Reduction of the three-dimensional Schrödinger equation for multilayered films

Abstract

In this paper we present a method for reducing the three dimensional Schrödinger equation to study confined metallic states, such as quantum well states, in a multilayer film geometry. While discussing some approximations that are employed when dealing with the three dimensionality of the problem, we derive a one dimensional equation suitable for studying such states using an envelope function approach. Some applications to the Cu/Co multilayer system with regard to spin tunneling/rotations and angle resolved photoemission experiments are discussed. Recent experimental studies of spin dependent, hot electron transmission such as those described in Ref. 4 and resonant tunneling through two discrete states (Ref. 5), have raised a number of interesting issues related to the ferromagnetic and insulating materials used, the nature of the electronic states that are involved in transmission and enhancements in spin filtering effects. Apart from the first principles based attempts which can be quite tedious, most theoretical studies of these spin dependent effects have used free electron band structures and other simplifications in the metallic as well as in the insulating regions. Our work, though motivated by free electron approaches such as those introduced by Slonczewski 6 , is an attempt to bring out a more realistic lateral dependence of the electronic states under consideration.This paper is organized as follows. First, starting from the three dimensional Schrödinger equation, we proceed to derive an envelope function approach suitable for multilayered films.This procedure will go beyond the free electron methods that have been commonly used in the past, making use of more realistic wave functions, but avoiding a full pledged ab-initio calculation when studying such systems. We introduce an approach which incorporates the two dimensional Bloch wave vector k || and show that the associated parallel band structure characteristic of the material being used, plays a major role in perpendicular transmission.Second, We will study spin tunneling and rotation effects (to be defined later) in a multilayer system with two ferromagnetic layers separated by a nonmagnetic metal such as Co/Cu/Co. We will also address some issues related to angle resolved photoemission and inverse photoemission experiments focused on confined states in metallic multilayers.

Model waveform accuracy standards for gravitational wave data analysis

“…II C and described in detail in Eqs. (18), (19), (23), (24), and (30)- (33). The table entries for phase error assume that the amplitude and phase errors are comparable.…”

“…While the accuracy standards derived here apply to any type of gravitational waveform, in practice we expect them to be most useful for broadband waveforms such as those from compact binary systems. Narrow band gravitational wave signals, such as from rotating neutron stars, are expected to be analyzed using modified versions of the data analysis methods described here, i.e., using the F -statistic matched filter or time-domain heterodyning [18]. The accuracy-standard analysis presented here could easily be extended to these cases [19,20].…”

Upper limits on gravitational wave emission from 78 radio pulsars

“…As of the beginning of 2005 the Australia Telescope National Facility (ATNF) online pulsar catalogue [13] listed 1 154 millisecond and young pulsars, all with rotation frequencies >25 Hz (gravitational wave frequency >50 Hz) that fall within the design band of the LIGO and GEO 600 interferometers, and the search for their gravitational waves has developed rapidly since the start of data-taking runs in 2002. Data from the first science run (S1) were used to perform a search for gravitational waves at twice the rotation frequency from PSR J1939 2134 [2]. Two techniques were used in this search: one a frequency domain, frequentist search, and the other a time domain, Bayesian search which gave a 95% credible amplitude upper limit of 1:4 10 ÿ22 , and an ellipticity upper limit of 2:9 10 ÿ4 assuming I zz 10 38 kg m 2 .…”

“…Our previous known pulsar searches [2,3] have excluded pulsars within binary systems, despite the majority of pulsars within our detector band being in such systems. To address this, we have included an additional time delay to transform from the binary system barycenter (BSB) to pulsar proper time, which is a stationary reference frame with respect to the pulsar.…”

“…The results will also be presented in terms of the pulsars' equatorial ellipticity ", which under the assumption of triaxiality is related to h 0 via Eq. (1.1) by " 0:237 h 0 10 ÿ24 r 1 kpc 1 Hz 2 10 38 kg m 2 I zz : (6.1)…”