Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces

The condition-based complexity analysis framework is one of the gems of modern numerical algebraic geometry and theoretical computer science. One of the challenges that it poses is to expand the currently limited range of random polynomials that we can handle. Despite important recent progress, the available tools cannot handle random sparse polynomials and Gaussian polynomials, that is polynomials whose coefficients are i.i.d. Gaussian random variables. We initiate a condition-based complexity framework based on the norm of the cube that is a step in this direction. We present this framework for real hypersurfaces and univariate polynomials. We demonstrate its capabilities in two problems, under very mild probabilistic assumptions. On the one hand, we show that the average run-time of the Plantinga-Vegter algorithm is polynomial in the degree for random sparse (alas a restricted sparseness structure) polynomials and random Gaussian polynomials. On the other hand, we study the size of the subdivision tree for Descartes' solver and run-time of the solver by Jindal and Sagraloff (arXiv:1704.06979). In both cases, we provide a bound that is polynomial in the size of the input (size of the support plus logarithm of the degree) for not only on the average, but all higher moments.Comment: 34 pages. Version 1, conference version; from version 2, journal versio

Fabrication and Characterization of Modulation-Doped ZnSe/(Zn,Cd)Se (110) Quantum Wells: A New System for Spin Coherence Studies

We describe the growth of modulation-doped ZnSe/(Zn,Cd)Se quantum wells on (110) GaAs substrates. Unlike the well-known protocol for the epitaxy of ZnSe-based quantum structures on (001) GaAs, we find that the fabrication of quantum well structures on (110) GaAs requires significantly different growth conditions and sample architecture. We use magnetotransport measurements to confirm the formation of a two-dimensional electron gas in these samples, and then measure transverse electron spin relaxation times using time-resolved Faraday rotation. In contrast to expectations based upon known spin relaxation mechanisms, we find surprisingly little difference between the spin lifetimes in these (110)-oriented samples in comparison with (100)-oriented control samples.Comment: To appear in Journal of Superconductivity (Proceedings of 3rd Conference on Physics and Applications of Spin-dependent Phenomena in Semiconductors

Simple and Nearly Optimal Polynomial Root-finding by Means of Root Radii Approximation

We propose a new simple but nearly optimal algorithm for the approximation of all sufficiently well isolated complex roots and root clusters of a univariate polynomial. Quite typically the known root-finders at first compute some crude but reasonably good approximations to well-conditioned roots (that is, those isolated from the other roots) and then refine the approximations very fast, by using Boolean time which is nearly optimal, up to a polylogarithmic factor. By combining and extending some old root-finding techniques, the geometry of the complex plane, and randomized parametrization, we accelerate the initial stage of obtaining crude to all well-conditioned simple and multiple roots as well as isolated root clusters. Our algorithm performs this stage at a Boolean cost dominated by the nearly optimal cost of subsequent refinement of these approximations, which we can perform concurrently, with minimum processor communication and synchronization. Our techniques are quite simple and elementary; their power and application range may increase in their combination with the known efficient root-finding methods.Comment: 12 pages, 1 figur

Electron Spin Injection at a Schottky Contact

We investigate theoretically electrical spin injection at a Schottky contact between a spin-polarized electrode and a non-magnetic semiconductor. Current and electron density spin-polarizations are discussed as functions of barrier energy and semiconductor doping density. The effect of a spin-dependent interface resistance that results from a tunneling region at the contact/semiconductor interface is described. The model can serve as a guide for designing spin-injection experiments with regard to the interface properties and device structure.Comment: 4 pages, 4 figure

Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces

International audienceThe condition-based complexity analysis framework is one of the gems of modern numerical algebraic geometry and theoretical computer science. One of the challenges that it poses is to expand the currently limited range of random polynomials that we can handle. Despite important recent progress, the available tools cannot handle random sparse polynomials and Gaussian polynomials, that is polynomials whose coefficients are i.i.d. Gaussian random variables. We initiate a condition-based complexity framework based on the norm of the cube, that is a step in this direction. We present this framework for real hypersurfaces. We demonstrate its capabilities by providing a new probabilistic complexity analysis for the Plantinga-Vegter algorithm, which covers both random sparse (alas a restricted sparseness structure) polynomials and random Gaussian polynomials. We present explicit results with structured random polynomials for problems with two or more dimensions. Additionally, we provide some estimates of the separation bound of a univariate polynomial in our current framework

Electric-field dependent spin diffusion and spin injection into semiconductors

We derive a drift-diffusion equation for spin polarization in semiconductors by consistently taking into account electric-field effects and nondegenerate electron statistics. We identify a high-field diffusive regime which has no analogue in metals. In this regime there are two distinct spin diffusion lengths. Furthermore, spin injection from a ferromagnetic metal into a semiconductor is enhanced by several orders of magnitude and spins can be transported over distances much greater than the low-field spin diffusion length.Comment: 5 pages, 3 eps figure

Optical Pumping in Ferromagnet-Semiconductor Heterostructures: Magneto-optics and Spin Transport

Epitaxial ferromagnetic metal - semiconductor heterostructures are investigated using polarization-dependent electroabsorption measurements on GaAs p-type and n-type Schottky diodes with embedded In1-xGaxAs quantum wells. We have conducted studies as a function of photon energy, bias voltage, magnetic field, and excitation geometry. For optical pumping with circularly polarized light at energies above the band edge of GaAs, photocurrents with spin polarizations on the order of 1 % flow from the semiconductor to the ferromagnet under reverse bias. For optical pumping at normal incidence, this polarization may be enhanced significantly by resonant excitation at the quantum well ground-state. Measurements in a side-pumping geometry, in which the ferromagnet can be saturated in very low magnetic fields, show hysteresis that is also consistent with spin-dependent transport. Magneto-optical effects that influence these measurements are discussed.Comment: PDF, 4 figures, 1 tabl

Spin diffusion and injection in semiconductor structures: Electric field effects

In semiconductor spintronic devices, the semiconductor is usually lightly doped and nondegenerate, and moderate electric fields can dominate the carrier motion. We recently derived a drift-diffusion equation for spin polarization in the semiconductors by consistently taking into account electric-field effects and nondegenerate electron statistics and identified a high-field diffusive regime which has no analogue in metals. Here spin injection from a ferromagnet (FM) into a nonmagnetic semiconductor (NS) is extensively studied by applying this spin drift-diffusion equation to several typical injection structures such as FM/NS, FM/NS/FM, and FM/NS/NS structures. We find that in the high-field regime spin injection from a ferromagnet into a semiconductor is enhanced by several orders of magnitude. For injection structures with interfacial barriers, the electric field further enhances spin injection considerably. In FM/NS/FM structures high electric fields destroy the symmetry between the two magnets at low fields, where both magnets are equally important for spin injection, and spin injection becomes locally determined by the magnet from which carriers flow into the semiconductor. The field-induced spin injection enhancement should also be insensitive to the presence of a highly doped nonmagnetic semiconductor (NS$^+$) at the FM interface, thus FM/NS$^+$/NS structures should also manifest efficient spin injection at high fields. Furthermore, high fields substantially reduce the magnetoresistance observable in a recent experiment on spin injection from magnetic semiconductors

Spin oscillations in transient diffusion of a spin pulse in n-type semiconductor quantum wells

By studying the time and spatial evolution of a pulse of the spin polarization in $n$-type semiconductor quantum wells, we highlight the importance of the off-diagonal spin coherence in spin diffusion and transport. Spin oscillations and spin polarization reverse along the the direction of spin diffusion in the absence of the applied magnetic field are predicted from our investigation.Comment: 5 pages, 4 figures, accepted for publication in PR

Quantum Information Processing with Ferroelectrically Coupled Quantum Dots

I describe a proposal to construct a quantum information processor using ferroelectrically coupled Ge/Si quantum dots. The spin of single electrons form the fundamental qubits. Small (<10 nm diameter) Ge quantum dots are optically excited to create spin polarized electrons in Si. The static polarization of an epitaxial ferroelectric thin film confines electrons laterally in the semiconductor; spin interactions between nearest neighbor electrons are mediated by the nonlinear process of optical rectification. Single qubit operations are achieved through "g-factor engineering" in the Ge/Si structures; spin-spin interactions occur through Heisenberg exchange, controlled by ferroelectric gates. A method for reading out the final state, while required for quantum computing, is not described; electronic approaches involving single electron transistors may prove fruitful in satisfying this requirement.Comment: 10 pages, 3 figure