Avg spin polarization density

[huckelbuckel]

The following script in the Graphene-bloch state tutorial  calculates avg spin pol density by calculating electron densities of both up and down-spin electrons:
http://quantumwise.com/documents/tutorials/latest/GrapheneBloch/index.html/examples/density_average.py

I was able to get colorful graphs for zigzag and armchair ribbons with an indication that spin pol density for zigzag ribbons is much higher at the edges in comparison to armchair.

(see dig at the bottom of the page : http://quantumwise.com/documents/tutorials/latest/GrapheneBloch/index.html/electrondensity.spin.html)
The 'y' and 'z' axis display the size of ribbon in angstrom.There is a vertical colored scale (from -8 to +8) at the right side of graph

I am wondering what is the unit of this scale.Is it dipole moment per unit vol??  or is it just a number showing relative ratio?What i should write down in my manuscript if i use such graph?

Also spin polarization is calculated by using  up and down spin electron densities through formula:
r = (n_u - n_d)/(n_u + n_d)


but in avg spin pol density calculation a factor + 1e-19 has been added in denominator why?

# Calculate the polarization density from the up/down densities
(n_u, n_d) = nlread('density.nc', ElectronDensity)
n_up = n_u[:,:,:].inUnitsOf(n_u.unit())
n_dn = n_d[:,:,:].inUnitsOf(n_d.unit())
n = (n_up - n_dn)/(n_up + n_dn + 1e-19)
av = numpy.array(n[:,:,:].sum(axis=0))

[zh]

From the definition of so-called "spin polarization", the number in the figure mentioned by you shows just the relative ratio.

"1e-19"---->0.

[Anders Blom]

The quantity you plot is dimensionless, it's the ratio of two densities: the difference vs. the sum of the spin up and down densities, or short just the "spin polarization". However, I must admit the numbers are misleading; when the plot was created the focus was on the qualitative picture (seen by the colors). To also get it quantitatively correct, a more appropriate way would be to average, not sum, the values over the X axis. To do so, you can change the line defining "av" to

Code: python

av = numpy.average(numpy.array(n[:,:,:]),axis=0)

If you do so, the color axis will run from about -6% to +6%, and now you have a relevant measure of the average spin polarization in the YZ plane. The 1e-19 is irrelevant, it's just a safeguard against division by zero. However, in the latest implementation we have included this protection behind the scenes and therefore this term can actually be excluded without problems.

[huckelbuckel]

I ran it for zigzag Si based nano ribbon.I am attaching the graph for both the old and new spin polarization.The difference is visible

Old graph is for:
av = numpy.array(n[:,:,:].sum(axis=0))

New one is for:
av = numpy.average(numpy.array(n[:,:,:]),axis=0) 

For the new graph, i am getting  values at the edges like  0.128, 0.096    without any % sign.
Does it mean these values are 12.8%  and 9.6% as a relative measure of spin polarization??.

of course for Armchair ribbon i am getting much lesser values like....0.000020, 0.000015 etc

[Anders Blom]

0.128=12.8% yes

[huckelbuckel]

Thanks 'zh' and Dr. Blom for the help