Dynamical matrix and dispersion

[Karthiksrao]

Dear all,

In this paper:
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.74.125402
In the appendix the author attempts to arrive at the spectral density of states of the surface of a half-space. To do this, he arrives at the Green's function of the surface atom of 1D atomic chain and then tries to extend it to the 3D problem with the following argument:

'In the case of a real 3D surface, we can split the problem
into multiple independent problems, each one corresponding
to a different wave vector parallel to the surface, q . Because
of parallel momentum conservation, each of these problems
is completely decoupled from the rest, and can be described
by an effective one-dimensional system. Now, instead of
having −2t for the diagonal elements, we must take into account
the dispersion parallel to the surface, so the diagonal
elements are kd=−2t+c^2 q^2 at the inner layers'

Here he is talking about the dynamical matrix of the 1D chain. On moving to 3D, he mentions that the diagonal elements should be added by an energy term c^2 q^2.

Is this obvious? I am not able to intuitively understand why the diagonal elements of the dynamic matrix will be shifted by this energy term.. Can anybody help me out with this? For e.g., Why just the diagonal terms? I tried to discuss this with the author, he mentioned - it is akin to the tight binding Hamiltonian for electrons, your diagonal element needs to be shifted by the energy corresponding to the momentum parallel to the surface.

I am afraid it is still not clear to me.

Thanks!

 

[eljose79]

Thank you for bringing up this interesting paper and question. I can understand your confusion about the addition of the energy term c^2 q^2 to the diagonal elements of the dynamical matrix in the 3D case. Let me try to explain why this is necessary.

In a 1D atomic chain, the dynamical matrix is a 2x2 matrix with the diagonal elements representing the energy of an atom at a specific position. This energy is simply given by -2t, where t is the hopping parameter between neighboring atoms. However, in a 3D surface, the atoms are not only connected to their neighbors in the chain, but also have a dispersion in the direction parallel to the surface, represented by the wave vector q. This means that the energy of an atom in the surface layer will also depend on this parallel momentum, and thus the diagonal elements of the dynamical matrix need to be modified accordingly.

Now, why is it only the diagonal elements that are affected by this energy term? This is because the off-diagonal elements in the dynamical matrix represent the coupling between different atoms in the chain, and this remains the same in both the 1D and 3D cases. The only difference is in the diagonal elements, where the energy of the atoms is now affected by the parallel momentum.

To give an analogy, think of a tight binding Hamiltonian for electrons in a 1D chain. The diagonal elements represent the on-site energy of an electron at a specific position, and this energy is shifted by an amount corresponding to the momentum parallel to the surface in the 3D case. This is similar to the diagonal elements in the dynamical matrix being shifted by the energy term c^2 q^2.

I hope this explanation helps clarify the reasoning behind adding the energy term to the diagonal elements in the 3D case. If you have further questions or concerns, please do not hesitate to reach out to the author or other experts in the field for more clarification. Science is a collaborative effort, and we are always happy to help each other understand complex concepts.