Density functional theory in the solid state

Abstract

Density functional theory (DFT) has been used in many fields of the physical sciences, but none so successfully as in the solid state. From its origins in condensed matter physics, it has expanded into materials science, high-pressure physics and mineralogy, solid-state chemistry and more, powering entire computational subdisciplines. Modern DFT simulation codes can calculate a vast range of structural, chemical, optical, spectroscopic, elastic, vibrational and thermodynamic phenomena. The ability to predict structure-property relationships has revolutionized experimental fields, such as vibrational and solid-state NMR spectroscopy, where it is the primary method to analyse and interpret experimental spectra. In semiconductor physics, great progress has been made in the electronic structure of bulk and defect states despite the severe challenges presented by the description of excited states. Studies are no longer restricted to known crystallographic structures. DFT is increasingly used as an exploratory tool for materials discovery and computational experiments, culminating in ex nihilo crystal structure prediction, which addresses the long-standing difficult problem of how to predict crystal structure polymorphs from nothing but a specified chemical composition. We present an overview of the capabilities of solid-state DFT simulations in all of these topics, illustrated with recent examples using the CASTEP computer program.

Keywords: computational chemistry; computational materials science; condensed matter theory; density functional theory; electronic structure theory.

Figure 1.

(a) The Coulomb potential (dashed line) and an example pseudo-potential (solid line) for a carbon atom, along with the corresponding radial components of the 2s-orbital. The potential and pseudo-potential match beyond r_c, as do the computed wave functions (b). (Online version in colour.)

Figure 2.

(a,b) The density of states around the Fermi level (dashed line) for the half-metal Heusler alloy CoFe_0.5Mn_0.5Si computed using PBE (a) and PBE+U (b). With the PBE exchange–correlation functional the system is predicted to be fully metallic, whereas the inclusion of a modest Hubbard U (U=2.1 eV) opens up a band gap for the minority spins. Adapted with permission from Hasnip et al. [30]. Copyright • 2013 American Institute of Physics.

Figure 3.

Defect levels in HfO₂ arising from oxygen vacancies (V) and interstitials (I) in a variety of charge states. The top valence band (VB) and bottom conduction band (CB) are shown for reference. The short lines show the energy level position within the gap and the dots show electron occupancies. On the x-axis, V indicates ‘vacancy’ and I ‘interstitial’ with the superscripts showing the defect charge state. Reproduced with permission from Xiong et al. [35]. Copyright • 2005 American Institute of Physics.

Figure 4.

A comparison between the computed band-gap from a screened-exchange (SX) DFT calculation (blue), an ordinary PBE DFT calculation (green) and the experimental band gap for a variety of materials. Reproduced with permission from Clark & Robertson [34]. Copyright • 2010 by the American Physical Society.

Figure 5.

Gap eigenstate of the charged oxygen vacancy and zinc vacancy in ZnO. The coloured contours show the charge density associated with the defect state ranging from low electron density (blue) through to high electron density (red). Reproduced with permission from Clark et al.[36]. Copyright • 2010 by the American Physical Society.

Figure 6.

The potential energy surface experienced by a hydrogen atom on an MgO (111) surface during layer-by-layer growth. MgO has a rocksalt structure, with an A-B-C stacking sequence along (111); initially the surface is OH-terminated (site B, with the Mg sublayer at site A), but as the next Mg layer is deposited (at site C) the OH dissociates and the H atoms rise to the new surface and move across to the potential energy minimum. This energy minimum is at site A, the next site in the MgO stacking sequence. Adapted with permission from Lazarov et al. [37]. Copyright • 2011 by the American Physical Society.

Figure 7.

Thermodynamically stable compositions can be predicted using a convex-hull plot (or Maxwell construction more generally). In this case, A is hydrogen and B is carbon. At ambient pressure, the stable compositions are predicted to be H₂ (x=0), CH₄ (x=0.2) and C (x=1, graphite). At 10 GPa, further compositions become thermodynamically stable, including polyethene and the predicted (and experimentally confirmed) graphane [–49]. (Online version in colour.)

Figure 8.

Discovery of novel magnetism in the elements. (a) Ferromagnetism is confined to the elements highlighted in red, promoted by narrow 3d and 4f bands. Many elements, such as aluminium [60], become ‘electrides’ at high pressures—the valence electrons are squeezed between the atomics cores, localizing them and leading to band narrowing. Could this localization lead to unexpected ferromagnetism? Extensive spin unrestricted searches over many elements in the periodic table, at a range of pressures, were performed. The alkali elements highlighted in blue were revealed to exhibit (weak) ferromagnetism. (b) Potassium (purple) adopts the simple cubic structure at about 20 GPa, and is a ferromagnetic electride [61]. The spin-density isosurface is shown in green. Nearly one electron spin-polarized is localized in the centre of the cubic cell, leading to a CsCl crystal structure with the electron playing the role of the anion. Adapted with permission from Pickard & Needs [61]. Copyright • 2011 by the American Physical Society.

Figure 9.

Visualization of the eigenvectors of a 31.7 cm⁻¹ phonon mode with a wavevector q=(1/3,1/3,1/3) from a DFPT phonon calculation of the Pa3 low-temperature phase of C₆₀. ‘External’ librational modes such as this one emerge from exactly the same crystalline periodic lattice dynamics formalism as ‘internal’ molecular deformation modes. Reproduced with permission from PCCP owner societies from Parker et al. [67]. (Online version in colour.)

Figure 10.

Inelastic neutron spectra of the internal modes of C₆₀ in the Pa3 phase recorded at the ISIS neutron scattering facility on the TOSCA instrument (blue), the MARI instrument (olive green and black) compared with the predicted spectrum from a DFPT calculation (red). Reproduced with permission from PCCP owner societies from Parker et al. [67].

Figure 11.

Calculated Raman spectrum of C₆₀ in the Fm-3 model structure (blue) compared with the measured spectrum at room temperature (red) corrected for instrumental factors.

Figure 12.

Trajectories of cold high-pressure structures of hydrogen obtained from simulations with classical and quantum nuclei at 80 GPa starting from the P2₁/c-24 structure. Yellow balls show the representative configurations of the centroids throughout the course of the simulation. The red rods show the static (geometry-optimized) structure. A conventional hexagonal cell containing 144 atoms was used. (a,c,e,g) show the z–x plane and (b,d,f,h) show the x–y plane of the hcp lattice. The four simulations are (i) MD with classical nuclei at 50 K (a,b), (ii) PIMD for D at 50 K (c,d), (iii) PIMD for H at 50 K (e,f) and (iv) PIMD for D at 150 K (g,h). In the MD simulation, the anisotropic intermolecular interaction outweighs the thermal and quantum nuclear fluctuations. Therefore, the molecular rotation is highly restricted. The thermal plus quantum nuclear fluctuations outweigh the anisotropic intermolecular interactions in the PIMD simulations of H at 50 K and D at 150 K. Reproduced with permission from Li et al. [70]. Copyright • 2013 the Institute of Physics.

Figure 13.

The structure of galactose as determined by comparison between the computed and observed NMR chemical shifts. (Left) The galactose structure showing hydrogen (white), carbon (grey) and oxygen (red), with hydrogen bonds shown in blue. (Right) The experimental ¹H NMR spectrum together with the spectra calculated using the two proposed hydrogen bond networks (see text). The atoms contributing to the diagnostic small downfield peak are highlighted. Reproduced with permission from Kibalchenko et al. [72]. Copyright • 2010 with permission from Elsevier.

Figure 14.

Hypothetical LDH structure (a) Atom colours are hydrogen (white), oxygen (red), nitrogen (blue), aluminium (purple) and magnesium (green). ¹H MAS NMR spectra of two Mg/Al LDH materials showing the proportions of the different hydroxyl groups present (b). Reproduced with permission from Cadars et al. [73]. Copyright • 2011 the American Chemical Society.

Figure 15.

(a) Medium angle annular dark field (MAADF) image of a nitrogen substitutional dopant atom in graphene; (b) model used for simulations and (c) experimental and calculated EELS data. Reproduced with permission from Nicholls et al. [76]. Copyright • 2013 the American Chemical Society.