Improved recursive Green's function formalism for quasi one-dimensional systems with realistic defects

We derive an improved version of the recursive Green's function formalism (RGF), which is a standard tool in the quantum transport theory. We consider the case of disordered quasi one-dimensional materials where the disorder is applied in form of randomly distributed realistic defects, leading to partly periodic Hamiltonian matrices. The algorithm accelerates the common RGF in the recursive decimation scheme, using the iteration steps of the renormalization decimation algorithm. This leads to a smaller effective system, which is treated using the common forward iteration scheme. The computational complexity scales linearly with the number of defects, instead of linearly with the total system length for the conventional approach. We show that the scaling of the calculation time of the Green's function depends on the defect density of a random test system. Furthermore, we discuss the calculation time and the memory requirement of the whole transport formalism applied to defective carbon nanotubes

QR Factorization of Tall and Skinny Matrices in a Grid Computing Environment

Previous studies have reported that common dense linear algebra operations do not achieve speed up by using multiple geographical sites of a computational grid. Because such operations are the building blocks of most scientific applications, conventional supercomputers are still strongly predominant in high-performance computing and the use of grids for speeding up large-scale scientific problems is limited to applications exhibiting parallelism at a higher level. We have identified two performance bottlenecks in the distributed memory algorithms implemented in ScaLAPACK, a state-of-the-art dense linear algebra library. First, because ScaLAPACK assumes a homogeneous communication network, the implementations of ScaLAPACK algorithms lack locality in their communication pattern. Second, the number of messages sent in the ScaLAPACK algorithms is significantly greater than other algorithms that trade flops for communication. In this paper, we present a new approach for computing a QR factorization -- one of the main dense linear algebra kernels -- of tall and skinny matrices in a grid computing environment that overcomes these two bottlenecks. Our contribution is to articulate a recently proposed algorithm (Communication Avoiding QR) with a topology-aware middleware (QCG-OMPI) in order to confine intensive communications (ScaLAPACK calls) within the different geographical sites. An experimental study conducted on the Grid'5000 platform shows that the resulting performance increases linearly with the number of geographical sites on large-scale problems (and is in particular consistently higher than ScaLAPACK's).Comment: Accepted at IPDPS10. (IEEE International Parallel & Distributed Processing Symposium 2010 in Atlanta, GA, USA.

A discrete Fourier transform for virtual memory machines

An algebraic theory of the Discrete Fourier Transform is developed in great detail. Examination of the details of the theory leads to a computationally efficient fast Fourier transform for the use on computers with virtual memory. Such an algorithm is of great use on modern desktop machines. A FORTRAN coded version of the algorithm is given for the case when the sequence of numbers to be transformed is a power of two

Improvements on non-equilibrium and transport Green function techniques: the next-generation transiesta

We present novel methods implemented within the non-equilibrium Green function code (NEGF) transiesta based on density functional theory (DFT). Our flexible, next-generation DFT-NEGF code handles devices with one or multiple electrodes ($N_e\ge1$) with individual chemical potentials and electronic temperatures. We describe its novel methods for electrostatic gating, contour opti- mizations, and assertion of charge conservation, as well as the newly implemented algorithms for optimized and scalable matrix inversion, performance-critical pivoting, and hybrid parallellization. Additionally, a generic NEGF post-processing code (tbtrans/phtrans) for electron and phonon transport is presented with several novelties such as Hamiltonian interpolations, $N_e\ge1$ electrode capability, bond-currents, generalized interface for user-defined tight-binding transport, transmission projection using eigenstates of a projected Hamiltonian, and fast inversion algorithms for large-scale simulations easily exceeding $10^6$ atoms on workstation computers. The new features of both codes are demonstrated and bench-marked for relevant test systems.Comment: 24 pages, 19 figure

Counting statistics of transport through Coulomb blockade nanostructures: High-order cumulants and non-Markovian effects

Recent experimental progress has made it possible to detect in real-time single electrons tunneling through Coulomb blockade nanostructures, thereby allowing for precise measurements of the statistical distribution of the number of transferred charges, the so-called full counting statistics. These experimental advances call for a solid theoretical platform for equally accurate calculations of distribution functions and their cumulants. Here we develop a general framework for calculating zero-frequency current cumulants of arbitrary orders for transport through nanostructures with strong Coulomb interactions. Our recursive method can treat systems with many states as well as non-Markovian dynamics. We illustrate our approach with three examples of current experimental relevance: bunching transport through a two-level quantum dot, transport through a nano-electromechanical system with dynamical Franck-Condon blockade, and transport through coherently coupled quantum dots embedded in a dissipative environment. We discuss properties of high-order cumulants as well as possible subtleties associated with non-Markovian dynamics.Comment: 27 pages, 8 figures, 1 table, final version as published in Phys. Rev.