A new numerical algorithm for solving the symmetric eigenvalue problem is presented. The technique deviates fundamentally from the traditional Krylov subspace iteration based techniques (Arnoldi and Lanczos algorithms) or other Davidson-Jacobi techniques, and takes its inspiration from the contour integration and density matrix representation in quantum mechanics. It will be shown that this new algorithm -named FEAST -exhibits high efficiency, robustness, accuracy and scalability on parallel architectures. Examples from electronic structure calculations of Carbon nanotubes (CNT) are presented, and numerical performances and capabilities are discussed.

Probing a single nuclear spin in a silicon single electron transistor Appl. Phys. Lett. 101, 072407 (2012) Characteristics of gate-all-around silicon nanowire field effect transistors with asymmetric channel width and source/drain doping concentration J. Appl. Phys. 112, 034513 (2012) Additional information on J. Appl. Phys. The silicon nanowire transistor (SNWT) is a promising device structure for future integrated circuits, and simulations will be important for understanding its device physics and assessing its ultimate performance limits. In this work, we present a three-dimensional (3D) quantum mechanical simulation approach to treat various SNWTs within the effective-mass approximation. We begin by assuming ballistic transport, which gives the upper performance limit of the devices. The use of a mode space approach (either coupled or uncoupled) produces high computational efficiency that makes our 3D quantum simulator practical for extensive device simulation and design. Scattering in SNWTs is then treated by a simple model that uses so-called Büttiker probes, which was previously used in metal-oxide-semiconductor field effect transistor simulations. Using this simple approach, the effects of scattering on both internal device characteristics and terminal currents can be examined, which enables our simulator to be used for the exploration of realistic performance limits of SNWTs.

Estimating the number of eigenvalues located in a given interval of a large sparse Hermitian matrix is an important problem in certain applications, and it is a prerequisite of eigensolvers based on a divide-andconquer paradigm. Often, an exact count is not necessary, and methods based on stochastic estimates can be utilized to yield rough approximations. This paper examines a number of techniques tailored to this specific task. It reviews standard approaches and explores new ones based on polynomial and rational approximation filtering combined with a stochastic procedure. We also discuss how the latter method is particularly wellsuited for the FEAST eigensolver.ESTIMATING EIGENVALUE COUNTS 675 count OEa; b in OEa; b. While this method yields an exact count, it requires two complete LDL T factorizations, and this can be quite expensive for realistic eigenproblems.This paper discusses two alternative methods that provide only an estimate for OEa; b , but which are relatively inexpensive. Both methods work by estimating the trace of the spectral projector P associated with the eigenvalues inside the interval OEa; b. This spectral projector is expanded in two different ways, and its trace is computed by resorting to stochastic trace estimators, for example, [10,11]. The first method utilizes filtering techniques based on Chebyshev polynomials. The resulting projector is expanded as a polynomial function of A. In the second method, the projector is constructed by integrating the resolvent of the eigenproblem along a contour in the complex plane enclosing the interval OEa; b. In this case, the projector is approximated by a rational function of A.For each of the aforementioned methods, we present various implementations depending on the nature of the eigenproblem (generalized versus standard) and cost considerations. Thus, in the polynomial expansion case, we propose a barrier-type filter when dealing with a standard eigenproblem and two high/low pass filters in the case of generalized eigenproblems. In the rational expansion case, we have the choice of using an LU factorization or a Krylov subspace method to solve linear systems. The optimal implementation of each method used for the eigenvalue count depends on the situation at hand and involves compromises between cost and accuracy. While it is not the aim of this paper to explore detailed analysis of these techniques, we will discuss various possibilities and provide illustrative examples.The polynomial and rational expansion methods are motivated by two distinct approaches recently suggested in the context of electronic structure calculations: (i) spectrum slicing and (ii) Cauchy integral eigen-projection. In the spectrum slicing techniques [1], the eigenpairs are computed by dividing the spectrum in many small subintervals, called 'slices' or 'windows'. For each window, a barrier function is approximated by Chebyshev-Jackson polynomials in order to select only the portion of the spectrum in the slice. In this method, it is important to determine an approximate c...

The FEAST method for solving large sparse eigenproblems is equivalent to subspace iteration with an approximate spectral projector and implicit orthogonalization. This relation allows to characterize the convergence of this method in terms of the error of a certain rational approximant to an indicator function. We propose improved rational approximants leading to FEAST variants with faster convergence, in particular, when using rational approximants based on the work of Zolotarev. Numerical experiments demonstrate the possible computational savings especially for pencils whose eigenvalues are not well separated and when the dimension of the search space is only slightly larger than the number of wanted eigenvalues. The new approach improves both convergence robustness and load balancing when FEAST runs on multiple search intervals in parallel.

In this letter, we report a three-dimensional (3D) quantum mechanical simulation to investigate the effects of surface roughness scattering (SRS) on the device characteristics of Si nanowire transistors (SNWTs). We treat the microscopic structure of the Si/SiO 2 interface roughness directly by using a 3D finite element technique. The results show that 1) SRS reduces the electron density of states in the channel, which increases the SNWT threshold voltage, and 2) the SRS in SNWTs becomes more effective when more propagating modes are occupied, which implies that SRS is more important in planar metal-oxide-semiconductor field-effect-transistors with many transverse modes occupied than in small-diameter SNWTs with few modes conducting.