[Wien] Transition probabilities and broadening

[Peter Blaha]

Hi,

First of all I suggest you read the "optic" paper of Claudia Draxl and 
J. Sofo, the authors of the optic module (see UG for reference).

> 1) I am interested in plotting the transition probability as function of 
> energy to understand the energy loss function of Bi2Te3. As I understood 
> it the imaginary part of the dielectric function is given by
> 
> Eps_im(omega, 0) = (constant/omega^2) times | Matrix elements |^2 times 
> JDOS (eq 2.18 on page 20 of the book “Excitation of Plasmons and 
> Interband Transitions by Electrons” by Heinz Raether)
> 
> So in principle, I could take the data from Eps_im and divide it by the 
> data in the JDOS file and multiply it by omega^2 (Energy^2 divided by 
> hbar^2), this way I have the transition probabilities as function of energy.
> Would this be a correct way of doing it?

Yes, but only in an average way. In your equation you miss that the 
matrix elements are k-point dependent.

At a certain energy several k-points and bandcombinations (i,j) may 
contribute to eps_2, and each of these (i,j) transitions may have a very 
different transition probability. It will certainly not be a smooth 
function.

While for core level spectra the transition probability is really a 
smooth function of energy and it makes sense to plot it, for optics it 
is more a property of the atom- and l-character of valence and 
conduction band with a certain energy difference between them.

> Alternatively, I understood that it’s possible to get the symmatrized 
> matrix elements in case.symmat as stated 
> here: https://www.mail-archive.com/wien@zeus.theochem.tuwien.ac.at/msg04663.html
> 
> However, when I look at case.symmat it is not completely clear to me as 
> to how I should interpret the columns in the file, so I would like some 
> help to be clear of the terms. For example the top part of the file has 
> the following output:
> ------------------------------------------------------------------------------------------------------------
> 2     Re <X><X>.         Re <Z><Z>                        ALL
> 
> KP:        1 NEMIN  NEMAX  :        16       99  dE:       -5.0       
>   3.0   K:                 1
> 
> ------------------------------------------------------------------------------------------------------------
> 
> And then here some columns. My assumption is that Re  <X><X> and  Re 
> <Z><Z> are the squared matrix elements (and so the transition 
> probabilities) along the x-axis and z-axis, respectively. I do not 
> understand what the “ALL” is for and in the second line I am guessing 
> that “KP” means k-point. I am guessing that the NEMIN and NEMAX 
> represent the lowest and highest band levels. dE is then the energy range.
> The last column “K:                 1” I am not sure how to interpret, 
> so some assistance on this would be highly appreciated.

Your interpretation is correct. "ALL" simply means it is the total 
matrix element (optic allows to compute eg. only the contributions from 
atom 1 or 2,.... , see UG for optics). The last column is just an index 
for the i^th k-point.
Important is, what you did not show: After the header comes:

       1   1 0.133101E-07 0.142705E-07
       1   2 0.683084E-12 0.415542E-13
       1   3 0.313927E-07 0.356868E-07
       ....
       2   2  ...
       2   3  ....
       ...
The first 2 columns are the band band indices i,j and the matrix 
elements are calculated between the 2 corresponding bands. You need eg. 
case.energy to read the corresponding eigenvalues and get the energy 
difference between them (+ the weights-files, which tell you if band i 
is occupied and j is unoccupied, since only those contribute to optics).
And the whole summation over k is done via the Tetrahedron method.

> 2) This question regards the “Gamma: broadening of interband spectrum” 
> that one introduces in the first line of the case.inkram file. Namely, 
> how is this broadening implemented? Is this a Lorentzian broadening? How 
> should I correctly interpret this broadening?

I'd think it is Lorentzian, but don't know for sure. It should be stated 
in the optics-paper. If not, one would have to check the code.
> 
> 
> Kind regards,
> Luigi Maduro
> 
> 
> 
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-- 
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Peter BLAHA, Inst.f. Materials Chemistry, TU Vienna, A-1060 Vienna
Phone: +43-1-58801-165300
Email: peter.blaha at tuwien.ac.at    WIEN2k: http://www.wien2k.at
WWW:   http://www.imc.tuwien.ac.at
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