question on transmission coefficient

[cherry]

hi, I have one question due to the transmission coefficient. My result of transmission coefficient of a two-probe system shows that the coefficient value can be larger than 1 at some energy levels. Although it says that T(k)>1 if several channels contribute to the transport at the same energy. However, based on the physical meaning (or definition) , this value should be always between o and 1.
Anyone can help me to understand this better? Thank you very much.

cherry

[Nordland]

The transmission probability for a single incomming electron wave is always between 0 and 1. The transmission spectrum is not a spectrum of the probability of a single electron being transmitted through the two-probe It is the sum of probability of all the single modes being transmitted through the two-probe, so if there. at a given energy,
is three block waves called A,B,C and each of these has a transmission probability between 0 and 1, then the transmission spectrum is sum of the probability of A, B & C added together.

In more mathematical terms, you can see it as this:
For each block wave in the electrode, define a stochastic variable [tex]X_i[/tex] with a probability [tex]t_i[/tex] for being [tex]1[/tex] and [tex]1-t_i[/tex] for being [tex]0[/tex]. Define a new stochastic variable as the sum of these stochastic variable [tex] X = \sum_i X_i[/tex].  The value in the transmission spectrum is the expectation value of this stochastic variable [tex]<X>[/tex], which equals [tex]<X> = \sum_i <X_i> = \sum_i t_i[/tex], and therefore even if [tex]t_i[/tex] always is between 0 and 1, then the sum is not.

I hope that I am making it more clear

[cherry]

Thanks a lot, Nordland.
How many waves for a 3-D electrode in a two-probe system? Should we use transmission eigenvalues or eigenstate to estimate it? Thanks.

[Nordland]

There is technical infinite number of modes in a 3D electrode.
In order to get the full transmission spectrum of a electrode you need to perform an integral/average over the entire parallel k-space, and for each
k-point in the parallel k-space, there is a finite number of modes, but there is finite number of modes for each k-point, but the there is an infinite numer of k-points.
Therefore when we calculate the transmission spectrum, we need to perform this integral over k-space, and this is done using a ri

The best way to predict the transmission spectrum is to calculate the bandstructure for the electrode, and count the number of bands at a given energy, however this is almost only possible for 1D systems.

[cherry]

Thanks a lot. Nordland

[Nordland]

You are most welcome