Ionic crystals such as solid electrolytes and complex oxides are central to modern energy storage, sensing, actuation, and other functional applications. A crucial fundamental issue in the atomic and quantum scale modeling of these materials is defining the macroscopic polarization. In a periodic crystal, the usual definition of the polarization as the first moment of the charge density in a unit cell depends qualitatively and quantitatively on the choice of unit cell; since there is no unique choice of unit cells, the dipole moment depends on the choice of the unit cell.

We examine this issue using a rigorous approach based on the framework of two-scale convergence. By examining the continuum limit of when the lattice spacing is much smaller than the characteristic dimensions of the body, we prove that accounting for the boundaries consistently provides a route to uniquely compute electric fields and potentials despite the non-uniqueness of the polarization. Specifically, different choices of the unit cell in the interior of the body leads to correspondingly different partial unit cells at the boundary; while the interior unit cells satisfy charge neutrality, the partial cells on the boundary typically do not, and the net effect is for these changes to compensate for each other.

In order to do away with this discrepancy related to the polarization definition, different authors have suggested different remedies to this problem. Notable among them is the "Modern Theory of Polarization" due to Resta-Vanderbilt, where they define the macroscopic polarization as the average current in the unit cell. This often goes by the "Berry phase" definition of polarization due to how the quantum mechanical current is calculated. Another definition of macroscopic polarization, popular in the continuum mechanics community, is the derivative of the free energy density with respect to the electric field. Unlike the former, the latter two definitions of polarization give us apparent unique quantities.     

    The objective of this work is to connect the three definitions of polarization presented above. We show that the physical quantities of interest arrived at using these three theories are the same. Having presented a solution to explain the discrepancy in the dipole definition of polarization, we use that result to connect the different definitions of polarization. We show that the dipole definition and the Berry phase definition of polarization compute the same bulk charge distribution. We then show that the free energy density definition has two parts, a surface and a bulk part, analogous to the surface and the bulk description presented in the 2-scale approach.

Here we show that the rate of change of polarization is not a unique quantity. We go through examples and calculations to understand why that is the case. We present simple counterexamples to disprove the claim.

Understanding the response of 2D materials for engineering purposes is of growing interest, and to this end, it becomes crucial to describe the macroscopic behavior of these materials for engineering purposes. Since the physics governing the stable configuration of the system is at the nanoscale while engineering applications are at a macroscopic scale, we can effectively homogenize over the small length scales to obtain an averaged model which would reasonably approximate the electronic nature at a macroscopic length scale. The idea of a thermodynamic limit keeping the overall energy density and charge density well defined is essential in achieving a macroscopic model.

    The interest in 2D materials is that they can exhibit large strain gradients, leading to significant flexoelectric response. From the point of view of applications, these 2D materials hold potential for developing sensors and energy harvesters.    

    Since flexoelectricity is the electro-mechanical coupling between strain gradients and polarization, extracting a well-defined polarization term from the overall energy becomes essential. We only consider the limit of the electronic contribution to the total energy neglecting the molecular contribution. This work considers the thermodynamic limit of 2D materials extracting a 2-scale limit from the kinetic and exchange energy functionals along with a polarization and surface charge term from the coulomb functional.

Here we want to analyze the Flexoelectric response of 2D Materials. Flexoelectricity is the field of coupling between strain gradients and polarization. We thus want to analyze the electronic nature of 2D materials when subject to such strain gradients; for computational purposes, we choose bending deformation of 2D materials. We will treat this as an application of the 2D Polarization Theorem by studying the system's energy against the radius of curvature and polarization in the system.

The Recursive Projection Method (RPM) is a method used for solving non-linear equations. The idea is to break up the original non-linear problem into two problems, one for which we can apply fixed-point iteration while another for which we can apply Newton-Rhapson. The reason for this is that Newton-Rhapson is computationally efficient, while fixed-point is numerically efficient. If we can identify the different regions where we can apply each of the methods, we can reduce the overall number of steps that each might take to reach the final solution.

The idea is to implement the RPM Method as an electronic solver(Mixing) for the LSMS, Ab-Init, etc. We hope to see that upon using many parallel processors, this reduces the number of steps considerably compared to simple mixing and Broyden mixing.

The field of Quantum Materials is characterized by two key aspects. One is a strong correlation and the other is topological properties. Kohn Sham equations make it possible to solve multi-electron Schrodinger's equation using the independent electron analysis. Most systems that can be approximated with this assumption have already been studied. The interest now is in materials where the independent electron analysis isn't valid. These materials are called Strongly Correlated materials. The quintessential example of this is superconductivity. This is when two electrons see each other. Due to the influence of some interactions (the only known example is phonon interaction), the Coulomb repulsion of these two electrons is overcome and they can be stabilized to form what is commonly known as Cooper pairs. It is the motion of these Cooper pairs that tend to show zero resistance when the material ti subject to a potential difference; hence the name superconductor. On the other hand, Topological materials are materials that tend to show topological properties. A quintessential example is Topological insulators where the bulk is insulating and the surface is conducting.

Continued below.

We have started the project on Quantum Materials by focusing on superconductivity.  There are two projects on superconductivity that I have been focusing on. The first is discovering a superconductor that exhibits a first-order phase transition. 

The first successful thermodynamic theory of superconductivity to describe the effect of a superconductor under an external magnetic field was by Ginzburg and Landau. They described the transition as a second-order phase transition that cemented its status in physics history as a second-order phase transition and all experimental measurements done in this field were that which corresponded to second-order phase transitions.

A second-order phase transition is a continuous phase transition where the thermodynamic phase of the state changes continuously across the transition temperature. 

A first-order phase transition has a discontinuous change in its thermodynamic state across the transition temperature and often has the absorption or release of latent heat (think of water boiling). There is often a sudden structural change in the material with tin being an example of a material where the structural change is high enough to disintegrate the material. The first-order nature of these transformations is often used to build switching devices since the transition requires some extra energy to initiate.

Our focus is finding superconducting materials that have been characterized as a second-order phase transition when in reality, they are first order. We use the thermodynamic energy minimization approach to describe the system's equilibrium. Using the ideas of Gibbs, we analyze the system under different loading devices. This gives us an estimate of the Helmholtz free energy of the material which can be used to make predictions about the material. Analyzing the experimental literature of different superconducting materials, we try to describe their behavior as a first-order phase transformation and make predictions to be validated by experimentalists.

One of the reasons to look at a material that would be superconducting under a first-order phase transition is that quite often a material that exhibits a first-order phase transition has a larger effect of stress on transition temperature.

Another example is to use applications of first-order phase transformation. In addition to their current application as superconductors, we could use the applications of materials that show a first-order phase transition.

The second project is focused on looking into a modification to the Ginzburg Landau theory of superconductivity due to Gurtin. 

Gurtin focused on the gauge-invariant real-space formulation of the Ginzburg Landau theory and noticed a term that is allowed by symmetry but was absent from the original Ginzburg Landau theory. This term is trivial in the real space gauge invariant formulation but is non-trivial when expressed in the original complex valued order parameter formulation. Our objective is to look at the implications of this term.