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# Correlated Materials Design: Prospects and Challenges

Abstract

The design of correlated materials challenges researchers to combine the maturing, high throughput framework of DFT-based materials design with the rapidly-developing first-principles theory for correlated electron systems. We review the physics of correlations, and propose strategies to capture their effects at each stage of the materials design workflow. We illustrate the workflow via examples in several materials classes, encompassing superconductors, charge ordering and Mott insulators, highlighting the interplay between theory and experiment and with a view towards finding new materials.

## Introduction

\label{sec:intro}

The ability to design new materials with desired properties is crucial to the development of new technology. The design of silicon and lithium-ion based materials are well known examples which led to the proliferation of consumer hand-held devices today. However, materials discovery has historically proceeded via trial and error, with a mixture of serendipity and intuition being the most fruitful path. For example, all major classes of superconductors–from elemental mercury in 1911, to the heavy fermions, cuprates and most recently, the iron-based superconductors–have been discovered by chance (Greene 2012).

The dream of materials design is to leverage, rather than ignore, our theories of electronic structure and combine them with our increasing computational ability to discover new materials. Beyond its technological implications, the challenge of materials design is also one of great intellectual depth. In principle, we know the fundamental equation needed to model the behavior of a material: it is the Schrödinger equation describing electrons moving in the potential of a periodic lattice, mutually interacting via the Coulomb repulsion. Solving this equation is another matter.

In practice, we can classify materials by how well we can solve their corresponding Schrödinger equation. For the class of compounds encompassing simple metals, insulators and semiconductors, termed weakly correlated materials, we have a well-developed theory of their excitation spectra called Fermi liquid theory. From a practical viewpoint, the theoretical framework of density functional theory (DFT) naturally lends itself to computational implementations for modeling properties. Materials which are not well-described by DFT are colloquially termed strongly correlated materials.

For weakly correlated materials, DFT has become the underlying workhorse of the scientific community. Extensive benchmarks of software implementations (Lejaeghere 2016) have shown that DFT reliably produces the total energy of a given configuration of atoms, enabling comparisons of stability between different chemical polymorphs. The maturity of DFT, combined with searchable repositories of experimental data (ICDD, ICSD, NIMS) data, has fostered the growth of databases of computed materials properties (Materials Project, AFLOWlib, NIMS). The field of weakly correlated systems has advanced to the point where one can successfully design materials (Fennie 2008, Gautier 2015, Fredeman 2011). A clear example that theoretical approaches are coming of age is the recent prediction of superconductivity in H$${}_{3}$$S under high pressure near 190 K (Duan 2014). Subsequently, hydrogen sulfide was observed to superconduct near 200 K, the highest temperature superconductor discovered so far (Drozdov 2015).

In contrast, materials design for strongly correlated systems is less mature, stemming from the fundamental challenge of understanding the physics of electron correlations. Correlated systems exhibit novel phenomena not observed in weakly-correlated materials: metal-insulator transitions, magnetic order and unconventional superconductivity are salient examples. While designing and optimizing materials with these properties would advance both technology and our understanding of the underlying physics, in practice we lack a tool akin to DFT capable of reliably modeling properties and scaling up to the thousands of calculations necessary.

In this article, we seek to summarize outstanding challenges in materials design as it pertains to correlated materials, and propose strategies to solve them. We begin by providing a practical definition of correlations (Sec. \ref{sec:correlations}), followed by a view of the workflow of materials design (Sec. \ref{sec:workflow}). Then we give four examples of materials design in correlated systems to illustrate the application of our ideas (Sec. \ref{sec:tlcscl3}-\ref{sec:bacoso}) and conclude with a brief outlook.

## What are correlated materials?

\label{sec:correlations}

The basic feature of correlated materials is their electrons cannot be described as non-interacting particles. Since the behavior of the constituent electrons are strongly coupled to one another, studying the behavior of individual particles generally provides little insight into the macroscopic properties of a correlated material. While this conceptual definition is valuable for understanding the fundamental physics, it is of less use to a DFT practitioner, who seeks to make predictions for comparison with experimental observations. To arrive at an operational definition of a correlated material, we examine DFT and how it relates to the observed electronic spectra.

The idea behind DFT is that for ground state properties, we can use the total electron density $$\rho(\vec{r})$$ as the fundamental variable in our equations, rather than the complicated quantum many-body wavefunction $$\Psi$$. Kohn and Sham (Kohn 1965) recast the theory in practical form: the proved that there exists a periodic potential $$V_{\text{KS}}(\vec{r})=V_{\text{KS}}[\rho](\vec{r})$$, which itself is a functional of the density, and that solving the problem of non-interacting electrons in the presence of this potential will give the exact density. The Kohn-Sham potential consists of three contributions: $$V_{\text{KS}}=V_{\text{ion}}+V_{\text{H}}+V_{\text{xc}}$$. The first is the one-body attractive potential of the nuclear ions. The other two arise from the electron-electron interaction: the (classical) Hartree component $$V_{\text{H}}$$ captures most of the Coulomb interaction, and the remaining contribution is contained in the exchange-correlation term $$V_{\text{xc}}$$. In practice, the exchange-correlation term is difficult to capture, and is generally modeled by approximations known as the local density approximation (LDA) or generalized gradient approximation (GGA).

Assuming we have the exact $$V_{\text{xc}}$$, DFT guarantees the correct ground state density and energy, but makes no claims about the electronic spectrum. For electrons moving in the lattice potential $$V_{\text{ion}}$$ of the nuclear ions, the general form of the Green’s function is

$$G(\omega)=\frac{1}{\omega+\nabla^{2}/2+\mu-V_{\text{ion}}-V_{\text{H}}-\Sigma(\omega)}.\\$$

We have used atomic units, included a chemical potential $$\mu$$, and separated the large Hartree component out from the self-energy $$\Sigma$$. The self-energy is generally frequency-dependent, and we have omitted the argument $$\vec{r}$$ from all quantities. The eigenenergies that are the result diagonalizing the Kohn-Sham hamiltonian $$H_{\text{KS}}=-\frac{1}{2}\vec{\nabla}^{2}+V_{\text{KS}}(\vec{r})$$ does not capture the frequency-dependent effects of many-body interactions, and should not be interpreted as physical eigenvalues. Nevertheless, we often ignore the lack of formal justification and compute the Green’s function using the Kohn-Sham solution anyway:

$$G_{\text{KS}}(\omega)=\frac{1}{\omega+\mu-H_{\text{KS}}}.\\$$

Thus, we find the self-energy in DFT $$\Sigma_{\text{KS}}=V_{\text{xc}}$$ is in fact frequency independent. In comparing with experiment, we always expect deviations at high frequencies because the non-interacting Kohn-Sham framework should only work well near the Fermi surface, where the quasiparticles of Fermi liquid theory are well-defined. However, at low frequencies (which for this section we take to mean $$\omega\lesssim 1$$ eV, a typical che