Tight Binding calculation of band structure

[mohsen2002]

Dear all,
I want to calculate the band structure of graphene for a unit cell with 8 atoms in the Tight Binding approximation. There is no problem about drawing the band structure for a unit cell with 2 atoms. But by increasing the unit cell size, first brillouin decrease and there is a gap in the band structure
Any suggestion appreciated.

Mohsen

 

[saaskis]

I thought that increasing the size of the unit cell simply folds the bandstructure? So I cannot see how this folding would lead to an opening of a gap, that does not sound very physical to me...

But for a unit cell of 8 atoms, you have to construct the effective Hamiltonian H_eff(k), which is an 8*8 matrix. The internal hopping matrix elements are simply "t", whereas the hoppings into or out of the unit cell get an exponential factor (this is where the wavevector "k" steps in).

This is explained e.g. in the book "Atom to Transistor" by S. Datta.

 

[mohsen2002]

Dear Saaskis,
Thanks for your reply.
could you please explain the exponential part of your reply or introduce a reference. I read Data, it only explain the subbands for nanotue. But i have problem with Graphene.
Zone folding concept is fully explain for nanotubes, But i can not find a good reference for graphene. Any suggestion?
Mohsen

 

[saaskis]

I don't have Datta now available, but if I remember correctly, he showed how to derive the "Bloch" Hamiltonian for general systems. If you understand what he does, you should be able to reproduce the same calculation for graphene. You know how to derive the bandstructure for a unit cell of two carbon atoms, right? Exactly the same principle holds if you choose a larger unit cell. I cannot give you any reference, since I doubt that anyone has published or showed such calculations just for fun... I understand that for example for systems with defects, it may be necessary to introduce a larger unit cell, but for pristine graphene, it does not really make any sense.

But to make sure that we understand what zone-folding means, let us briefly study the simplest example: a 1D TB chain with N atoms and periodic boundary conditions. The discrete Schrödinger equation for each site is

$$-t\psi_{j-1}-t\psi_{j+1} = E \psi_j,$$

which can be solved using the Bloch ansatz $$\psi_j=ue^{ikj}$$, leading to

$$E_k=-t(e^{-ik}+e^{ik})=-2t\cos(k).$$

The distinct k-values are $$k_n=2\pi n/N, \ n=0,1,\dots,N-1$$. The Bloch Hamiltonian is a 1*1 matrix

$$H_{eff}(k)=-t(e^{-ik}+e^{ik}),$$

since from any site one can hop to the right (exponential factor e^ik from the Bloch Ansatz) or to the left (exponential factor e^-ik from the Bloch Ansatz).

Now suppose that we use a unit cell of two atoms. We know that this should lead to a identical bandstructure, but with "folded zones". The number of unit cells is then N/2, and our Ansatz would be

$$\left( \begin{array}{c} \psi_j \\ \psi_{j+1} \\ \end{array} \right) = \left( \begin{array}{c} u_1 \\ u_2\\ \end{array} \right) e^{ik(j+1)/2}, \ j=1,3,5,\dots,N-1,$$

where I chose the exponential factor simply so that the first unit cell gets factor e^ik, the second e^2ik etc. The distinct values of k are now (assuming N is even for this to make sense) $$k_n=4\pi n/N,\ n=0,1,\dots,N/2-1$$. As expected, the number of allowed k-values in the Brillouin zone was halved due to the bigger unit cell. This leads to the Schrödinger equation

$$\left( \begin{array}{cc} 0 & -t(1+e^{ik}) \\ -t(1+e^{-ik}) & 0 \\ \end{array} \right) \left( \begin{array}{c} u_1 \\ u_2\\ \end{array} \right) = E \left( \begin{array}{c} u_1 \\ u_2\\ \end{array} \right).$$

Diagonalization gives the eigenvalues

$$E_k=\pm t \sqrt{(1+\cos(k))^2+\sin(k)^2}=\pm t\sqrt{2+2\cos(k)}=\pm 2t\cos(k/2).$$

(Here one should be a little careful with the trigonometry, I guess.) As expected, we got two branches that meet each other at k=pi. Rescaling the new k-variable by a factor of two, it would be clear how the original bandstructure E_k=-2tcos(k) changes to E_k=pm 2tcos(k): plotting shows that the old bandstructure in the Brillouin zone (-pi,pi) got folded with respect to the points -pi/2 and pi/2. This scaling also halves the Brillouin zone to (-pi/2,pi/2).

Now, one can directly see how this can be generalized to more complicated systems: the only problem is that in higher dimensions and in more complicated lattices, the intuition of folding is easily lost. But in principle, the analysis should be straightforward. Just try to do it yourself and ask if you run into any problems.

 

[mohsen2002]

Thanks for your reply. I use this method for graphene (it is not easy) and it works properly.
Thanks.

Mohsen

 

[wind001001]

I don't have Datta now available, but if I remember correctly, he showed how to derive the "Bloch" Hamiltonian for general systems. If you understand what he does, you should be able to reproduce the same calculation for graphene. You know how to derive the bandstructure for a unit cell of two carbon atoms, right? Exactly the same principle holds if you choose a larger unit cell. I cannot give you any reference, since I doubt that anyone has published or showed such calculations just for fun... I understand that for example for systems with defects, it may be necessary to introduce a larger unit cell, but for pristine graphene, it does not really make any sense.

But to make sure that we understand what zone-folding means, let us briefly study the simplest example: a 1D TB chain with N atoms and periodic boundary conditions. The discrete Schrödinger equation for each site is

$$-t\psi_{j-1}-t\psi_{j+1} = E \psi_j,$$

which can be solved using the Bloch ansatz $$\psi_j=ue^{ikj}$$, leading to

$$E_k=-t(e^{-ik}+e^{ik})=-2t\cos(k).$$

The distinct k-values are $$k_n=2\pi n/N, \ n=0,1,\dots,N-1$$. The Bloch Hamiltonian is a 1*1 matrix

$$H_{eff}(k)=-t(e^{-ik}+e^{ik}),$$

since from any site one can hop to the right (exponential factor e^ik from the Bloch Ansatz) or to the left (exponential factor e^-ik from the Bloch Ansatz).

Now suppose that we use a unit cell of two atoms. We know that this should lead to a identical bandstructure, but with "folded zones". The number of unit cells is then N/2, and our Ansatz would be

$$\left( \begin{array}{c} \psi_j \\ \psi_{j+1} \\ \end{array} \right) = \left( \begin{array}{c} u_1 \\ u_2\\ \end{array} \right) e^{ik(j+1)/2}, \ j=1,3,5,\dots,N-1,$$

where I chose the exponential factor simply so that the first unit cell gets factor e^ik, the second e^2ik etc. The distinct values of k are now (assuming N is even for this to make sense) $$k_n=4\pi n/N,\ n=0,1,\dots,N/2-1$$. As expected, the number of allowed k-values in the Brillouin zone was halved due to the bigger unit cell. This leads to the Schrödinger equation

$$\left( \begin{array}{cc} 0 & -t(1+e^{ik}) \\ -t(1+e^{-ik}) & 0 \\ \end{array} \right) \left( \begin{array}{c} u_1 \\ u_2\\ \end{array} \right) = E \left( \begin{array}{c} u_1 \\ u_2\\ \end{array} \right).$$

Diagonalization gives the eigenvalues

$$E_k=\pm t \sqrt{(1+\cos(k))^2+\sin(k)^2}=\pm t\sqrt{2+2\cos(k)}=\pm 2t\cos(k/2).$$

(Here one should be a little careful with the trigonometry, I guess.) As expected, we got two branches that meet each other at k=pi. Rescaling the new k-variable by a factor of two, it would be clear how the original bandstructure E_k=-2tcos(k) changes to E_k=pm 2tcos(k): plotting shows that the old bandstructure in the Brillouin zone (-pi,pi) got folded with respect to the points -pi/2 and pi/2. This scaling also halves the Brillouin zone to (-pi/2,pi/2).

Now, one can directly see how this can be generalized to more complicated systems: the only problem is that in higher dimensions and in more complicated lattices, the intuition of folding is easily lost. But in principle, the analysis should be straightforward. Just try to do it yourself and ask if you run into any problems.

Hi, could you explain the physical significant of the folding? What does it imply?

 

[mohsen2002]

the bandstructure is periodic in the inverse (fourier) space, but in the band structure is ploted in the first brillouin zone, by increasing the real space unit cell the inverse unit cell decreas. For plotting the new bandstructure you should move the old branch of bandstructure to new brillouin zone, this is zone folding.
i hope it helps.