Real-space Finite-difference Method Research Articles

GPU acceleration of conjugate gradient method obtaining Green's function for transport-property calculation

We present a graphics processing unit (GPU) acceleration procedure for the transport-property calculation of the RSPACE code, which uses density functional theory and the real-space finite-difference method. Porting the RSPACE code to run on a GPU using Open Accelerator and libraries results in the improvement of the computational speed and efficiency compared with a central processing unit. Furthermore, we achieved a speed up of ∼ 4x compared with a single GPU and effective processing in a multi-GPU environment by parallelizing in a way that minimizes communication and synchronization. Our GPU-acceleration procedure is demonstrated by the calculation of electron-transport properties of (4,4) carbon nanotubes (CNTs) with a 5-8-5 defect for which the calculations are suitable for parallel processing on GPUs. It is found that the information loss is strongly affected by the defect geometry when electron waves propagating in CNTs are utilized as the information transfer signal.

Real-space method for first-principles electron transport calculations: Self-energy terms of electrodes for large systems

We present a fast and stable numerical technique to obtain the self-energy terms of electrodes for first-principles electron-transport calculations. Although first-principles calculations based on the real-space finite-difference method are advantageous for execution on massively parallel computers, large-scale transport calculations are hampered by the computational cost and numerical instability of the computation of the self-energy terms. Using the orthogonal complement vectors of the space spanned by the generalized Bloch waves that actually contribute to transport phenomena, the computational accuracy of transport properties is significantly improved with a moderate computational cost. To demonstrate the efficiency of the present technique, the electron-transport properties of a Stone-Wales (SW) defect in graphene and silicene are examined. The resonance scattering of the SW defect is observed in the conductance spectrum of silicene since the $\sigma^\ast$ state of silicene lies near the Fermi energy. In addition, we found that one conduction channel is sensitive to a defect near the Fermi energy, while the other channel is hardly affected. This characteristic behavior of the conduction channels is interpreted in terms of the bonding network between the bilattices of the honeycomb structure in the formation of the SW defect. The present technique enables us to distinguish the different behaviors of the two conduction channels in graphene and silicene owing to its excellent accuracy.

ATLAS: A real-space finite-difference implementation of orbital-free density functional theory

Orbital-free density functional theory (OF-DFT) is a promising method for large-scale quantum mechanics simulation as it provides a good balance of accuracy and computational cost. Its applicability to large-scale simulations has been aided by progress in constructing kinetic energy functionals and local pseudopotentials. However, the widespread adoption of OF-DFT requires further improvement in its efficiency and robustly implemented software. Here we develop a real-space finite-difference (FD) method for the numerical solution of OF-DFT in periodic systems. Instead of the traditional self-consistent method, a powerful scheme for energy minimization is introduced to solve the Euler–Lagrange equation. Our approach engages both the real-space finite-difference method and a direct energy-minimization scheme for the OF-DFT calculations. The method is coded into the ATLAS software package and benchmarked using periodic systems of solid Mg, Al, and Al3Mg. The test results show that our implementation can achieve high accuracy, efficiency, and numerical stability for large-scale simulations.

Numerical solver for first-principles transport calculation based on real-space finite-difference method.

We propose an efficient procedure to obtain Green's functions by combining the shifted conjugate orthogonal conjugate gradient (shifted COCG) method with the nonequilibrium Green's function (NEGF) method based on a real-space finite-difference (RSFD) approach. The bottleneck of the computation in the NEGF scheme is matrix inversion of the Hamiltonian including the self-energy terms of electrodes to obtain the perturbed Green's function in the transition region. This procedure first computes unperturbed Green's functions and calculates perturbed Green's functions from the unperturbed ones using a mathematically strict relation. Since the matrices to be inverted to obtain the unperturbed Green's functions are sparse, complex-symmetric, and shifted for a given set of sampling energy points, we can use the shifted COCG method, in which once the Green's function for a reference energy point has been calculated the Green's functions for the other energy points can be obtained with a moderate computational cost. We calculate the transport properties of a C(60)@(10,10) carbon nanotube (CNT) peapod suspended by (10,10)CNTs as an example of a large-scale transport calculation. The proposed scheme opens the possibility of performing large-scale RSFD-NEGF transport calculations using massively parallel computers without the loss of accuracy originating from the incompleteness of the localized basis set.

First-principles transport calculation method based on real-space finite-difference nonequilibrium Green's function scheme

We demonstrate an efficient nonequilibrium Green's function transport calculation procedure based on the real-space finite-difference method. The direct inversion of matrices for obtaining the self-energy terms of electrodes is computationally demanding in the real-space method because the matrix dimension corresponds to the number of grid points in the unit cell of electrodes, which is much larger than that of sites in the tight-binding approach. The procedure using the ratio matrices of the overbridging boundary-matching technique [Y. Fujimoto and K. Hirose, Phys. Rev. B 67, 195315 (2003)], which is related to the wave functions of a couple of grid planes in the matching regions, greatly reduces the computational effort to calculate self-energy terms without losing mathematical strictness. In addition, the present procedure saves computational time to obtain the Green's function of the semi-infinite system required in the Landauer-B\"uttiker formula. Moreover, the compact expression to relate Green's functions and scattering wave functions, which provide a real-space picture of the scattering process, is introduced. An example of the calculated results is given for the transport property of the BN ring connected to (9,0) carbon nanotubes. The wave-function matching at the interface reveals that the rotational symmetry of wave functions with respect to the tube axis plays an important role in electron transport. Since the states coming from and going to electrodes show threefold rotational symmetry, the states in the vicinity of the Fermi level, the wave function of which exhibits fivefold symmetry, do not contribute to the electron transport through the BN ring.

Novel Computational Methods for Nanostructure Electronic Structure Calculations

Experimentally relevant nanocrystals often contain a few thousands to hundreds of thousands of atoms. Yet, to understand their electronic structures, surface and impurity effects, atomic relaxations, interior electric fields, carrier dynamics, and transports, it is often necessary to carry out atomistic simulations. Owing to the advance of recent algorithm developments and improved supercomputer powers, it is now possible to calculate such nanocrystals based on ab initio methods. In this review, we discuss the numerical algorithms (the plane-wave pseudopotential method and the real-space finite-difference method) used in conventional density-functional-theory calculations, which enable the simulations of systems up to one or two thousand atoms. We also introduce methods designed specifically for nanostructure calculations. These methods [the charge-patching method (CPM) and the linear scaling three-dimensional fragment method (LS3DF)] can be used to calculate systems with hundreds of thousands of atoms. Whereas CPM is an approximation with ab initio quality, the LS3DF method is an O(N) method with essentially the same results as the direct methods. The computational aspects of the algorithms, especially for their parallelization scalability, are also emphasized in the review.

A massively-parallel electronic-structure calculations based on real-space density functional theory

Based on the real-space finite-difference method, we have developed a first-principles density functional program that efficiently performs large-scale calculations on massively-parallel computers. In addition to efficient parallel implementation, we also implemented several computational improvements, substantially reducing the computational costs of O(N3) operations such as the Gram–Schmidt procedure and subspace diagonalization. Using the program on a massively-parallel computer cluster with a theoretical peak performance of several TFLOPS, we perform electronic-structure calculations for a system consisting of over 10,000 Si atoms, and obtain a self-consistent electronic-structure in a few hundred hours. We analyze in detail the costs of the program in terms of computation and of inter-node communications to clarify the efficiency, the applicability, and the possibility for further improvements.

Order-Nfirst-principles calculation method for self-consistent ground-state electronic structures of semi-infinite systems

We present an efficient and highly accurate first-principles calculation method with linear system-size scaling to determine the self-consistent ground-state electron-charge densities of nanostructures suspended between semi-infinite bulks by directly minimizing the energy functional. By making efficient use of the advantages of the real-space finite-difference method, we can impose arbitrary boundary conditions on models and employ spatially localized orbitals. These advantages enable us to calculate the ground-state electron-charge densities in semi-infinite systems. Examples of electronic structure calculations for a one-dimensional case and a conductance calculation for sodium nanowires are presented. The calculated electronic structure of the one-dimensional system agrees well with the exact analytical solution, and the conduction properties of the sodium nanowires are consistent with experimental and other theoretical results. These results imply that our procedure enables us to accurately compute self-consistent electronic structures of semi-infinite systems.

Real space Hartree-Fock configuration interaction method for complex lateral quantum dot molecules

We present unrestricted Hartree-Fock method coupled with configuration interaction (CI) method (URHF-CI) suitable for the calculation of ground and excited states of large number of electrons localized by complex gate potentials in quasi-two-dimensional quantum dot molecules. The method employs real space finite difference method, incorporating strong magnetic field, for calculating single particle states. The Hartree-Fock method is employed for the calculation of direct and exchange interaction contributions to the ground state energy. The effects of correlations are included in energies and directly in the many-particle wave functions via CI method using a limited set of excitations above the Fermi level. The URHF-CI method and its performance are illustrated on the example of ten electrons confined in a two-dimensional quantum dot molecule.

Real-space electronic-structure calculations with a time-saving double-grid technique

We present a set of efficient techniques in first-principles electronic-structure calculations utilizing the real-space finite-difference method. These techniques greatly reduce the overhead for performing integrals that involve norm-conserving pseudopotentials, solving Poisson equations, and treating models which have specific periodicities, while keeping a high degree of accuracy. Since real-space methods are inherently local, they have a lot of advantages in applicability and flexibility compared with the conventional plane-wave approach, and promise to be well suited for large and accurate {\it ab initio} calculations. In order to demonstrate the potential power of these techniques, we present several applications for electronic structure calculations of atoms, molecules and a helical nanotube.

First-principles study on field evaporation of surface atoms from W(0 1 1) and Mo(0 1 1) surfaces

The simulations of field-evaporation processes for surface atoms on W(0 1 1) and Mo(0 1 1) surfaces are implemented using first-principles calculations based on the real-space finite-difference method. The threshold values of the external electric field for evaporation of the surface atoms, which are ∼6 V/Å for tungsten and ∼5 V/Å for molybdenum, are in agreement with the experimental results. While the threshold value of the electric field and the local-field enhancement around the evaporating atoms agree with those expected from the conclusion of the previous study using structureless jellium, the induced charge around the surface atom has a significant difference from that obtained by the jellium model.

First-Principles Study on the Scanning Tunneling Microscopy Image of H-Adsorbed Si(001)

The tunneling current flowing between the tip and H-adsorbed Si(001) surface in Scanning Tunneling Microscopy (STM) is investigated using the first-principles calculations based on the real-space finite-difference method. The resultant current map is consistent with the STM image in which H-adsorbed dimer looks geometrically lower than the bare dimer. Although the isosurface of the local density of states above the H-terminated Si dimer, which are mainly attributed by the Si-H σ bonding states, locate itself higher than that above the bare Si buckled dimer, these bonding states do not contribute to the tunneling current. On the other hand, many electrons tunnel from π bond of the unreacted dimer into the tip. Accordingly, the H-adsorbed dimer appears geometrically lower than the bare dimer in the STM, since the tip must approach closer to the sample surface in order to achieve the constant tunneling current.

半無限電極に挟まれたナノジャンクションの第一原理電子状態計算法

We develop a method for high-speed and high-accuracy first-principles calculations to derive the ground-state electronic structure by directly minimizing the energy functional. Making efficient use of the advantages of the real-space finite-difference method, we apply arbitrary boundary conditions and employ spatially localized orbitals. These advantages enable us to calculate the ground-state electronic structure of a nanostructure sandwiched between crystalline electrodes. The framework of this method and numerical examples for metallic nanowires are presented.

First-principles study on field evaporation for silicon atom on Si(001) surface

The simulations of field-evaporation processes for silicon atoms on various Si(001) surfaces are implemented using the first-principles calculations based on the real-space finite-difference method. We find that the atoms which locate on atomically flat Si(001) surfaces and at step edges are easily removed by applying an external electric field, and the threshold value of the external electric field for evaporation of atoms on atomically flat Si(001) surfaces, which is predicted between 3.0 and 3.5 V/Å, is in agreement with the experimental data of 3.8 V/Å. In this situation, the local field around an evaporating atom does not play a crucial role. This result is instead interpreted in terms of the bond strength between an evaporating atom and surface.

First-principles Calculation Method for Electronic Structures of Nanojunctions Suspended between Semi-infinite Electrodes

We develop a method for high-speed and high-accuracy first-principles calculations to derive the ground-state electronic structure by directly minimizing the energy functional. Making efficient use of the advantages of the real-space finite-difference method, we apply arbitrary boundary conditions and employ spatially localized orbitals. These advantages enable us to calculate the ground-state electronic structure of a nanostructure sandwiched between crystalline electrodes. The framework of this method and numerical examples for metallic nanowires are presented.

A coherent relation between structure and conduction of infinite atomicwires

We demonstrate a theoretical analysis concerning the geometrical structures andelectrical conduction of infinite monatomic gold and aluminium wires in theprocess of their elongation, based on first-principles molecular-dynamicssimulations using the real-space finite-difference method. Our study predicts thatthe single-row gold wire ruptures up to form a dimer coupling structure when theaverage interatomic distance increases up to more than 3.0 Å, and that the wire isconductive before breaking but changes to an insulator at the rupturing point. Inthe case of the aluminium wire, it exhibits a magnetic ordering due to the spinpolarization, and even when stretched up to the average interatomic distance of3.5 Å, a dimerization does not occur and the wire keeps a metallic nature.

Complex photonic band structures of a conductive metal lattice by a quadratic eigensystem

For conductive periodic media, Maxwell’s equations can be transformed into quadratic eigensystems by the plane-wave expansion method. We computed complex photonic band structures for the two-dimensional square lattice with conductive cylinders by solving the quadratic eigensystem. The photonic band structures obtained for metal cylinders with finite conductivity closely resemble those computed for perfectly conductive cylinders by a real space finite-difference method. However, some modes can become unphysical in the finitely conductive metal lattice.