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Register a new userA Nested Dissection Approach to Modeling Transport in Nanodevices: Algorithms and Applications
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Modeling nanoscale devices quantum mechanically is a computationally challenging problem where new methods to solve the underlying equations are in a dire need. In this paper, we present an approach to calculate the charge density in nanoscale devices, within the context of the non equilibrium Greens function approach. Our approach exploits recent advances in using an established graph partitioning approach. The developed method has the capability to handle open boundary conditions that are represented by full self energy matrices required for realistic modeling of nanoscale devices. Our method to calculate the electron density has a reduced complexity compared to the established recursive Greens function approach. As an example, we apply our algorithm to a quantum well superlattice and a carbon nanotube, which are represented by a continuum and tight binding Hamiltonian respectively, and demonstrate significant speed up over the recursive method.
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The Hierarchical Schur Complement method (HSC), and the HSC-extension, have significantly accelerated the evaluation of the retarded Greens function, particularly the lesser Greens function, for two-dimensional nanoscale devices. In this work, the HSC-extension is applied to determine the solution of non-equilibrium Greens functions (NEGF) on three-dimensional nanoscale devices. The operation count for the HSC-extension is analyzed for a cuboid device. When a cubic device is discretized with $N times N times N$ grid points, the state-of-the-art Recursive Green Function (RGF) algorithm takes $mathcal{O}(N^7)$ operations, whereas the HSC-extension only requires $mathcal{O}(N^6)$ operations. %Realistic operation counts also depend on the system dimensions in $xyz$-directions and the form of contact self-energy matrix. Operation counts and runtimes are also studied for three-dimensional nanoscale devices of practical interest: a graphene-boron- nitride-graphene multilayer system, a silicon nanowire, and a DNA molecule. The numerical experiments indicate that the cost for the HSC-extension is proportional to the solution of one linear system (or one LU-factorization) and that the runtime speed-ups over RGF exceed three orders of magnitude when simulating realistic devices, such as a graphene-boron nitride-graphene multilayer system with 40,000 atoms.
Multiscale models allow for the treatment of complex phenomena involving different scales, such as remodeling and growth of tissues, muscular activation, and cardiac electrophysiology. Numerous numerical approaches have been developed to simulate multiscale problems. However, compared to the well-established methods for classical problems, many questions have yet to be answered. Here, we give an overview of existing models and methods, with particular emphasis on mechanical and bio-mechanical applications. Moreover, we discuss state-of-the-art techniques for multilevel and multifidelity uncertainty quantification. In particular, we focus on the similarities that can be found across multiscale models, discretizations, solvers, and statistical methods for uncertainty quantification. Similarly to the current trend of removing the segregation between discretizations and solution methods in scientific computing, we anticipate that the future of multiscale simulation will provide a closer interaction with also the models and the statistical methods. This will yield better strategies for transferring the information across different scales and for a more seamless transition in selecting and adapting the level of details in the models. Finally, we note that machine learning and Bayesian techniques have shown a promising capability to capture complex model dependencies and enrich the results with statistical information; therefore, they can complement traditional physics-based and numerical analysis approaches.
First-principles calculations combining density-functional theory and continuum solvation models enable realistic theoretical modeling and design of electrochemical systems. When a reaction proceeds in such systems, the number of electrons in the portion of the system treated quantum mechanically changes continuously, with a balancing charge appearing in the continuum electrolyte. A grand-canonical ensemble of electrons at a chemical potential set by the electrode potential is therefore the ideal description of such systems that directly mimics the experimental condition. We present two distinct algorithms, a self-consistent field method (GC-SCF) and a direct variational free energy minimization method using auxiliary Hamiltonians (GC-AuxH), to solve the Kohn-Sham equations of electronic density-functional theory directly in the grand canonical ensemble at fixed potential. Both methods substantially improve performance compared to a sequence of conventional fixed-number calculations targeting the desired potential, with the GC-AuxH method additionally exhibiting reliable and smooth exponential convergence of the grand free energy. Finally, we apply grand-canonical DFT to the under-potential deposition of copper on platinum from chloride-containing electrolytes and show that chloride desorption, not partial copper monolayer formation, is responsible for the second voltammetric peak.
Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential off-diagonal decay (nearsightedness) for the density matrix of gapped systems at zero electronic temperature in both orthogonal and non-orthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for non-metallic systems. We further discuss the case of density matrices for metallic systems at positive electronic temperature. A few other possible applications are also discussed.
We report on charge transport and current fluctuations in a single bacteriorhodpsin protein in a wide range of applied voltages covering direct and injection tunnelling regimes. The satisfactory agreement between theory and available experiments validates the physical plausibility of the model developed here. In particular, we predict a rather abrupt increase of the variance of current fluctuations in concomitance with that of the I-V characteristic. The sharp increase, for about five orders of magnitude of current variance is associated with the opening of low resistance paths responsible for the sharp increase of the I-V characteristics. A strong non-Gaussian behavior of the associated probability distribution function is further detected by numerical calculations.
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