Plane wave density functional theory codes generally assume periodicity in all three dimensions. This causes difficulties when studying charged systems, for instance energies per unit cell become infinite, and, even after being renormalised by the introduction of a uniform neutralising background, are very slow to converge with cell size. The periodicity introduces spurious electric fields which decay slowly with cell size and which also slow the convergence of other properties relating to the ground state charge density. This paper presents a simple self-consistent technique for producing rapid convergence of both energies and charge distribution in the particular geometry of 2D periodicity, as used for studying surfaces.
Solution of the Poisson equation for two dimensional periodic structures (slabs) in an overlapping localized site density scheme
Bertaut的等效电密度概念（E. F. Bertaut，Journal de Physique { \bf 39}，1331（1978））应用于二维周期性连续电荷密度分布的情况。下面的推导与Bertaut所介绍的不同。该方法求解了（$x,y$）平面上周期性边界条件和垂直于$z$方向的一般有限电压边界条件下局域密度重叠方案的Poisson方程，并在Fourier空间中计算了长程势。对于$K_{||}\ne 0$情况，傅立叶变换有助于计算三维周期意义上的解，而对于$K_{||}=0$情况，所需的电荷中性是起点。对于这两种情况下，需要适当的球谐函数表示，以达到便于数值实现的表达式。在这个局域密度方案中，可以导出在$z$方向上的有限电压与偶极子密度的$z$分量之间的显式关系。
Supercell models are often used to calculate the electronic structure of local perturbations from the ideal periodicity in the bulk or on the surface of a crystal or in wires. When the defect or adsorbent is charged, a jellium counter charge is applied to maintain overall neutrality, but the interaction of the artificially repeated charges has to be corrected, both in the total energy and in the one-electron eigenvalues and eigenstates. This becomes paramount in slab or wire calculations, where the jellium counter charge may induce spurious states in the vacuum. We present here a self-consiste…
TiO2 is a technologically important material. In particular, its anatase polymorph plays a major role in photocatalysis, which can also accommodate charged and neutral vacancies. There is, however, scant theoretical work on the vacancy charge and associated diffusion from surface to subsurface and bulk in the literature. Here, we aim to understand +2 charge and neutral vacancy diffusion on anatase (101) surface using 72 and 216 atoms surface slabs employing a semi-local density functional and the Hubbard model. The activation-relaxation technique nouveau (ARTn) coupled with Quantum Espresso is…
When a system which contains a dipole, and whose dimensionality is less than three, is studied in a code which imposes periodic boundary conditions in all three dimensions, an artificial electric field arises which keeps the potential periodic. This has an impact on the total energy of the system, and on any other attribute which would respond to an electric field. Simple corrections are known for 0D systems embedded in a cubic geometry, and 2D slab systems. This paper shows how the 0D result can be extended to tetragonal geometries, and that for a particular $c/a$ ratio the correction is zero…
Attaining accurate average structural properties in a molecular simulation should be considered a prerequisite if one aims to elicit meaningful insights into a system's behavior. For charged surfaces in contact with an electrolyte solution, an obvious example is the density profile of ions along the direction normal to the surface. Here we demonstrate that, in the slab geometry typically used in simulations, imposing an electric displacement field $D$ determines the integrated surface charge density of adsorbed ions at charged interfaces. This allows us to obtain macroscopic surface charge den…
There has been much interest over many years in studying charged systems after the artificial imposition of periodic boundary conditions, and correcting for the resulting divergence of the electrostatic energy density. A correction for cubic cells was derived by Makov and Payne in 1995, and its leading error term is of the form $L^{-5}$ for a cube of side $L$. Most modern Density Functional Theory codes use a different treatment of the `Z$α$' energy term to that used by Makov and Payne, resulting in an error term of the form $L^{-3}$ if their correction is used unmodified. This paper shows how…
Charged point defects in materials are widely studied using Density Functional Theory (DFT) packages with periodic boundary conditions. The formation energy and defect level computed from these simulations need to be corrected to remove the contributions from the spurious long-range interaction between the defect and its periodic images. To this effect, the CoFFEE code implements the Freysoldt-Neugebauer-Van de Walle (FNV) correction scheme. The corrections can be applied to charged defects in a complete range of material shapes and size: bulk, slab (or two-dimensional), wires and nanoribbons.…
The catalyst development for N2 electroreduction reaction (NRR) with low onset potential and high Faradaic efficiency is highly desired, but remains challenging. Machine learning (ML) recently emerged as a complementary tool to accelerate material discovery; however a ML model for NRR has yet to be developed. Here, we develop and report slab-graph convolutional neural network (SGCNN), an accurate and flexible ML model that is applicable to catalytic surface reactions. With the self-accumulated database of 2,699 surface calculations, SGCNN predict binding energies, ranging over 8 eV, of five ke…
Surfaces of ionic solids interacting with an ionic solution can build up charge by exchange of ions. The surface charge is compensated by a strip of excess charge at the border of the electrolyte forming an electric double layer. These electric double layers are very hard to model using the supercells methods of computational condensed phase science. The problem arises when the solid is an electric insulator (as most ionic solids are) permitting a finite interior electric field over the width of the slab representing the solid in the supercell. The slab acts as a capacitor. The stored charge i…
Rashba-like spin-orbit interaction at oxide heterostructures emerges as a much sought-after feature in the context of oxide spintronics and spin-orbitronics. KTaO$_3$ (KTO) is one of the best substrates available for the purpose, owing to its strong spin-orbit interaction and alternating $+1|-1$ charged layers along the (001) direction. Employing first-principles calculations within density functional theory (DFT) and proposing a possible electrostatic model for charge transfer to the surfaces of KTO slabs, we comprehensively analyze Rashba-like spin-orbit interaction with the help of three-di…
