A comprehensive study of popular eigenvalue methods employed for quantum calculation of energy eigenstates in nanostructures using GPUs

Rodrigues, Walter; Pecchia, Alessandro; Maur, Matthias Auf der; Carlo, Aldo Di

doi:10.1007/s10825-015-0695-z

Cited by 6 publications

(4 citation statements)

References 30 publications

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“…Shift‐and‐invert spectral transformation and FEAST methods are also popularly used for diagonalizations. Previous studies however reported that they are not suitable to solve degenerate systems, or do not necessarily show better performance than Lanczos iterations . The effect of a multigrid preconditioner is also investigated with ML package, where parameters for the preconditioner are set as default values for a 2‐level domain decomposition and details of these default values can be found in the document of ML package available online .…”

Section: Resultsmentioning

confidence: 99%

On the achievement of high fidelity and scalability for large‐scale diagonalizations in grid‐based DFT simulations

Choi

Kim

Yeom

et al. 2018

Int J of Quantum Chemistry

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Recent advance in high performance computing (HPC) resources has opened the possibility to expand the scope of density functional theory (DFT) simulations toward large and complex molecular systems. This work proposes a numerically robust method that enables scalable diagonalizations of large DFT Hamiltonian matrices, particularly with thousands of computing CPUs (cores) that are usual these days in terms of sizes of HPC resources. The well-known Lanczos method is extensively refactorized to overcome its weakness for evaluation of multiple degenerate eigenpairs that is the substance of DFT simulations, where a multilevel parallelization is adopted for scalable simulations in as many cores as possible. With solid benchmark tests for realistic molecular systems, the fidelity of our method are validated against the locally optimal block preconditioned conjugated gradient (LOBPCG) method that is widely used to simulate electronic structures. Our method may waste computing resources for simulations of molecules whose degeneracy cannot be reasonably estimated. But, compared to LOBPCG method, it is fairly excellent in perspectives of both speed and scalability, and particularly has remarkably less (< 10%) sensitivity of performance to the random nature of initial basis vectors. As a promising candidate for solving electronic structures of highly degenerate systems, the proposed method can make a meaningful contribution to migrating DFT simulations toward extremely large computing environments that normally have more than several tens of thousands of computing cores. K E Y W O R D S degenerate eigenpairs, density functional theory, high performance computing, Lanczos iterations, large-scale electronic structures 1 | I N TR ODU C TI ON Density functional theory (DFT) calculations are popular to predict various quantum phenomena in solid and molecular systems such as reaction mechanisms, charge transports, phase transitions, and vibrational excitations. [1-3] Theoretical understanding of those natural phenomena ofteninvolves simulations of systems that contain thousands of electrons, which require a huge computational cost that cannot be afforded by a single personal computer any more. Fortunately, the possibility to tackle such computationally expensive problems within reasonable times is opened by high performance computing (HPC) resources that normally consist of many computing cores. As the size of computing resources has been continuously increased with device-downscaling, nowadays it is no more difficult to find HPC clusters that have more than 10 5 cores. [4] Consequently, various implementations and methodologies are evolved and proposed to simulate large-scale electronic structures by leveraging HPC resources with parallel computing. [5][6][7] Real-space approaches are regarded as the reasonable strategy to explore large-scale electronic structures, [8][9][10][11] since they show better systemsize scalability and higher parallel efficiency than atom-centered basis set approaches. Packages of electronic structur...

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Section: Resultsmentioning

confidence: 99%

On the achievement of high fidelity and scalability for large‐scale diagonalizations in grid‐based DFT simulations

Choi

Kim

Yeom

et al. 2018

Int J of Quantum Chemistry

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“…The atomic positions in the atomistic structure have been relaxed using a modified Keating's valence force field (VFF) model [21]. Then we calculated the first four spin degenerate electron and hole states from an empirical tight-binding (ETB) hamiltonian, employing an sp 3 d 5 s * parametrization [22] and using a GPU accelerated Jacobi-Davidson algorithm [23]. From these states we obtained the momentum matrix elements, spontaneous emission spectra and the carrier densities, using Fermi-Dirac statistics and fixed quasi Fermi levels.…”

Section: Methodsmentioning

confidence: 99%

Unraveling the "Green Gap" problem: The role of random alloy fluctuations in InGaN/GaN light emitting diodes

der Maur,

Pecchia,

Penazzi

et al. 2015

Preprint

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White light emitting diodes based on III-nitride InGaN/GaN quantum wells currently offer the highest overall efficiency for solid state lighting applications. Although current phosphor-converted white LEDs have high electricity-to-light conversion efficiencies, it has been recently pointed out that the full potential of solid state lighting could be exploited only by color mixing approaches without employing phosphor-based wavelength conversion. Such an approach requires direct emitting LEDs of different colors, in particular in the green/yellow range ov the visible spectrum. This range, however, suffers from a systematic drop in efficiency, known as the "green gap", whose physical origin has not been understood completely so far. In this work we show by atomistic simulations that a consistent part of the "green gap" in c-plane InGaN/GaN based light emitting diodes may be attributed to a decrease in the radiative recombination coefficient with increasing Indium content due to random fluctuations of the Indium concentration naturally present in any InGaN alloy.

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“…Also, due to the statistical randomness of the single structures, a relatively big super-cell has to be chosen (containing roughly 100,000 atoms in this example), a sufficient number of states and k-points needs to be considered, and all calculations have to be done on a statistic ensemble of 30 or more random samples. Therefore, the numerical solvers like the eigenvalue solver have to be optimised [52], and parallelization schemes have to be employed. For example, different random samples and k-points can be calculated in parallel.…”

Section: Optical Properties In Iii-nitride Ledsmentioning

confidence: 99%

Multiscale Approaches for the Simulation of Optoelectronic Devices

Maur

2016

JGE

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Multiscale approaches for electronic device simulation have become a subject of high interest during the last decade. In this article we will give an overview on the current activities in the field. We will provide some practical examples from optoelectronics where multiscale simulations can be useful. Basic coupling schemes are discussed, and approaches for the combination of continuous models and models with atomistic resolution are described.

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A comprehensive study of popular eigenvalue methods employed for quantum calculation of energy eigenstates in nanostructures using GPUs

Cited by 6 publications

References 30 publications

On the achievement of high fidelity and scalability for large‐scale diagonalizations in grid‐based DFT simulations

On the achievement of high fidelity and scalability for large‐scale diagonalizations in grid‐based DFT simulations

Unraveling the "Green Gap" problem: The role of random alloy fluctuations in InGaN/GaN light emitting diodes

Multiscale Approaches for the Simulation of Optoelectronic Devices

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