Efficient Methods for Multidimensional Global Polynomial Approximation with Applications to Random PDEs

In this work, we consider several ways to overcome the challenges associated with polynomial approximation and integration of smooth functions depending on a large number of inputs. We are motivated by the problem of forward uncertainty quantification (UQ), whereby inputs to mathematical models are considered as random variables. With limited resources, finding more efficient and accurate ways to approximate the multidimensional solution to the UQ problem is of crucial importance, due to the “curse of dimensionality” and the cost of solving the underlying deterministic problem. The first way we overcome the complexity issue is by exploiting the structure of the approximation schemes used to solve the random partial differential equations (PDE), thereby significantly reducing the overall cost of the approximation. We do this first using multilevel approximations in the physical variables, and second by exploiting the hierarchy of nested sparse grids in the random parameter space. With these algorithmic advances, we provably decrease the complexity of collocation methods for solving random PDE problems. The second major theme in this work is the choice of efficient points for multidimensional interpolation and interpolatory quadrature. A major consideration in interpolation in multiple dimensions is the balance between stability, i.e., the Lebesgue constant of the interpolant, and the granularity of the approximation, e.g., the ability to choose an arbitrary number of interpolation points or to adaptively refine the grid. For these reasons, the Leja points are a popular choice for approximation on both bounded and unbounded domains. Mirroring the best-known results for interpolation on compact domains, we show that Leja points, defined for weighted interpolation on R, have a Lebesgue constant which grows subexponentially in the number of interpolation nodes. Regarding multidimensional quadratures, we show how certain new rules, generated from conformal mappings of classical interpolatory rules, can be used to increase the efficiency in approximating multidimensional integrals. Specifically, we show that the convergence rate for the novel mapped sparse grid interpolatory quadratures is improved by a factor that is exponential in the dimension of the underlying integral

Initial stage of the 2D-3D transition of a strained SiGe layer on a pit-patterned Si(001) template

We investigate the initial stage of the 2D-3D transition of strained Ge layers deposited on pit-patterned Si(001) templates. Within the pits, which assume the shape of inverted, truncated pyramids after optimized growth of a Si buffer layer, the Ge wetting layer develops a complex morphology consisting exclusively of {105} and (001) facets. These results are attributed to a strain-driven step-meandering instability on the facetted side-walls of the pits, and a step-bunching instability at the sharp concave intersections of these facets. Although both instabilities are strain-driven, their coexistence becomes mainly possible by the geometrical restrictions in the pits. It is shown that the morphological transformation of the pit surface into low-energy facets has strong influence on the preferential nucleation of Ge islands at the flat bottom of the pits.Comment: 19 pages, 7 figure

Spin-Valley Kondo Effect in Multi-electron Silicon Quantum Dots

We study the spin-valley Kondo effect of a silicon quantum dot occupied by $\mathcal{N}$ electrons, with $\mathcal{N}$ up to four. We show that the Kondo resonance appears in the $\mathcal{N}=1,2,3$ Coulomb blockade regimes, but not in the $\mathcal{N}=4$ one, in contrast to the spin-1/2 Kondo effect, which only occurs at $\mathcal{N}=$ odd. Assuming large orbital level spacings, the energy states of the dot can be simply characterized by fourfold spin-valley degrees of freedom. The density of states (DOS) is obtained as a function of temperature and applied magnetic field using a finite-U equation-of-motion approach. The structure in the DOS can be detected in transport experiments. The Kondo resonance is split by the Zeeman splitting and valley splitting for double- and triple-electron Si dots, in a similar fashion to single-electron ones. The peak structure and splitting patterns are much richer for the spin-valley Kondo effect than for the pure spin Kondo effect.Comment: 8 pages, 4 figures, in PRB format. This paper is a sequel to the paper published in Phys. Rev. B 75, 195345 (2007

Giant g factor tuning of long-lived electron spins in Ge

Control of electron spin coherence via external fields is fundamental in spintronics. Its implementation demands a host material that accommodates the highly desirable but contrasting requirements of spin robustness to relaxation mechanisms and sizeable coupling between spin and orbital motion of charge carriers. Here we focus on Ge, which, by matching those criteria, is rapidly emerging as a prominent candidate for shuttling spin quantum bits in the mature framework of Si electronics. So far, however, the intrinsic spin-dependent phenomena of free electrons in conventional Ge/Si heterojunctions have proved to be elusive because of epitaxy constraints and an unfavourable band alignment. We overcome such fundamental limitations by investigating a two dimensional electron gas (2DEG) confined in quantum wells of pure Ge grown on SiGe-buffered Si substrates. These epitaxial systems demonstrate exceptionally long spin relaxation and coherence times, eventually unveiling the potential of Ge in bridging the gap between spintronic concepts and semiconductor device physics. In particular, by tuning spin-orbit interaction via quantum confinement we demonstrate that the electron Land\'e g factor and its anisotropy can be engineered in our scalable and CMOS-compatible architectures over a range previously inaccessible for Si spintronics