[gpaw-users] Dielectric function and polarizability of low dimension materials

[Tian Tian]

Hi Thomas,

Thanks a lot. Your explanation helped me a lot.

Just curious, if the macroscopic dielectric response tensor \epsilon of a 2D material in a superlattice (full Coulombic), will it be possible to deduce from \alpha?
i.e.

1) \epsilon_M(L) = 1 + 4 \pi \alpha / L, where L is the length of the superlattice in z-direction. If this is true such relation is valid for all directions.

I found this relation also proposed for epsilon_xx and epsilon_yy for 2D materials, (Cudazzo et al. Phys. Rev. B 84, 85406 (2011).) while for epsilon_zz some literature gives a different relation:

2) \alpha \propto (1 - 1/\epsilon) (Tóbik, J. et al. & Dal Corso, J. Chem. Phys. 120, 9934–9941 (2004).)

In order to reconstruct epsilon for a superlattice 2D material from \alpha, which approach would be correct for the z-direction then?

Thank you and best wishes
Tian


在 2018年8月3日，上午11:47，Thomas Olsen <tolsen at fysik.dtu.dk<mailto:tolsen at fysik.dtu.dk>> 写道：

Hi Tian

Yes that is correct. For example for a 2D system, the dipole moment per volume does not make much sense so instead one can consider the 2D polarization (dipole moment per area) and the 2D polarizability defined by

P^{2D} = \alpha^{2D} E,

which has the unit of length. \alpha^{2D} is well-defined for a 2D system, but \epsilon_M is not and will depend on the length of the unit cell in the non-periodic direction.

/Thomas
________________________________________
Fra: Tian  Tian <tian.tian at chem.ethz.ch<mailto:tian.tian at chem.ethz.ch>>
Sendt: 3. august 2018 11:26
Til: Thomas Olsen
Cc: gpaw-users at listserv.fysik.dtu.dk<mailto:gpaw-users at listserv.fysik.dtu.dk>
Emne: Re: [gpaw-users] Dielectric function and polarizability of low    dimension materials

Hi Olsen,

Thanks a lot for your prompt response. I see your point. So by its definition, the \alpha calculated here is closer to the electric susceptibility (the usually referred \chi in electrostatics, i.e. P = \chi E)?

Also from the equation Im(\eps) = 4\pi Im(\alpha), it seems that \eps and \alpha has the same unit, while according to the manuscript the unit given by get_polarizability has the unit \AA^(non-periodic directions). So should the equation between epsilon_M and \alpha be actually Im(\epsilon_M) = 4\pi * \alpha / V_{non-pbc} ?

Best
Tian


在 2018年8月3日，上午10:38，Thomas Olsen <tolsen at fysik.dtu.dk<mailto:tolsen at fysik.dtu.dk><mailto:tolsen at fysik.dtu.dk>> 写道：

Hi Tian

The formula follows from the relation between E and P and the definition of the polarizability: P = \alpha E

\epsilon E =  4\pi D = E + 4 \pi P = (1+4\pi\alpha) E

so \epsilon = 1 + 4\pi \alpha

The \alpha appearing in the Claussius-Mossotti formula is the atomic (or molecular) polarizability of the constituent atoms and not the bulk polarizability.
I dont think it is a good approximation fro covalently bonded solids.

/Thomas
________________________________________
Fra: gpaw-users-bounces at listserv.fysik.dtu.dk<mailto:gpaw-users-bounces at listserv.fysik.dtu.dk><mailto:gpaw-users-bounces at listserv.fysik.dtu.dk> <gpaw-users-bounces at listserv.fysik.dtu.dk<mailto:gpaw-users-bounces at listserv.fysik.dtu.dk><mailto:gpaw-users-bounces at listserv.fysik.dtu.dk>> på vegne af Tian Tian via gpaw-users <gpaw-users at listserv.fysik.dtu.dk<mailto:gpaw-users at listserv.fysik.dtu.dk><mailto:gpaw-users at listserv.fysik.dtu.dk>>
Sendt: 2. august 2018 17:46
Til: gpaw-users at listserv.fysik.dtu.dk<mailto:gpaw-users at listserv.fysik.dtu.dk><mailto:gpaw-users at listserv.fysik.dtu.dk>
Emne: [gpaw-users] Dielectric function and polarizability of low        dimension materials

Dear All,

I have one question concerning the polarizability of calculated by gpaw. In the gpaw tutorial,  https://wiki.fysik.dtu.dk/gpaw/tutorials/dielectric_response/dielectric_response.html, there is an equation relating the imaginary part of dielectric function and polarizability as:
[cid:81EE823F-EA6A-46CF-BBFD-E08100329181 at ethz.ch]
which is also called by the function gpaw.response.df.get_polarizability. I wonder if there is a reference to the concrete proof of this equation? For instance in the Claussius-Mossotti equation of bulk materials, such relation seems not valid:
[cid:CD7E206C-39F0-493B-B3F0-27E94C68C93E at ethz.ch]

I appreciate much your help concerning this issue, and wish there is any publications I can refer to.

Best wishes
Tian
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