, let's say, set ky=0 and plug that into your dispersion equation. If you are plotting along the zone diagonal, set kx=ky and off you go. Do the same for other high-symmetry directions.

Zz.
Best wishes and Thank you again.
I have Hamiltonian matrix and overlap matrix. How to calculate tight binding dispersion for graphene in matlab?
I would be appreciated if could some one give me a hand on my problem.
 
But ZapperZ told you already. Just take kx=ky=t and plot E(t).
 
Could you elaborate a little more? Is your question about the physics of tight binding calculations or about implementing tight binding calculations in MATLAB?
 
ConradAveraging said:
Could you elaborate a little more? Is your question about the physics of tight binding calculations or about implementing tight binding calculations in MATLAB?
implementing tight binding calculation in matlab.
matlab cose in the following path: Γ→K→M→Γ is:
Nk=20;
t=-2.550;
for ik=1:Nk+1
k_x=2*pi*(ik-1)/Nk*0;
k_y=2*pi*(ik-1)/Nk*0.667;
H=zeros(2);
H(1,2)=t*(exp(-1i*k_x/sqrt(3))+exp(1i/(2*sqrt(3))*((k_x+sqrt(3)*k_y)))+exp(1i/(2*sqrt(3))*(k_x-sqrt(3)*k_y)));
H(2,1)=t*(exp(1i*k_x/sqrt(3))+exp(-1i/(2*sqrt(3))*((k_x+sqrt(3)*k_y)))+exp(-1i/(2*sqrt(3))*(k_x-sqrt(3)*k_y)));
[V,D]=eig(H)
E_1(ik)=D(1,1)
E_2(ik)=D(2,2)
end
plot(E_1)
hold on
plot(E_2)
Nk=20;
t=-2.550;
for ik=21:Nk+10
k_x=2*pi*(ik-1)/Nk*0;
k_y=2*pi*(ik-1)/Nk*-0.667;
H=zeros(2);
H(1,2)=t*(exp(-1i*k_x/sqrt(3))+exp(1i/(2*sqrt(3))*((k_x+sqrt(3)*k_y)))+exp(1i/(2*sqrt(3))*(k_x-sqrt(3)*k_y)));
H(2,1)=t*(exp(1i*k_x/sqrt(3))+exp(-1i/(2*sqrt(3))*((k_x+sqrt(3)*k_y)))+exp(-1i/(2*sqrt(3))*(k_x-sqrt(3)*k_y)));
[V,D]=eig(H)
E_1(ik)=D(1,1)
E_2(ik)=D(2,2)
end
plot(E_1)
hold on
plot(E_2)
Nk=20;
t=-2.550;
for ik=31:Nk+20
k_x=2*pi*(ik-1)/Nk*0.557;
k_y=2*pi*(ik-1)/Nk*0;
H=zeros(2);
H(1,2)=t*(exp(-1i*k_x/sqrt(3))+exp(1i/(2*sqrt(3))*((k_x+sqrt(3)*k_y)))+exp(1i/(2*sqrt(3))*(k_x-sqrt(3)*k_y)));
H(2,1)=t*(exp(1i*k_x/sqrt(3))+exp(-1i/(2*sqrt(3))*((k_x+sqrt(3)*k_y)))+exp(-1i/(2*sqrt(3))*(k_x-sqrt(3)*k_y)));
[V,D]=eig(H)
E_1(ik)=D(1,1)
E_2(ik)=D(2,2)
end
plot(E_1)
hold on
plot(E_2)
but in the following path: K→Γ→M→K not true.
Nk is The number of points between two points K and Γ.
 
It looks like your not selecting the correct k points on your last for loop. If you're mainly interested in the band structure and not the wavefunctions you can cut through Γ,K, and M in one line, with M being in 2BZ, which is a direction similar to your first for loop. But if you want to look at different symmetry points, I usually define the coords of Γ,K,M, then define a vector that connects them like v1=K-Γ. I can't tell you if you've selected the correct symmetry points because that depends how you build your lattice
 
graphene lattice is:
upload_2015-4-22_11-6-50.png
 

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  • #10
ok. So if you set kx=0, and iterate ky you will plot through all the symmetry points. Also, K=(b1-b2)/3, K'=(2*b2+b1)/3, M=(b2+b1)/2. I would also suggest that you plot your path in k-space as a check.
 
  • #11
ConradAveraging said:
ok. So if you set kx=0, and iterate ky you will plot through all the symmetry points. Also, K=(b1-b2)/3, K'=(2*b2+b1)/3, M=(b2+b1)/2. I would also suggest that you plot your path in k-space as a check.
Thank you very much.
I'm able to plot the 2D for graphene on matlab.
matlab code is:
close all
clear all
clc
t=-2.550;
acc=1.44e-10;
a=1.732*acc;

s=0:0.02:1
k_x=(2*pi/(sqrt(3)*a))*(1-s)
k_y=(2*pi/(3*a))*(1-s)
E1=t.*sqrt(1+4.*cos((sqrt(3).*a/2).*k_x).*cos((a/2).*k_y)+4.*cos((a/2).*k_y).^2)
E2=-E1;
plot(E1,'*')
hold on
plot(E2,'*')

s=0:0.02:1
k_x=(2*pi/(sqrt(3)*a))*s
k_y=0
E1=t.*sqrt(1+4.*cos((sqrt(3).*a/2).*k_x).*cos((a/2).*k_y)+4.*cos((a/2).*k_y).^2)
E2=-E1;
plot(E1,'*')
hold on
plot(E2,'*')

s=0:0.02:1
k_x=(2*pi/(sqrt(3)*a))
k_y=(2*pi/(3*a))*s
E1=t.*sqrt(1+4.*cos((sqrt(3).*a/2).*k_x).*cos((a/2).*k_y)+4.*cos((a/2).*k_y).^2)
E2=-E1;
plot(E1,'*')
hold on
plot(E2,'*')
but I can not draw diagrams tandem.
 
  • #12
Just concatenate E = [E1 E2 E3]
 
  • #13
ConradAveraging said:
Just concatenate E = [E1 E2 E3]
The problem was solved
thanks
 
  • #14
Hi dear friends.
I'm able to plot graphene banstructure in 2D, by this code:
*****************************************************************************************
%Plot of Energy dispersion trial1.m
x=linspace(-2.5,2.5,50); y=linspace(-9,10,20);
[a,b]=meshgrid(x,y)
z= 3.033*sqrt(1+4*cos(b*0.1*pi).*cos(a*1.2283)+4*(cos(a*1.2283)).^2);
zz=- 3.033*sqrt(1+4*cos(b*0.1*pi).*cos(a*1.2283)+4*(cos(a*1.2283)).^2);
figure,plot(x,z)
title('Energy dispersion relation of graphene')
xlabel('K(1/A)'),ylabel(' Energy of graphene(ev)')
hold on
plot(x,zz)
***************************************************************************************
But now I'm trying to have graphene bandstructure in 3D like this:
Graphene_pi_bandstructure.png

But I can't do this.
Can you give me the MATLAB code for 3D-Bandstructure?
 
  • #15
@sina67 ... here's a modified code for 3d plot:
%Plot of Energy dispersion trial1.m
x=linspace(-2.5,2.5,50);
y=linspace(-9,10,20);
[a,b]=meshgrid(x,y)
z= 3.033*sqrt(1+4*cos(b*0.1*pi).*cos(a*1.2283)+4*(cos(a*1.2283)).^2);
zz=- 3.033*sqrt(1+4*cos(b*0.1*pi).*cos(a*1.2283)+4*(cos(a*1.2283)).^2);
figure
surf(a,b,z)
title('Energy dispersion relation of graphene')
xlabel('K(1/A)'),ylabel(' Energy of graphene(ev)')
hold on
surf(a,b,zz)
 
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  • #16
Can anyone write a MATLAB code for the 3D band structure from the Hamiltonian matrix? Not the one from the explicit expression for the eigenvalue, but just from the Hamiltonian to 3D plot. Thanks a lot!
 
  • #17
just a thought
Years ago I was working on graphite intercalation compounds. Did Fermi surface study. Then, made 2-D band structure calculations and found out that to get a good fit I had to include second nearest neighbour interactions within the tight binding model.
 
  • #18
Barana, what is a and acc?
 

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