Transmission Eigenvalues revisited

[srdguezb]

Hello

I know there are several posts about transmission eigenvalues. Particularly good explained is http://quantumwise.com/forum/index.php?topic=349.0. However I have a basic question and I need some support (replies or bibliography) to understand this issue.

My question is: "As far as I understand the concept of "transmission eigenchannel" it refers to a set of k-states in the left electrode that are mapped to the same set of k-states into the right electrode. The procedure to obtain them should be the following: For a fixed energy, we need to calculate for each k-state in the left contact its composition in k-states in the right contact. Once calculated the above for every k-state in the left contact we obtain 1 (and only 1) transmission matrix, which depends on energy but not in k. The diagonalization of that matrix provides a several sets of k-states.

However, atk calculates the transmission eigenvalues depending on energy and left k-point, T(E,k), and you wrote that for a particular energy E and k-point k, you take the trace of tt^\dag. So my questions are the following : for every E, and every k, do you calculate one transmission matrix? The procedure is not to build one transmission matrix which contains every k-point for a fixed energy and calculate the trace?


If this is correct, which is the relation between 'my' transmission eigenchannel and yours?"

PD: We're considering 3-D contacts.

[zh]

The transmission eigenvalues (or transmission spectrum) can be obtained by NEGF technique or the scattering-state approach. The details of the procedure for calculating the transmission matrix can be found in references:
Th. Martin and R. Landauer , Wave-packet approach to noise in multichannel mesoscopic systems, Phys. Rev. B 45, 1742–1755 (1992)
Mads Brandbyge, et al, Density-functional method for nonequilibrium electron transport, Phys. Rev. B 65, 165401 (2002)
Hyoung Joon Choi, et al, First-principles scattering-state approach for nonlinear electrical transport in nanostructures, PHYSICAL REVIEW B 76, 155420 (2007).

The understanding on how the transmission eigenvalues can be reduced to a simple problem: consider how an electron  is transfered in a sandwich system with the following potential: i) V (left electrode region) = v1 ( a constant value); ii) V(center region) = v2; iii) V(right electrode region) = v1, where v1 [tex] \neq [/tex] v2.  In quantum mechanics, the transmission coefficients of electron in such sandwich system can be obtained by solving the Schrödinger equation  with the continuity of wavefunctions and derivatives at boundaries.

[srdguezb]

Thank you for your quick reply.

I think I have understood the process that takes place if we have a sandwich-like system with the characteristics you describe. However, I think the key of my question could be in the 3D character of the contacts. In those systems, for a fixed energy E0, it is possible to have lots of k-states in the left contact. When we speak about 'transmission' we can speak of several different, but related, issues:

Transmission spectrum for E0: The whole transmission for that energy, taken into account the contribution of all the left k-states with E0. Of course, to calculate the transmission of each of those left k-states we must consider the hamiltonian of the system.

Transmission eigenchannel: A set of left k-states that when travelling through the central zone of the system produce a wavefunction in the right electrode that only have components belonging to the same set of k-states.

T(E,k): (Atomistix toolkit): I think (suppose) it is the k-state components of the wavefunction in the right electrode if we have a pure k-state in the left electrode.

If all the above is true. My question is: ¿is there a relation between T(E,k) and the transmission eigenchannel as I unsderstand them?

Thank you again

I will read carefully the references you sent. My ideas about eigenchannel transmission have been taken from Physical Review B, 76, 115117 (2007)

[Anders Blom]

Since (kx,ky) is a conserved quantum number, it doesn't matter if you treat the full system, with all k-points, or you do each one separately.

The relation is quite trivially that the sum of the transmission eigenvalues is equal to T(E,k)

[srdguezb]

Hello Anders,

My system has an infinite 3D electrode + molecule (central region) + infinite 3D electrode. Do you mean that in this kind of system (kx,ky) is conserved?

Best regards

[Anders Blom]

This is a fundamental assumption, yes. You would need phonon scattering or similar to change these quantum numbers.

[srdguezb]

Hi again,

Anders, even when an external voltage is applied, is (kx,ky) conserved? I mean, if the energy is fixed and the fermi levels are different in the contacts, is that assumption  retained?

I think that since the energy difference from E0 to the conduction band minimum is different (kx,ky) should be also different in each contact.

I also think that, as you say, phonon scattering (or other scattering that affects to the energy of the electron) would affect to the energy and thus to (kx,ky) but it is not the unique reason for changing (kx,ky) as I arguing above.

Finally, in a 3d contact- molecule- 3d contact (kx,ky) can change although the energy is constant.

Then, I am still confused.

Thank you for your interest

[Anders Blom]

It is actually really so, that even with different electrodes, (kx,ky) are conserved. Quantum numbers, and in particular conserved ones, are always a result of a particular symmetry in the system, and (kx,ky) correspond directly to the periodic boundary conditions of the system in the X/Y directions.

(Btw, it's not really kx,ky but rather kA,kB, since the two unit cell axes A and B, which are perpendicular to the transport direction and determine the k-points, are not necessarily perpendicular to each other or aligned with X/Y.)

[pengdou]

Dear Doc.Anders Blom,
Would you please kindly show me some references about this point?

This is a fundamental assumption, yes. You would need phonon scattering or similar to change these quantum numbers.

[Anders Blom]

This is, if you wish, just Bloch's theorem, which is described in any basic textbook on solid state physics.