Hi Joe,
Thanks for getting back so quickly!
I realized after sending my initial email that I made a mistake in the ordering
of the Python functions. You are indeed correct that I should be summing after
taking the modulus squared of the transmission coefficient matrix elements.
Before we finish the discussion, I just wanted to verify that the indices of
the modes I am finding match up with the indices of the submatrix. You
mentioned that I had the right idea, but did not comment on that. I just want
to be sure.
Thanks again,
Kevin
> On Aug 1, 2018, at 7:45 AM, Joseph Weston <joseph.westo...@gmail.com> wrote:
>
> Hi Kevin,
>
> Thanks for the well-posed and focused question!
>
>> I would like to calculate the valley polarized conductance for a graphene
>> nanoribbon with 2 leads (let’s say in the x-direction). My current approach
>> is this:
>>
>> 1) Get the S-matrix for a certain energy
>> 2) Find the indices of the modes with positive velocities
>> 3) Separate these modes by positive and negative momenta
>> 4) Get the sub matrix from the S-matrix
>> 5) Sum the transmission coefficients for positive and negative momenta
>
> This is the right idea.
>
>> 2) Find the indices of the modes with positive velocities
>
> We only care about selecting (by their valley) the correct *incoming*
> modes. In Kwant's convention these are the modes with *negative*, not
> positive, velocity.
>
> This is in an obscure part of the documentation [1], where we say
>
>> positive velocity and momentum directions are defined with respect to
> the translational symmetry direction of the [lead]
>
> When leads are attached to a scattering region, their symmetry direction
> points *away* from the scattering region "towards infinity", which means
> that the incoming modes have negative velocity in Kwant's convention.
>
>
>> The code below illustrates these 5 steps:
>>
>> import progressbar
>> def valleyPolarizability(syst, energies, lead_start=0, lead_end=1):
>> KPs = []
>> Ks = []
>> bar = progressbar.ProgressBar()
>> for energy in bar(energies):
>> smatrix = kwant.smatrix(syst, energy, params={'phi': phi})
>> positives = np.where(smatrix.lead_info[lead_start].velocities >= 0)[0]
>> momentas = smatrix.lead_info[lead_start].momenta[positives]
>> K_prime_indices = np.where(momentas < 0)[0]
>> K_indices = np.where(momentas >= 0)[0]
>> submatrix = smatrix.submatrix(lead_end, lead_start)
>> K_prime_T = np.absolute(np.sum(submatrix[:, K_prime_indices]))**2
>> K_T = np.absolute(np.sum(submatrix[:, K_indices]))**2
>> KPs.append(K_prime_T)
>> Ks.append(K_T)
>> return KPs, Ks
>
> This is almost correct except for a couple of things. Firstly we should
> select the modes that have negative velocity, as already discussed.
> Secondly:
>
>> K_prime_T = np.absolute(np.sum(submatrix[:, K_prime_indices]))**2
>
> Here you're summing the scattering *amplitudes*, when you really want to
> sum the scattering *probabilities*:
>
>> K_prime_T = np.sum(np.abs(submatrix[:, K_prime_indices]))**2
>
> i.e. you need to take the absolute square before summing.
>
>
> Happy Kwanting,
>
> Joe
>
>
> [1]:
> https://kwant-project.org/doc/1.3/reference/generated/kwant.physics.PropagatingModes
>
>
