Transmission of periodic systems

[asanchez]

Hi,

I've been reading around and it is quite clear to me that for perfectly periodic systems the transmission at a certain energy & transverse k-point must be an integer. But should the total transmission at a certain energy be an integer always?

In order to investigate this issue I have been trying to calculate the TransmissionSpectrum of bulk copper. I can see with the transmission analyzer that the number of channels at a particular (E,k) is always an integer ("numerical" integers anyway, e.g. 0.999999...) but then the "total" transmission can be non-integer. Is the total transmission a brillouin zone integration of the  values along the transverse directions and can be non-integer?

It isn't clear to me whether this is a physical result or it just means my calculation isn't  converged (my electrode is too small or not enough k-points, etc). I tried calculating this TransmissionSpectrum in both Device & Bulk Configurations. In both cases I'm getting non-integers. The bulk configuration I used was just one conventional cell and the device configuration included 2x conventional cells along the Z direction as electrodes and 3x as the central region. In all cases the size of both transverse directions was just 1 conventional cell.

Thanks in advance

[asanchez]

Could anyone clarify this for me? My question is:

Can a bulk crystalline system have non-integer transmission?

In case the previous post was too long. Thanks in advance.

[Jess Wellendorff]

Sorry for the waiting time, we are working on an answer for this post.

[Anders Blom]

Your analysis is correct. T(E,k) can only be integer if there is no scattering, but when this is integrated over the 2D zone, the values will not, in general, be integer. The statement about "integer transmission" is typically made for 1D systems, where the integration is not needed.

[asanchez]

Thank you Dr. Blom, much appreciated.