Supercell Approximation Research Articles

Electrostatic interactions in finite systems treated with periodic boundary conditions: Application to linear-scaling density functional theory

We present a comparison of methods for treating the electrostatic interactions of finite, isolated systems within periodic boundary conditions (PBCs), within density functional theory (DFT), with particular emphasis on linear-scaling (LS) DFT. Often, PBCs are not physically realistic but are an unavoidable consequence of the choice of basis set and the efficacy of using Fourier transforms to compute the Hartree potential. In such cases the effects of PBCs on the calculations need to be avoided, so that the results obtained represent the open rather than the periodic boundary. The very large systems encountered in LS-DFT make the demands of the supercell approximation for isolated systems more difficult to manage, and we show cases where the open boundary (infinite cell) result cannot be obtained from extrapolation of calculations from periodic cells of increasing size. We discuss, implement, and test three very different approaches for overcoming or circumventing the effects of PBCs: truncation of the Coulomb interaction combined with padding of the simulation cell, approaches based on the minimum image convention, and the explicit use of open boundary conditions (OBCs). We have implemented these approaches in the ONETEP LS-DFT program and applied them to a range of systems, including a polar nanorod and a protein. We compare their accuracy, complexity, and rate of convergence with simulation cell size. We demonstrate that corrective approaches within PBCs can achieve the OBC result more efficiently and accurately than pure OBC approaches.

Variational solution of the three-dimensional Schrödinger equation using plane waves in adaptive coordinates

A series of improvements for the solution of the three-dimensional Schrödinger equation over a method introduced by Gygi [F. Gygi, Europhys. Lett. 19, 617 (1992); F. Gygi, Phys. Rev. B 48, 11692 (1993)] are presented. As in the original Gygi's method, the solution (orbital) is expressed by means of plane waves in adaptive coordinates u, where u is mapped from Cartesian coordinates, u=f(r). The improvements implemented are threefold. First, maps are introduced that allow the application of the method to atoms and molecules without the assistance of the supercell approximation. Second, the electron-nucleus singularities are exactly removed, so that pseudo-potentials are no longer required. Third, the sampling error during integral evaluation is made negligible, which results in a true variational, second-order energy error procedure. The method is tested on the hydrogen atom (ground and excited states) and the H(2)(+) molecule, resulting in milli-Hartree accuracy with a moderate number of plane waves.

Possibility of transforming the electronic structure of one species of graphene adatoms into that of another by application of gate voltage: First-principles calculations

Graphene provides many advantages for controlling the electronic structure of adatoms and other adsorbates via gating. Using the projected density of states and charge density obtained from first-principles density-functional periodic supercell calculations, we investigate the possibility of performing ``alchemy'' of adatoms on graphene, i.e., transforming the electronic structure of one species of adatom into that of another species by application of a gate voltage. Gating is modeled as a change in the number of electrons in the unit cell, with the inclusion of a compensating uniform background charge. Within this model and the generalized gradient approximation to the exchange-correlation functional, we find that such transformations are possible for K, Ca, and several transition-metal adatoms. Gate control of the occupation of the $p$ states of In on graphene is also investigated. The validity of the supercell approximation with uniform compensating charge and the model for exchange and correlation is also discussed.

Supercell approximation for an incommensurately modulated crystal of profilin:actin

Supercell approximation for an incommensurately modulated crystal of profilin:actin

Improved fake mode free plane wave expansion method

We analyze the origin of the fake modes introduced by the plane wave expansion method with three-dimension (3D) supercell approximation. Through the detailed analysis of the energy distribution of fake modes and real modes, we propose the plane wave expansion-three planar-slab waveguides method to remove the fake modes and obtain the fake mode free band structure of a two-dimensional air hole photonic crystal slab. To the best of our knowledge, this is the first time that such a fake mode free photonic crystal band structure is presented. Our method is also definitely useful in designing other 3D devices.

Influence of the alkyl and alkoxy side chains on the electronic structure and charge-transport properties of polythiophene derivatives

Density Functional Theory has been used to study the structural, electronic and charge-transport properties of two regio-regular head-to-tail polythiophene derivatives, i.e. poly(3-hexyl-thiophene), P3HT, and poly(3-oxyhexyl-thiophene), P3OHT. The effect of substituents on the electronic structure was analyzed by means of bandwidth, bandgap, effective mass, total and partial densities of states and crystal orbital overlap populations. Electronic couplings were estimated from band diagrams as the splitting of the valence band. The neutral and cationic states of isolated oligomers were optimized using the supercell approximation. The hole-transfer rates and mobilities were evaluated according to Marcus's theory. Results provide a compelling illustration of the effect of side chains on the crystal packing, electronic structure and charge-transport properties. Thus, the hole mobility calculated for the alkyl derivative was 0.15 cm(2) V(-1) s(-1) (experimental mobility is 0.10 cm(2) V(-1) s(-1)), while the alkoxy derivative has a theoretical mobility of 0.49 cm(2) V(-1) s(-1). The obtained results hopefully could motivate experimentalists to try out P3OHT for an improved charge carrier mobility.

Ab initio characterization of the electronic properties of PbTe quantum dots embedded in a CdTe matrix

The electronic properties of PbTe nanocrystals (NCs) embedded in a CdTe matrix are investigated using an ab initio pseudopotential method and a repeated supercell approximation. The asymmetric distribution of polar dot-matrix interfaces leads to the presence of an electrostatic field within the NC region. The consequences of this field on the electronic properties of the dot-matrix system are discussed in detail. The resulting spatial separation of electrons and holes inside the NC region gives rise to a quantum-confined Stark effect. It influences the photoluminesence of the NCs.

Evanescent modes in sonic crystals: Complex dispersion relation and supercell approximation

Evanescent modes in complete sonic crystals (SCs) and SC with point defects are reported both theoretically and experimentally in this paper. Plane wave expansion (PWE) and in general, ω(k) methods have been used to calculate band structures showing gaps that have been interpreted as ranges of frequencies where no real k exists. In this work, we extend PWE to solve the complex k(ω) problem applied to SC, introducing the supercell approximation for studying one vacancy. Explicit matrix formulation of the equations is given. This k(ω) method enables the calculation of complex band structures, as well as enabling an analysis of the propagating modes related with real values of the function k(ω), and the evanescent modes related with imaginary values of k(ω). This paper shows theoretical results and experimental evidences of the evanescent behavior of modes inside the SC band gap. Experimental data and numerical results using the finite elements method are in very good agreement with the predictions obtained using the k(ω) method.

Propagating and evanescent properties of double-point defects in sonic crystals

Complex band structures and multiple scattering theory have been used in this paper to analyze the overlapping of the evanescent waves localized in point defects in sonic crystals (SCs). The extended plane wave expansion (EPWE) with supercell approximation gives the imaginary part of the Bloch vectors that produces the decay of the localized modes inside the periodic system. Double cavities can present a coupling between the evanescent modes localized in the defect, showing a symmetric or antisymmetric mode. When point defects are close, the complex band structures reveal a splitting of the frequencies of the localized modes. Both the real part and the imaginary values of k of the localized modes in the cavities present different values for each localized mode, which gives different properties for each mode. The novel measurements, in very good agreement with analytical data, show experimental evidence of the symmetric and antisymmetric localized modes for a double-point defect in SCs. The investigation of the localization phenomena and the coupling between defects in periodic systems has fundamental importance in both pure and applied physics.

The effect of Stone-Wales defect orientations on the electronic properties of single-walled carbon nanotubes

Density functional theory (DFT) calculations of the Stone-Wales defected (S-WD) single-walled carbon nanotube (CNT) (10, 0) are carried out to understand the effect of S-WD orientations on the electronic properties of CNT. We have considered the influence of super-cell approximation on the defect formation energy and the electronic properties of both circumferential and axial S-W defects in CNT. We found that the probability of S-WD orientation depends on the defect concentration. Density of states of defected CNTs calculations have been applied to indicate the effect of S-WD concentration on the semiconducting energy gap. Utilizing the local density of states investigation, also, we explain the reasons for the adsorption of foreign atoms and molecules on S-WDs.

Electronic structure, localization, and spin-state transition in Cu-substitutedFeSe:Fe1−xCuxSe

We report density-functional studies of the ${\text{Fe}}_{1\ensuremath{-}x}{\text{Cu}}_{x}\text{Se}$ alloy done using supercell and coherent-potential approximation methods. Magnetic behavior was investigated using the disordered local moment approach. We find that Cu occurs in a nominal ${d}^{10}$ configuration and is highly disruptive to the electronic structure of the Fe sheets. This would be consistent with a metal-insulator transition due to Anderson localization. We further find a strong crossover from a weak moment itinerant system to a local moment magnet at $x\ensuremath{\approx}0.12$. We associate this with the experimentally observed jump near this concentration. Our results are consistent with the characterization of this concentration-dependent jump as a transition to a spin glass.

Effects of disorder in FeSe: an ab initio study

Using the coherent-potential approximation, we have studied the effects of excess Fe, Sedeficiency and substitutions of S and Te on the Se sub-lattice and Co, Ni and Cu on the Fesub-lattice in FeSe. Our results show that (i) a small amount of excess Fe substantiallydisorders the Fe-derived bands while an Se deficiency affects mainly the Se-derived bands,(ii) the substitution of S or Te enhances the possibility of Fermi surface nesting, especially inFeSe0.5Te0.5, in spite of disordering the Se-derived bands and (iii) the electron doping through Co, Ni orCu disorders the system and pushes down the Fe-derived bands, thereby destroying thepossibility of Fermi surface nesting. A comparison of these results with the rigid-band,virtual-crystal and supercell approximations reveals the importance of describing disorderwith the coherent-potential approximation.

Density functional theory calculations of defect energies using supercells

Reliable calculations of defect properties may be obtained with density functional theory (DFT) using the supercell approximation. We systematically review the known sources of error and suggest how to perform calculations of defect properties in order to minimize errors. We argue that any analytical error-correction scheme relying on electrostatic considerations alone is not appropriate to derive reliable defect formation energies, certainly not for relaxed geometries. Instead we propose finite size scaling of the calculated defect formation energies, and compare the application of this with both fully converged and ‘Gamma’ (Γ) point only k-point integration. We provide a recipe for practical DFT calculations which will help to obtain reliable defect formation energies and demonstrate it using examples from III–V semiconductors.

Simple intrinsic defects in gallium arsenide

We investigate the structural properties and energy levels of simple intrinsic defects in gallium arsenide. The first-principles calculations (1) apply boundary conditions appropriate to charge defects in supercells and enable quantitatively accurate predictions of defect charge transitions with a supercell approximation, (2) are demonstrated to be converged with respect to cell size and (3) assess the sensitivity to model construction to Ga pseudopotential construction (3d core or 3d valence) and density functionals (local density or generalized gradient approximation). With these factors controlled, we present the first quantitatively reliable survey of defect levels in GaAs, reassess the available literature and begin to decipher the complexity of GaAs defect chemistry. The computed defect level spectrum spans the experimental GaAs band gap, defects exhibit multiple bistabilities with (sometimes overlapping) negative-U systems, express more extensive charge states than previously anticipated and collectively suggest that our atomistic understanding of GaAs defect physics needs to be reassessed.

DFT+U Calculations of Transition Metal Doped AlN

The electronic structure of transition metal (TM = V, Cr, Mn, Fe) doped aluminium nitride was presented. The calculations were made within the DFT and DFT+U approach and the supercell approximation. The eective interaction parameter Hubbard U for mentioned compounds was determined within linear response approach in contrast to the other type of calculations which treated this factor as an adjustable parameter.

Characteristic Energies and Shifts in Optical Spectra of Colloidal IV−VI Semiconductor Nanocrystals

We investigate structural, electronic, and optical properties of colloidal IV-VI semiconductor quantum dots (QDs) using an ab initio pseudopotential method and a repeated supercell approximation. In particular, rhombo-cuboctahedral quantum dots consisting of PbSe, PbTe, and SnTe with a pseudohydrogen passivation shell are investigated for different QD sizes. The obtained dependence of the confinement energy on the QD size questions the use of three-dimensional spherical potential well models for small QD structures. The predicted band gaps are almost in agreement with measured values. The calculated Franck-Condon shifts vary significantly with the QD size. Only for PbSe they may explain the Stokes shift between optical absorption and emission. For the tellurides, spectral properties such as the oscillator strength are more important.

Roles of sodium induced defects in CuInSe2 by first principles calculation

In order to investigate the roles of sodium in CuInSe 2 absorber layer, the formation energy of sodium induced point defects and bound defects in CuInSe 2 are calculated using first principles calculations within supercell approximation approach. Neutral defect Na Cu and acceptor defect Na In are found to form easily, which explains the enhanced p -type conductivity of CuInSe 2 by sodium incorporation. More than that, sodium induced defects are energetically favorable to form bound defects with intrinsic defects (i.e. V Cu , Cu i , In Cu , and V Se ), which is responsible for the increased range of composition. The contour plot of total valence charge density suggests the chemical bonding between Na In and Se atoms is stronger than that of Na i and Na Cu .

FullyAb InitioFinite-Size Corrections for Charged-Defect Supercell Calculations

In ab initio theory, defects are routinely modeled by supercells with periodic boundary conditions. Unfortunately, the supercell approximation introduces artificial interactions between charged defects. Despite numerous attempts, a general scheme to correct for these is not yet available. We propose a new and computationally efficient method that overcomes limitations of previous schemes and is based on a rigorous analysis of electrostatics in dielectric media. Its reliability and rapid convergence with respect to cell size is demonstrated for charged vacancies in diamond and GaAs.

Interplay of shape, interface structure, and electrostatic fields of ionic nanodots embedded in a polar semiconductor matrix

The structural, electrostatic, and electronic properties of PbTe nanocrystals (NCs) embedded in a CdTe matrix are investigated using an ab initio pseudopotential method and a repeated supercell approximation. The calculated dot-matrix interface geometries are in excellent agreement with the plane PbTe/CdTe interfaces. The constructed rhombo-cubo-octahedron shape of the NCs corresponds to the equilibrium crystal shape and transmission-microscopy observations. We demonstrate that polar interfaces with either cation or anion termination appear predominantly in one-half space with [111] or $[\overline{1}\overline{1}\overline{1}]$ orientation. This leads to the occurrence of a macroscopic electrostatic field, which superimposes that of the atoms. As a consequence an internal quantum-confined Stark effect is predicted. Its influence on the electronic states is discussed.

X-ray magnetic circular dichroism inCeFe2: First-principles calculations

The x-ray magnetic circular dichroism (XMCD) spectra of ${\text{CeFe}}_{2}$ at the $\text{Ce}\text{ }{L}_{2,3}$, ${M}_{4,5}$, $\text{Fe}\text{ }K$, and ${L}_{2,3}$ edges are investigated theoretically from first principles, using the fully relativistic Dirac linear muffin-tin orbital band-structure method. The electronic structure is obtained with the local spin-density approximation. The origin of the XMCD spectra in the compound is examined. The core hole effect in the final states has been investigated using a supercell approximation. It improves the agreement between the theory and the experiment at the $\text{Ce}\text{ }{M}_{5}$ edge. However, it has a minor influence on the shape of the $\text{Ce}\text{ }{L}_{2,3}$ XMCD spectra.