How to Calculate Hopping Parameters and Plot Band Structure in MATLAB?

[anahita]

Who can help me in writing the following program:

phase(i) = exp(1i*dot(k(:,index),R(:,i)))

This gives you the phase you're going to multiply with the hopping parameter. You'd have to write a loop that runs from i = 1:3 in order to multiply each contribution to the hopping parameter such that the product of all 3 would get you the total hopping parameter. k and R here are matrices that contain generated wavevectors and the coordinates for the atoms, respectively.

Es = -4.2;
Ep = 1.715;
orbitals = [Es Ep Ep Ep];
Vpp_sigma = 2.72;
Vpp_pii = -0.72;
Vss_sigma = -2.08;
Vsp_sigma = 2.48;

a = 3.86;
a1 = (a/2) * [sqrt(3), -1, 0];
a2 = (a/2) * [sqrt(3), 1, 0];

coordinates = [4.45714 0 0; %B
2.22857 0 0.46152]; %A

R1 = coordinates(2, :)-coordinates(1,:);
R2 = R1 + a1;
R3 = R1 + a2;
R = [R1' R2' R3'];

n = dot(R,[ 0 0 1])/norm(R); %direction cosine
l = dot(R,[1 0 0])/norm(R);
m = dot(R, [0 1 0])/norm(R);
tss = Vss_sigma;
txs = l*Vss_sigma;
tys = m*Vss_sigma;
tzs = n*Vss_sigma;
txy = l*m*Vpp_sigma - l*m*Vpp_pii;
txz = l*n*Vpp_sigma - l*n*Vpp_pii;
tyz = m*n*Vpp_sigma - m*n*Vpp_pii;
tzz = n^2 *Vpp_sigma+(1-n^2)*Vpp_pii;
txx = l^2 *Vpp_sigma+(1-l^2)*Vpp_pii;
tyy = m^2 *Vpp_sigma+(1-m^2)*Vpp_pii;

These will be your matrix elements that you'll want. Since the Hamiltonian matrix takes a block-diagonal form, you can create an A block and B block and set the Hamiltonian equal to [A B; B A]. Once you run the loop to get all the hopping parameters correct you can simply plot the eigenvalues of the full Hamiltonian matrix across the desired path to obtain the full band structure.

 

[eljose79]

Hi there,

I can definitely help you with writing this program. Let's start by breaking down the steps you'll need to take in order to get the total hopping parameter:

1. Create a loop that runs from i = 1:3. This will allow you to loop through each contribution to the hopping parameter.

2. Within the loop, use the formula provided to calculate the phase for each contribution. This will involve using the exp() and dot() functions, as well as the k and R matrices.

3. Multiply the phase with the corresponding contribution to the hopping parameter. This can be done using the * operator.

4. Keep track of the total hopping parameter by adding each contribution to a sum variable.

5. Once the loop has completed, the sum variable will contain the total hopping parameter.

As for creating the Hamiltonian matrix and plotting the eigenvalues, here are some steps you can follow:

1. Create two matrices, A and B, with dimensions of 2x2. These will represent the A and B blocks of the Hamiltonian matrix.

2. Use the matrix elements provided in the code snippet to fill in the corresponding entries in the A and B matrices.

3. Use the MATLAB function diag() to create a diagonal matrix with the total hopping parameter as the diagonal entries.

4. Use the MATLAB function blkdiag() to create the full Hamiltonian matrix by combining the A and B matrices with the diagonal matrix.

5. Use the MATLAB function eig() to calculate the eigenvalues of the Hamiltonian matrix.

6. Plot the eigenvalues across the desired path to obtain the full band structure.

I hope this helps! Let me know if you have any further questions or need clarification on any of the steps. Good luck with your program!