questions about transmission eigenvalue and eigenstate

[yongjunwinwin]

I don't quite understand the concept of transmission eigenvalue and transmission eigenstate. Could anyone explain to me these two concepts and how to make use of them to do analysis about transport properties?

[zh]

Please see here:
http://quantumwise.com/forum/index.php?topic=349.0

[yongjunwinwin]

1.Thanks for the reference! So though the transmission eigenstate we can view the transmission channel and the sum of the transmission eigenvalue equal to the transmission coefficient for a specific energy level (for example the Fermi level), Is what I summarize right?

2.And another question is that in the transmission eigenstate panel we have to choose quantum number in addition to energy level? As far as I know quantum number relates to a specific band(is that right?) So how do we choose quantum number? Does it have something with the energy level?

[zh]

1) For the point mentioned by you, it is right.
2) The quantum number in the TransmissionEigenstate() is the number of desired transmission eigenstate, i.e., this parameter will mean which transmission eigenstate will be calculated.

[yongjunwinwin]

1.Could you explain more clearly about quantum number? I majored in electronics engineering without a solid basis of chemistry and physics. So please explain in some more detailsabout the number of the transmission eigenstate?

2. I have attached the result of the transmission path and transmission eigenstate(energy level=0 eV, quantum number=0) of N=5 zigzag graphene nanoribbon. From the transmission path I can conclude that the electron mainly travel through the edge of the ribbon.
But I cannot interpret the iso-surface of transmission eigenstate, could anyone give me some advice about what I get from the transmission eigenstate.

Thanks in advance for any helpful responses.

[zh]

1. For example, the quantum number of 0 in the TransmissionEigenstate() means that the transmission eigenstate of the first transmission eigenvalue will be calculated.  
  quantum number         transmission eigenvalue
          1                            the 2nd
          2                            the 3rd
           ..................
2. For the zigzag graphene nanoribbon, maybe you did the non-spin-polarized calculation. In this situation the edge state exactly appears at the Fermi level (i.e., energy level = 0 eV). This is the reason why the transmission eigenstate for E=0 and quantum number of 0 mainly appears along the edges.

[yongjunwinwin]

1.Then, generally speaking, Did we need to calculate all the eigenstate of the specific energy level?

2.Yes,I did the non-spin-polarized calculation. You mean if I did the spin-polarized calculation, then the edge state will not appear at the Fermi level?

[zh]

1). No necessary to calculate all the transmission eigenstate for a specific energy level. It is more meaningful to analyze the transmission eigenvalues that have significant contributions to the transmission coefficient.
2). Yes, the edge state of zigzag graphene nanoribbon will become spin-splitting in the spin-polarized calculations and a small gap appears. For the edge states of zigzag graphene nanoribbons, it is well studied in literature. For more details of the edge states, please check the literature.

[yongjunwinwin]

Another question is that should spin-polarized calculation be adopted for all the calculation about zigzag graphene nanoribbon? I mean if I conducted the calculation of my model of zGNR using tranditional LDA, will this affect the acceptance of my paper for publication?

[zh]

It depends on the problem under study. There are many published papers that just did the non-spin-polarized calculations for discussing only the electron transport properties of zGNRs. If the spin-related transport properties (e.g., spin-flipping) are discussed, one must carry out the spin-polarized calculations.

[yongjunwinwin]

Thanks for your reply!