Quasiparticle (GW) or BSE spectrum for anisotropic crystal

[cantele]

Dear all,
my question(s) concerns the GW correction applied to an anisotropic crystal.

i) we expect an electric-field-direction dependent correction, right? In other words setting the electric
field direction as 100, 010 or 001 should give different results. That means a quasiparticle gap
which depends on the polarization of the perturbing electromagnetic field.

ii) the coordinates used by default for the electric field by Yambo are Cartesian, right? That means
that 100 means electromagnetic field with electric field along the x axis.
For non-orthogonal crystals, should it be more meaningful to use the direction of the direct or of the
reciprocal lattice vectors?

iii) let us assume that now the experimental result (e.g. absorption coefficient) has been obtained
with a sample composed of randomly oriented crystallites. I can compute three independent quasi-particle-corrected
dielectric functions (or else BSE spectra). Whichever are the three independent directions, what should better
fit the experiments? The arithmetical average of the three?

Thanks for help!

Giovanni

[andrea marini]

i) we expect an electric-field-direction dependent correction, right? In other words setting the electric
field direction as 100, 010 or 001 should give different results. That means a quasiparticle gap
which depends on the polarization of the perturbing electromagnetic field.

This a tricky point. QPs are dressed electronic states due to screening created by the other particles. it is a sort of improved mean-field approximation. QPs exist in any media regardless of the presence of an external perturbation. So, formally speaking, there should not be any Electric field dependence of the QP properties.

Nevertheless QP corrections may depend on the anysotropy effects on the dielectric screening, that in practice it is observed as a tensor-like structure of the inverse dielectric matrix when q->0. This can be easily verified by calculating the EELS along the three carthesian directions. If the EELS is difference you may expect some problem.

However, much care must be taken in analyzing these effects, because:

1. In 0D systems (molecule, atom) the q->0 contribution is negligible so no electric-field dependence
2. Large anysotropy in 1D 2D and 3D systrems means a lot of reciprocal space vectors, so also in these cases the q->0 contribution should be small

So, again, if there is any effect, it is actually related to the anisotropy in the dielectric screening and not to the presence of an Electric field.

Giovanni did you calculate the QP corrections along X,Y and Z ? What are the differences in teh gap ?

ii) the coordinates used by default for the electric field by Yambo are Cartesian, right? That means
that 100 means electromagnetic field with electric field along the x axis.
For non-orthogonal crystals, should it be more meaningful to use the direction of the direct or of the
reciprocal lattice vectors?

Yes electric field coordinates (remember that the modulus is meaningless) are in chartesian coordinates.
I would use physical arguments to decide the direction. If the system is a surface I would check that for polarization on the plane there is no difference while there is the polarization is perpendicular to the plane. What is your system ?

iii) let us assume that now the experimental result (e.g. absorption coefficient) has been obtained
with a sample composed of randomly oriented crystallites. I can compute three independent quasi-particle-corrected
dielectric functions (or else BSE spectra). Whichever are the three independent directions, what should better
fit the experiments? The arithmetical average of the three?

It depends on the constitution of the sample. The most general expression for the absorption should be a statistical average of the different dielectric absorptions as a function of the polarization direction multilpied by the distribution function of the crystallites. Can you estimate this distribution ? An erithmetical average would correspond approximately to take an uniform distribution. In this paper, for example, we used a gaussian-like distribution sampled for a few values of the polarization direction.

Andrea

[cantele]

i) we expect an electric-field-direction dependent correction, right? In other words setting the electric
field direction as 100, 010 or 001 should give different results. That means a quasiparticle gap
which depends on the polarization of the perturbing electromagnetic field.

This a tricky point. QPs are dressed electronic states due to screening created by the other particles. it is a sort of improved mean-field approximation. QPs exist in any media regardless of the presence of an external perturbation. So, formally speaking, there should not be any Electric field dependence of the QP properties.

Nevertheless QP corrections may depend on the anysotropy effects on the dielectric screening, that in practice it is observed as a tensor-like structure of the inverse dielectric matrix when q->0. This can be easily verified by calculating the EELS along the three carthesian directions. If the EELS is difference you may expect some problem.

So, the presence of this line

% LongDrXp
0.000000 | 0.000000 | 1.000000 | # [Xp] [cc] Electric Field

in the yambo input for the quasiparticle GW correction is needed for the calculation of screening, isn't it?

However, much care must be taken in analyzing these effects, because:

1. In 0D systems (molecule, atom) the q->0 contribution is negligible so no electric-field dependence
2. Large anysotropy in 1D 2D and 3D systrems means a lot of reciprocal space vectors, so also in these cases the q->0 contribution should be small

So, again, if there is any effect, it is actually related to the anisotropy in the dielectric screening and not to the presence of an Electric field.

Therefore, if I see a "polarization-dependent" spectrum in a 0D system, like in M. Palummo et al, J. Chem Phys. 131, 084102 (2009), it is because
it is a spectrum, right? I mean that there is only one quasi-particle gap, but a spectrum for each polarization because the spectrum is the response to
the external polarization

Giovanni did you calculate the QP corrections along X,Y and Z ? What are the differences in teh gap ?

If all of the above is right, this means that something wrong is occurring in my calculation! I have a kind of organic crystal with monoclinic cell,
and I tried to make the quasi-particle correction, by changing the direction of the electric field (100, 010, 001, even though these are NOT the
direction of the primitive lattice vectors). It was very surprising that, by looking to the o.qp files in the three cases the 2.285 eV DFT gap was reverted to
6.245 eV, 4.118 eV and 4.141 eV for the three polarizations, respectively!!!! I explained the fact that y and z electric fields produce "the same quasiparticle gap"
by considering that the primitive vectors of the lattice are:
crystal axes: (cart. coord. in units of a_0)
a(1) = ( 1.000000 0.000000 0.000000 )
a(2) = ( 0.000000 0.412396 0.000000 )
a(3) = ( -0.676389 0.000000 1.888873 )

reciprocal axes: (cart. coord. in units 2 pi/a_0)
b(1) = ( 1.000000 0.000000 0.358091 )
b(2) = ( 0.000000 2.424855 0.000000 )
b(3) = ( 0.000000 0.000000 0.529416 )
so, y and z directions of the electric field are parallel to two reciprocal lattice vectors, therefore orthogonal to two lattice planes families. I didn't try the b(1) direction (thinking to a wave polarization, I choosed three orthogonal directions), but now I'm convinced it is worth!!!

ii) the coordinates used by default for the electric field by Yambo are Cartesian, right? That means
that 100 means electromagnetic field with electric field along the x axis.
For non-orthogonal crystals, should it be more meaningful to use the direction of the direct or of the
reciprocal lattice vectors?

Yes electric field coordinates (remember that the modulus is meaningless) are in chartesian coordinates.
I would use physical arguments to decide the direction. If the system is a surface I would check that for polarization on the plane there is no difference while there is the polarization is perpendicular to the plane. What is your system ?

I explained it above, for a surface or a quantum wire it is easier to figure out what are the "right" directions, for an anisotropic crystal not so much, but maybe from the above discussion a choice could be the reciprocal lattice primitive vectors.

iii) let us assume that now the experimental result (e.g. absorption coefficient) has been obtained
with a sample composed of randomly oriented crystallites. I can compute three independent quasi-particle-corrected
dielectric functions (or else BSE spectra). Whichever are the three independent directions, what should better
fit the experiments? The arithmetical average of the three?

It depends on the constitution of the sample. The most general expression for the absorption should be a statistical average of the different dielectric absorptions as a function of the polarization direction multilpied by the distribution function of the crystallites. Can you estimate this distribution ? An erithmetical average would correspond approximately to take an uniform distribution. In this paper, for example, we used a gaussian-like distribution sampled for a few values of the polarization direction.
Andrea

This is a very helpful suggestion!!!! Maybe the other point of view is: a single crystal experiment but with unpolarized light. In this case, maybe one should use any three orthogonal directions and then take the average.

Sorry for too many questions!!

Giovanni

[Daniele Varsano]

Dear Giovanni,

his is a very helpful suggestion!!!! Maybe the other point of view is: a single crystal experiment but with unpolarized light. In this case, maybe one should use any three orthogonal directions and then take the average.

That's right! What we are calling average in reality is the trace of the polarizability tensor. You get the expression
1/3Tr{alpha_ij} performing an "isotropic" average (i.e. assuming you have molecule randomly distributed, for instance in
solution) and you can see that the out-of diagonal component sum to zero. In cases where you have a well oriented sample,
and polarized light, thinks changes, and you may look at the different absorption with different polarization (i.e. perpendicular
and parallel), and for instances took the differences of the two. That's something experimentalists do in the linear dichroism.
In presence of external magnetic fields thinks changes again, the out-of diagonal parts of the tensor does not averages to zero,
and this is the principle of the circular magnetic dichroism.

Hope it makes thinks more clear,

Daniele

[nayam]

Very informational thread on quasiparticle (GW) or BSE spectrum for anisotropic crystal. It is very useful to me for my thesis.