# A question about band structure silicene

• anahita
In summary, the conversation revolved around calculating the cosine of direction in the context of atomic orbitals and hopping parameters. The speaker was seeking help with finding the phase factors and direction cosine for the nearest neighbors in a silicene lattice. They were also discussing the use of MATLAB for plotting the band structure. The conversation concluded with a summary of the code that could be used for this calculation.
anahita
Dear forum people
I consider four atomic orbitals of the silicon atom: s, px , py and pz , thus Hi,j is a 4*4 matrix.
Hi,i is the expression of the diagonal matrix and Hi,j is the expression off-diagonal matrix where the expression for Hi,i and Hi,j is attached.
But I can not calculate the cosine of direction.
Who can help me find the cosine of direction?

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• matrix elements.docx
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Could you attach a pdf?

DrDu said:
Could you attach a pdf?

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• matrix elements.pdf
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So what is your problem? l, m, n are just the cartesian x, y and z coordinates of the distance vector of the two atoms.

DrDu said:
So what is your problem? l, m, n are just the cartesian x, y and z coordinates of the distance vector of the two atoms.
Position vectors of the 1 NN for silicene as follows:
R1=(a/√3,0,0) , R2=(-a/2√3,a/2,0) , R3=(-a/2√3,-a/2,0)
but n, l,m for the nearest neighbours is three values as we must have a value for the nearest neighbours.

Well, for example you take ##r_i= R0=(0,0,0)^T## which has the three nearest neighbours you specified. So e.g. for ##r_j=R2##:
##l=R1_x-R0_x=a/\sqrt{3}##.

DrDu said:
Well, for example you take ##r_i= R0=(0,0,0)^T## which has the three nearest neighbours you specified. So e.g. for ##r_j=R2##:
##l=R1_x-R0_x=a/\sqrt{3}##.

Dear DrDu
Thank you very much.

If you're trying to plot this in something like MATLAB you can form the two block matrices and run a loop that adds the phase factors for the hopping parameters and then plot over those points if you know the region it spans (which in your case is just the Brillouin zone)

nuclearpasta said:
If you're trying to plot this in something like MATLAB you can form the two block matrices and run a loop that adds the phase factors for the hopping parameters and then plot over those points if you know the region it spans (which in your case is just the Brillouin zone)
I can't calculate phase factors for hopping parameter. can help me?

It is just the product of the phase factors of the two orbitals involved.

DrDu said:
It is just the product of the phase factors of the two orbitals involved.
In graphene
<si|H|sj>=t1*g0 and <si|H|p(x)j>=t2*g1
where
g0= 1+exp(ik.R(B))+exp(ik.R(C)) and g1= 1-(1/2)(exp(ik.R(B))+exp(ik.R(C)))
How to calculate g0 and g1?

g0 and g1 are the sum of three hopping parameters. In g0, the first term is the hopping between the atoms in the same cell. So the phase factor for both orbitals is 1 and the contribution is t1*1*1. The other two hopping terms go to orbitals with phase factors exp(ik.R(B)) and exp(ik.R(C), respectively. So the second factor is t1*1*exp(ik.R(B)) and analogously for C.
For the p_x orbitals, g also contains the direction cosine, which is 1 for A and cos(+-120 deg) =-1/2 for B and C.

I'll post some snippets of what code in MATLAB would look like for a calculation like this.

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.

Code:
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.

nuclearpasta said:
I'll post some snippets of what code in MATLAB would look like for a calculation like this.

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.

Code:
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.
In the l,m,n inner product matrix and vector is not possible so n,l,m how obtained?

nuclearpasta said:
I'll post some snippets of what code in MATLAB would look like for a calculation like this.

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.

Code:
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.
Hi
In MATLAB program, cosine directions (n,l,m) not true because inner product R and vectors is impossible.
I can't construction matrix Hamiltonian because txy txz , etc are 1*3 vectors while tss is 1*1 vector.
can help me?

## 1. What is silicene?

Silicene is a 2-dimensional allotrope of silicon, similar to graphene but made of silicon atoms instead of carbon atoms. It is a single layer of silicon atoms arranged in a honeycomb lattice structure.

## 2. What is band structure?

Band structure refers to the arrangement of energy levels or bands within a material, particularly in relation to the movement of electrons. In other words, it shows how electrons are able to move through a material and where they are allowed to exist in terms of energy.

## 3. How is band structure of silicene different from graphene?

The band structure of silicene is not exactly the same as that of graphene due to the difference in atomic structure. Silicene has a larger band gap compared to graphene, meaning there is a larger energy difference between the valence band (where electrons are bound to atoms) and the conduction band (where electrons are free to move).

## 4. Why is understanding band structure of silicene important?

Understanding the band structure of silicene is important because it can provide insights into its electronic and optical properties, which can have potential applications in electronics and optoelectronics. It can also help in the development of new materials with desired properties.

## 5. What methods are used to study the band structure of silicene?

There are several methods used to study the band structure of silicene, including angle-resolved photoemission spectroscopy, scanning tunneling microscopy, and density functional theory calculations. These methods allow for the visualization and analysis of the electronic structure of silicene at different energy levels.

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