The Impact of Density Functional Theory on Materials Research

www.mrs.org/bulletin functional. This functional (i.e., a function whose argument is another function) de- scribes the complex kinetic and energetic interactions of an electron with other elec- Toward Computational trons. Although the form of this functional that would make the reformulation of the many-body Schrödinger equation exact is Materials Design: unknown, approximate functionals have proven highly successful in describing many material properties. Efficient algorithms devised for solving The Impact of the Kohn–Sham equations have been imple- mented in increasingly sophisticated codes, tremendously boosting the application of DFT methods. New doors are opening to in- Density Functional novative research on materials across physics, chemistry, materials science, surface science, and nanotechnology, and extend- ing even to earth sciences and molecular Theory on Materials biology. The impact of this fascinating development has not been restricted to aca- demia, as DFT techniques also find application in many different areas of in- Research dustrial research. The development is so fast that many current applications could Jürgen Hafner, Christopher Wolverton, and not have been realized three years ago and were hardly dreamed of five years ago. Gerbrand Ceder, Guest Editors The articles collected in this issue of MRS Bulletin present a few of these suc- cess stories. However, even if the compu- Abstract tational tools necessary for performing The development of modern materials science has led to a growing need to complex quantum-mechanical calculations relevant to real materials problems understand the phenomena determining the properties of materials and processes on are now readily available, designing a an atomistic level. The interactions between atoms and electrons are governed by the series of simulations that meaningfully laws of quantum mechanics; hence, accurate and efficient techniques for solving the represent a physical or chemical property basic quantum-mechanical equations for complex many-atom, many-electron systems or model a process (e.g., a chemical reac- must be developed. Density functional theory (DFT) marks a decisive breakthrough in tion or phase transformation in a mate- these efforts, and in the past decade DFT has had a rapidly growing impact not only on rial), remains a challenging task, requiring fundamental but also industrial research. This article discusses the fundamental a judicious choice of approximations and principles of DFT and the highly efficient computational tools that have been developed an expert handling of these tools. for its application to complex problems in materials science. Also highlighted are state-of- the-art applications in many areas of materials research, such as structural materials, Designing DFT Calculations for catalysis and surface science, nanomaterials, and biomaterials and geophysics. Materials Science The successful application of DFT to a Keywords: computation, density functional theory, modeling, nanoscale, simulation. materials problem involves three distinct steps: (1) translation of the engineering problem to a computable atomistic model, (2) computation of the required physico- Introduction chemical properties, and (3) validation of During the past decade, computer sim- equation for the complex many-atom, the simulation results by confrontation ulations based on a quantum-mechanical many-electron system is not analytically with laboratory experiments. description of the interactions between elec- solvable, and numerical approaches have trons and atomic nuclei have had an in- become invaluable for physics, chemistry, Translation creasingly important impact on materials and materials science. A breakthrough in A theory that relates computable quan- science, not only in fundamental under- these computational efforts was realized tities to macroscopic engineering behavior standing but also with a strong emphasis in 1964 when Walter Kohn and co- is a prerequisite in order for DFT calcula- toward materials design for future tech- workers developed the density functional tions to be relevant. Before any computa- nologies. The simulations are performed theory (DFT), a theory based on electron tion can be started, significant effort often with atomistic detail by solving the density, which is a function of only three must be expended to translate an engi- Schrödinger equation to obtain energies spatial coordinates.1 The Kohn–Sham neering problem into questions and chal- and forces, require only the atomic num- equations of DFT cast the intractable com- lenges that can be addressed with DFT. bers of the constituents as input, and plexity of the electron–electron interactions This initial step requires considerable in- should describe the bonding between the into an effective single-particle potential sight into the materials science and engi- atoms with high accuracy. The Schrödinger determined by the exchange-correlation neering aspects of the problem. MRS BULLETIN • VOLUME 31 • SEPTEMBER 2006 659 The Impact of Density Functional Theory on Materials Research The translation step may be illustrated by differ from the computed result because of LDA and GGA are by far the most com- the application of DFT to the design of other factors that are not characterized monly used functionals today. All func- better materials for rechargeable lithium well in the problem translation. However, tionals exist in spin-degenerate and batteries. Some properties, such as the it should be emphasized that one of the spin-polarized versions. Quite generally, electrode potential for a new material, can most important roles of computational GGA calculations represent a systematic be directly related to total energy differ- modeling is to provide information that is improvement, compared with the LDA, ences2 and are therefore easy to compute. not accessible (or accessible only with great although for heavy elements, a certain But another critical property, such as cycle difficulty) by laboratory experiments. tendency of the GGA to overcorrect the life (the loss of capacity after repeated LDA error is observed. In certain cases charge and discharge), is more difficult to Hierarchy of Exchange-Correlation (prominent examples are the structural predict with DFT, because it can be caused Functionals for DFT and magnetic ground state of iron and the by a variety of poorly identified phenom- DFT reformulates the many-body potential-energy landscape for molecular ena, such as reaction of the material with Schrödinger equation in the form of a set dissociation on a metal surface) the differ- the electrolyte to form an insulating layer, of coupled one-electron equations (the ence is not merely a quantitative one: only mechanical degradation with resulting Kohn–Sham equations), where all of the GGA calculations11 predict iron to be bcc loss of electrical contact, and phase trans- many-body interactions are contained in and magnetic, whereas the LDA predicts formations in the material. Each of these the exchange-correlation functional the ground state to be hexagonal and non- factors requires a different computation. magnetic. Only GGA calculations lead to ៬ Exc [n(r)], (1) the correct height and location of the dis- Computation sociation barrier for a diatomic molecule In many cases, solving the Schrödinger where n(r៬) is the electron density distribu- in contact with the surface of a metal.12 equation is not enough. The length and tion in the crystal. (For a modern intro- Climbing the density functional ladder time scales accessible by DFT calculations duction into DFT, see, for example, further to the meta-GGA or hyper-GGA are defined by the maximal system size Dreizler and Gross3 or the recent book by leads to only moderate progress, com- (typically a few hundred atoms; up to a Martin,4 which also provides a thorough pared with the GGA.13 Hybrid functionals few thousand in special cases) and by the introduction to the theory and practice of enjoy enormous popularity in molecular maximum time span that can be covered electronic structure calculations.) However, chemistry, but in materials science they by ab initio molecular dynamics (MD) sim- the exact form of this functional for which are merely at the test stage. Preliminary re- ulations (several tens of picoseconds). A the Kohn–Sham equations would give ex- sults show great promise for insulating strategy of multiscale simulations must be actly the same ground-state answer as the and semiconducting systems, but also used to translate the results of atomistic many-body Schrödinger equation is not show a need to develop new hybrid func- DFT simulations to real-world scales. As known, and approximations must be used. tionals better suited to describing metals.14 an example, the elementary steps in the Perdew5 has referred to the hierarchy of catalytic oxidation of CO on a metallic sur- approximations developed during the last Basis Sets face (adsorption/desorption/diffusion of two decades as the “Jacob’s ladder of DFT.” Practically, the effective one-electron ᭿ CO and O2, surface oxidation, dissociation In the local density approximation Kohn–Sham equations are nonlinear of oxygen, reaction CO + O → CO2, des- (LDA), the local exchange-correlation en- partial differential equations that are itera- orption of CO2, etc.) at fixed chemical ergy density is taken to be the same as in a tively solved by representing the electronic potentials of the reactants may be treated uniform electron gas of the same density. wave functions by a

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