PHYSICS TODAY SEPI[MBER isso COMPUTATIONAL MATERIALS SCIENCE COMPUTATIONAL MATERIALS SCIENCE: THE ERA OF APPLIED QUANTUM MECHANICS or many centuries, mate- The properties of new and artificially istry.) The impact of DFT on Frials were discovered, structured materials can be predicted and physics, however, was cer- mined, and processed in a tainly not smaller than its impact on chemistry. DFT largely serendipitous way. explained entirely by computations, using proves that the ground-state However, the characteriza- atomic numbers as the only input. tion of the atom and the pro- energy of an M-electron sys- tem is a function only of the gress made in x-ray diffrac- tion during the early years of Jerzy Bernholc electron density p(r). In DFT, this century started a quest the electrons are represented for a theory of materials in by one-body wavefunctions terms of their atomic constituents. Later decades saw iQ, which satisfy the Schrodinger-like equations (in Rydbergs): scientists developing many qualitative and semi-quantita- tive models that explained the principles of atomic cohe- sion and the basic properties of semiconductors, metals, and salts. Considering their simplicity, some ofthe models were surprisingly accurate and led to remarkable progress. KV2+VN(r)+ However, for most materials of current interest, the in- 12p(r’)dr +/¾c[P(r)]J a~~(r) = teratomic interactions are intricate enough to require i=1,. , M. fairly elaborate models. Fortunately, we are entering an The first term represents the kinetic energy; the second era in which high-performance computing is coming into is the potential due to all nuclei; the third is the classical its own, allowing true predictive simulations of complex electron—electron repulsion potential; and the fourth, the materials to be made from information on their individual so-called exchange and correlation potential, accounts for atoms. the Pauli exclusion principle and spin effects. The exact Methods for computing the properties of materials form of the exchange—correlation term is unknown, but a can be divided into two classes: those that do not use any local approximation, in which the exchange—correlation empirically or experimentally derived quantities, and potential of a homogeneous electron gas of density p(r) is those that do. The former are often called ab initio, or used at each point, has proved highly successful. DFT first-principles methods, while the latter are called em- generally predicts lattice constants, atomic positions, elas- pirical or semi-empirical. The ab initio methods are par- tic properties and phonon frequencies with errors smaller ticularly useful in predicting the properties of new mate- than a few percent. For example, my research group at rials or new complex material structures, and for predict- North Carolina State University computed the radial dis- ing trends across a wide range of materials. The tribution function of the then new solid C60 (see figure 1) semi-empirical methods excel at interpolating and ex- six months before the first neutron scattering data, yet trapolating from known properties. This article focuses on the result of2theIn fact,computationthe theoreticalagreed resultsalmost perfectlywere usedwithfor the ab initio methods, which retain their predictive power theexperiment.initial calibration of the experimental setup. even when experimental data are scarce or unavailable. Density functional theory predictions of cohesive en- ergies used to be less accurate, but including terms de- Methodology of ab initio calculations pendent on the gradient of the electron density has sub- As is well known, the binding in molecules and solids is stantially improved the agreement with experiment and due to the Coulomb forces between electrons and nuclei. with high-level quantum chemistry calculations.3 The exact solution of the full, many-body Schrodinger Large-scale computations equation describing a material is, of course, impossible, but one can make surprisingly accurate approximations Because the electronic structure calculations described above are computationally demanding, progress in the of a system’s ground state, and such approximations are field depends in perhaps equal measure on advances in widely used in condensed-matter physics. These approxi1- mations are based on density functional theory (DFT), theoretical methods and on advances in computer tech- which was developed at the University of California, San nology. For simple materials, such as silicon with only two Diego, by Walter Kohn, Pierre Hohenberg, and Lu Sham atoms in a periodically repeated unit cell, the computa- (and for which Kohn shared the 1998 Nobel Prize in tional effort required has become so modest now that the chemistry with John Pople, a pioneer in quantum chem- calculations can be carried out on any contemporary per- sonal computer. However, understanding the growth and properties of silicon devices requires calculating the char- acteristics of silicon surfaces. Because silicon is a cova- lently bonded material, the creation of a surface leads to 30 SEPTEMBER 1999 PHYSICS TODAY © 1999 American Institute of Physics, S-0031-9228-9909-020-1 FIGURE 1. CALCULATED electron distribution in C 60 at 1000 K, obtained from quantum molecular2 Yellow,dynamicsgreen, simulations.and blue denoteregions of successivelygreater electron density. The atomic structure of C 60 (background) consists of five- and six-membered nngs arranged in the shape of a soccerball. the well-known fast Fourier transform (FFT) algorithm, which works very quickly on vector su- percomputers and modern workstations. However, be- cause the FFT is a global operation, its performance slows down on massively parallel computers. For a large problem, the number of plane waves can be 50 000 or more, and one must use iterative diagonaliza- tion methods that mostly work with the occupied subspace. A particularly ef- fective approach was first developed by Roberto Car and Michele Parrinello (then at the International School for Advanced Stud- ies in Trieste), who com- bined the solution of the electronic structure prob- lem with molecular dy- namics for the atoms.5 Another approach is to solve itera- broken bonds. This process is energetically unfavorable, tively for the electronic wavefunctions, compute the forces, so some atoms will move to rebond, forming a recon- and move the atoms by a large step.6 In both cases the structed surface with less symmetry. (See the article by atoms follow Newton’s equations of motion with ab initio John J. Boland and John H. Weaver, PHYsIcs TODAY, interatomic forces. These methods are called ab initio, or August 1998, page 34.) Depending on the complexity of quantum molecular, dynamics. the resulting structure, the number of atoms N that one Ab initio calculations have long been useful in ex- must consider in the unit cell will range from 16 to 400. plaining experimental results and providing unique in- Because2, andtheasymptoticallysize of the computationaleven as N3, effortit is nogrowswonderroughlythat sights. The recent advances make it possible to predict asprogressN in computational materials physics is closely tied the properties of materials with complex atomic arrange- to progress in methodology and computers. ments, whose study would have been prohibitively expen- The largest ab initio calculations usually make use sive just a few years ago. The examples below illustrate of functions called “pseudopotentials,” which replace the the role that accurate and quantitatively reliable calcula- nuclear potential and the chemically inert core electrons with tions can play in modern condensed matter physics and an effective potential, so that only valence electrons are materials science, while also highlighting the advantages explicitly included in the calculations. (See the article by of collaboration and close interaction between theorists Marvin L. Cohen, PHYsIcs TODAY, July 1979, page 40; also and experimenters. see ref. 4.) The pseudopotentials are derived from atomic calculations that use atomic numbers as the only input. Solid C36 Because pseudo wavefunctions are smooth and nodeless, Some of the most exciting new materials discovered in the plane waves can be used as a basis set. This offers three last decade are the fullerenes. Solid C major advantages: 60, once it was > Plane waves do not depend on the atomic positions, so produced in quantity, was shown to have a number of using them makes the results more precise. remarkable properties, including superconductivity after > The accuracy of the result is determined by a single intercalation with alkali metal atoms. In fact, its transition parameter, the highest kinetic energy of the waves in- temperatures T~ are approaching 40 K, second only to cluded in the calculations. high-Ta oxides. The relatively high T~ of C60 is due to the > The kinetic energy (— V2) is diagonal in Fourier space, strong electron—phonon interaction in curved fullerenes. whereas the potential is diagonal in real space. This has stimulated Marvin Cohen, Steven Louie, and The transformation between the two spaces occurs via their coworkers at the University of California, Berkeley,7 In- to examine fullerenes with even greater curvatures. SEPTEMBER 1999 PHYSICS TODAY 31 at a grain boundary the coordination constraint becomes weaker. There is more room for relaxation, and arsenic atoms can assume their preferred three- fold coordination. Wouldn’t that behavior break bonds and make some silicon atoms very unhappy? Not if arsenic atoms act in pairs,