Transmission eigenvalues for one mode larger than 1

[Jenny]

Hi, everyone.

When I tried to calculate the transmission eigenvalues at Fermi level for my one-dimensional system, it came out with the result as below. The weird thing is that I got the first four channels that have eigenvalues larger than 1. I thought the range for the eigenvalues should be 0 to 1. Why would this happen? 


+------------------------------------------------------------------------------+
| Transmission Eigenvalues Report                                              |
| ---------------------------------------------------------------------------- |
| Left electrode Fermi level   = -4.316601e+00 eV                              |
| Right electrode Fermi level  = -4.316601e+00 eV                              |
| Energy zero                  = -4.316601e+00 eV                              |
| Energy                       =  0.000000e+00 eV                              |
| ---------------------------------------------------------------------------- |
| Number of transmission modes = 49                                            |
+------------------------------------------------------------------------------+
Eigenvalues (Up):
  1.376425e+00
  1.142384e+00
  1.046487e+00
  1.003812e+00
  9.181272e-01
  8.580740e-01
  8.237050e-01



The parameters I used to do the calculation is like this :
    energy=0*eV,
    k_point=[0, 0],
    energy_zero_parameter=AverageFermiLevel,

[Anders Blom]

My spontaneous reaction is that this could have something to do with a factor 2 for spin... If you sum the eigenvalues, does it match the value of T(E) at this k?

[Jenny]

Hi, Dr. Blom.

I used the default setting spin.up. So I guess it should do nothing with the spin. And I check the sum of eigenvalues is 11.218 while T=10.6307 for the transmission spectrum at [0,0]. I thought this might related to the k-point sampling for the transmission spectrum calculation. So I performed another calculation with k-point sampling 21*21 for transmission coefficient calculation at Fermi level. I got T=9.36476 which is even smaller . I tried to plot the k-dependent transmission coefficient and attached here. I found that T is largest at [0,0], which is about 21.4. I think it is with a factor 2 for spin. The code I used to plot it is attached(same as tutorial https://www.quantumwise.com/documents/tutorials/latest/MolecularDevice/index.html/chap.analysis.html). why it didn't show the factor 2 problem in the tutorial? BTW, I'm using atk-13.8.1 version.

And back to my question, I still didn't understand why the transmission eigenvalue for one channel would be larger than 1?

[Anders Blom]

13.8 is old, there is no detailed support for that. Actually you should run 13.8.2, that's the correct 13.8 version, with all bug  fixes.

T(E,k) does not depend on the k-point sampling for the transmission, so there is no point to try higher or lower if you just want to understand one single k-point - just set kpoints = [[0,0]] for the Gamma point, for instance.

I am not immediately aware of any old problems with transmission eigenvalues, but again the code is so old - almost 3 years ago - so it might be. You could try to compare Recursion and Direct and Krylov methods...

Maybe your calculation hasn't converged properly - I suggest running the tutorial with exactly the same system and compare that you get the same results as in the tutorial.

[Jenny]

Hi, Dr. Blom.

I'll try to run the tutorial and compare. Thank you for your suggestion.
But in the tutorial, it said "The contour plot of the k-dependent transmission above has a pronounced peak at (kA,kB)=(0.18,0.0), with a transmission coefficient of ~2. In the computation of the transmission spectrum earlier, we did not have a dense enough k-point sampling, and hence the transmission was underestimated, i.e. we obtained 1.45 while the 21x21 mesh gives 1.58". It indicates that T(E,k) did depend on the k-point sampling for the transmission right? Also, from my results, T shows different value for different k-point sampling at Fermi level. I'm confused why you said T does noe depend on the k-point sampling?

Jenny

[Jess Wellendorff]

Hi Jenny. What Anders is saying is that the transmission at some particular k-point k, T(E,k), does not depend on the number of sampled k-points. T(E,k) is just the transmission T(E) in that particular k-point. However, the total (k-averaged) transmission, T, does depend on the density of k-points, just like you observe.