# How can I prove that a material is ferroelectric using DFT?

I would like to prove that a 2D material (Say ZnO) is ferroelectric. Since Ferroelectric materials are a subset of piezoelectric materials. How should I proceed to prove that the polarization is indicative of ferroelectric nature?

One way would be to show that the spontaneous polarization is reversed by reversing the applied electric field.

But this seems not to be the case, as the Q&A which I've found on ResearchGate states :

How does confirm ferroelectricity from Density Functional Theory ? Question

Ferroelectric materials give spontaneous electric polarization? Is there any limit of polarization value for ferroelectric materials? How does differ (1) dielectric polarization (2) paraelectric polarization and (3) ferroelectric polarization on the basis of DFT calculations.

Answer This is not a simple question! I will deal with the easier parts first.

1. Your definition of ferroelectricity is incomplete. Yes, there should be a spontaneous polarization, but it must be reversible under a sufficiently large electric field.
2. The difference between ferroelectric polarization and dielectric or paraelectric polarisation is that in the latter two cases you need to apply an electric field to observe the polarisation. The main differences between paraelectric and dielectric polarization are context - a paraelectric is the state just above the Curie temperature of a ferroelectric material and as such will normally exhibit a temperature dependence of permittivity that obeys the Curie Weiss law, which is evidence of a soft phonon mode. Given the temperature constraints of DFT, this is a difficult issue to address. Back to your first question; with DFT you can determine whether your material has a dipole moment by summing the Born effective charges in the unit cell. However you need to be careful about whether you are looking a single unit cell or a large supercell to determine whether the dipole moment is coherent across all unit cells - that will then confirm whether you have a spontaneous polarization and not an antipolar or randomly polar material. BUT proving that the spontaneous polarisation is reversible is the problem. Essentially the definition of a ferroelectric is a phenomenological one - it relies on the observation of a field reversible polarisation, which does cause some issues. In experiments, you may have a material that from its crystal structure looks as though it may have a spontaneous polarization, but for which the coercive field is beyond the breakdown field of the material. You cannot prove this is ferroelectric. However, your problem is slightly different and comes down to how well equipped is your implementation of DFT to calculate structures under applied electric fields, i.e. so that you can prove reversibility of the dipole moment under a sufficiently high field.

Apart from this is there anything I should keep in mind while working with 2D materials?

It would be nice to hear some thoughts on the same!

As a good starting, you may take a look at this classical paper and its supplementary material: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.117.097601

Here I just show some general steps:

1. Calculate the polarization (I assume that you can use the VASP package):

2. Calculate the $$A, B, C, D$$ of Landau-Ginzburg expansion:

$$E= \sum_i \dfrac{A}{2}(P_i^2)+\dfrac{B}{4}(P_i^4)+\dfrac{C}{6}(P_i^6)+\sum_{\langle i,j\rangle}\dfrac{D}{2}(P_i-P_j)^2$$

1. Calculate the Curie temperature with AIMD or Monte-Carlo method (optional for your question).

Hope it helps.