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We have extended an earlier two-dimensional (2-D) analysis to provide a relatively simple approach to calculating the transmission of magnetic fields in three-dimensional (3-D) multilayered structures. The approach accommodates complex structures composed of an arbitrary number of layers, each of which may have an arbitrary magnetization direction and anisotropic permeability. The procedure relies on an initial transformation into the 2-D spatial frequency domain and then the use of transmission matrices to relate the fields in the different layers. The approach is general in that it provides a method for finding the field configuration for any set of 3-D magnetic sources embedded in a layered magnetic structure with a linear B-H relationship. In this paper, we concentrate on the analytical formulation and we illustrate the method by using it to determine both the external and internal fields of some relatively simple finite 3-D geometric shapes. We also derive new 3-D expressions for the flux density at the surface of a multilayered magnetic recording medium. In a companion paper, the procedure is applied to examine readback responses for a variety of 3-D magnetic recording configurations. Index Terms--Anisotropic permeability, frequency response, layered structures, magnetic fields, magnetic recording, 3-D transmission.

We have applied a technique for modeling the transmission of three-dimensional (3-D) magnetic fields through an arbitrary number of layers of a layered structure to the specific case of a recording medium in order to study and explain the readback response for a variety of perpendicular recording configurations. We examine in particular the low-frequency response associated with flux-flow in the soft underlayer and the behavior of the main response for finite read and write widths. Our work leads to a useful analytic approximation for the 3-D on-track readback response. Index Terms--Anisotropic permeability, DC null, frequency response, layered media, magnetic recording, 3-D response.

By matching a singular function approximation to the Fourier solution at the air-bearing surface (ABS), we have derived a three-dimensional (3-D) analytic model of a shielded giant magnetoresistive (GMR) head, with side shields, for perpendicular replay. We present an explicit expression for the potential in the ABS and give values for the parameters in that expression for a range of practical head dimensions. Using only a few terms of this singular potential model, we show that the vertical field is accurate to within 1% of the sensor potential in the region of the medium for the majority of head dimensions suitable for magnetic recording. Index Terms--Distal recording, magnetic recording/reading heads, mathematics, perpendicular recording.

We investigated the effects of transition curvature on [T.sub.50] and [V.sub.p-p] for a shielded giant magnetoresistive (GMR) element reading a single, isolated transition recorded on a perpendicular medium in the presence of a soft magnetic underlayer. We considered displacement and rotation of the read head relative to the recorded track and a range of read head geometries. The curvature of the transition adversdy affects [T.sub.50], which also increases with azimuth angle. Both [T.sub.50] and [V.sub.p-p] decrease as either the GMR element or the element-side shield spacing is reduced. Index Terms--Digital recording, magnetic recording/reading heads, output measures, perpendicular recording.

Using the Fourier method, we have derived a three-dimensional, fully analytic model of a shielded magnetoresistive or giant magnetoresistive head for perpendicular replay. The head may include side shields. The field and the spectral response function are expressed in closed form. Here, we use the model to show the effect of varying the sensor-shield spacings and the air-bearing-surface--underlayer separation on the field and response of the sensor at high areal density. Index Terms--Digital recording, magnetic recording/reading heads, mathematics, perpendicular recording.

In a move to extend the storage capabilities of magnetic storage systems beyond 1 Tb/[in.sup.2], the use of patterned media has often been cited. Here, recorded domains are constrained by the geometry of the magnetic island and not the geometry of the recording head. Conventional two-dimensional readout modeling techniques, using the reciprocity integral, rely on the assumption that the across-track medium magnetization is uniform under the giant magnetoresistive replay head. However, in the case of a geometrically constrained medium this is not the case. This paper investigates the effect that the island geometry has on the characteristics of the replay signal in perpendicular patterned magnetic media storage through the extension of the reciprocity integral to three dimensions. The paper describes replay pulses that offer different characteristics from those obtained by conventional two-dimensional techniques. The origins of these differences are explained by the variation in medium magnetization across the track. Index Terms--Magnetic recording, patterned media, perpendicular magnetization, reciprocity.

We have used a series of singular expansion functions to represent the potential across the gap of a shielded pole head in the presence of a highly permeable underlayer. This method of analysis reduces the size of the system of equations to be solved for the series coefficients to the number of coefficients that will be used. Just two series terms suffice to evaluate the magnetic potential to within 1% of the pole potential at any point on the air-bearing surface for a large range of head dimensions. Here, we express the surface field spectrum analytically. Including just two series terms provides an excellent estimate. Index Terms--Digital recording, magnetic recording/reading heads, magnetoresistive head, mathematics, perpendicular recording.

This paper describes a relatively simple approach to calculating the frequency response for a two-dimensional (2-D) magnetic recording system with a medium that can include an arbitrary number of layers. Each layer may have an arbitrary magnetization direction, anisotropic permeability, and exchange coupling. The approach relies on an initial transformation into the spatial frequency domain and then the use of transmission matrices to relate the fields in the different layers. The approach is general in that it provides a method for finding the field configuration for any set of 2-D magnetic sources embedded in a layered magnetic medium with a linear B-H relationship. Here, we focus on calculating the readback response for a variety of longitudinal and perpendicular recording configurations. Since the permeability may have any wavelength dependence, we can easily include the effect of exchange coupling in the underlayer, Index Terms--Anisotropic permeability, exchange coupling, frequency response, layered media, magnetic recording.

We present an accurate method of obtaining approximate field components for a two-dimensional perpendicular magnetic recording head in the presence of a soft magnetic underlayer. It is easy to use and may be applied to any configuration of rectangular poles, shields, and gaps. Formulas for the potential at the air-bearing surface across a pole-shield gap or beyond a pole edge are obtained by assuming nonlinear magnetic potential drops both from a pole corner and from a shield corner to the underlayer. The fast Fourier transform then gives the potential and field components elsewhere. We demonstrate the accuracy of the field component estimates for an unshielded single pole and for a symmetrically shielded single pole. Index Terms--Approximations, digital recording, magnetic recording/reading heads, mathematics.

We studied the dynamic switching time in two classes of media by considering two different particle orientation distribution functions. We calculated the switching constant directly from the Landau-Lifshitz-Gilbert equation of motion, which was chosen to simulate the dynamic properties of the media. A strong linear relation between the reciprocal of the switching time and the difference between the applied and anisotropy fields is illustrated. In media for which experimental results are available, the values we obtained here agree within a factor of 2. Index Terms--Magnetic recording, magnetization switching, media.

This paper presents a new, entirely analytic solution for the magnetic field of a thin-film head of arbitrary dimensions in the presence of a soft magnetic underlayer. The magnetic potential on each pole is assumed to be a known constant. Fourier coefficients and field components are given for two particular cases of practical interest. The paper demonstrates the accuracy of approximations that require no Fourier coefficients, but that must be evaluated numerically. Index Terms--Approximations, digital recording, Laplace equations, magnetic head fields, magnetic recording/reading heads, mathematics.

Solutions for the magnetic potential in rectangular regions with simple boundary conditions are used to approximate the two-dimensional potential in the head face plane of a recording head. This method of approximation can be applied to any recording head with rectangular corners, in the presence of an underlayer. The slot approximation is demonstrated for ring heads and shielded (magnetoresistive) pole heads. The quarter plane approximation is given for application to finite-length heads. Estimates of the field components and the spectral response function are derived from the approximate potential. The simple slot approximation is shown to be at least as accurate as other, often more complicated, methods of approximation. Most significantly, both the slot and the quarter-plane approximations are easy to apply and need no prior knowledge of the head field of the particular geometry under consideration. Index Terms - Approximations, digital recording, magnetic recording/reading heads, mathematics.

An investigation is presented of the number of exact Fourier terms needed to evaluate the field components of a ring head, a slotted plane, and planar heads of various dimension ratios to specified accuracy. Simple approximate formulas for the coefficients of a ring head and a slotted plane are given, and two approximations to the coefficients of a planar head of given dimensions are provided. The accuracy of the field components and the spectral response function, each evaluated with the approximate coefficients, is used to assess the usefulness of these approximations. Both of the approximations for the coefficients of planar heads lead to excellent estimates of the field components. Index Terms - Approximations, digital recording, magnetic recording/reading heads, mathematics.

Three-dimensional (3-D) finite element models have been utilized to simulate electromagnetic behaviors in spintunneling random access memory (STram). The most significant contributors have been identified. Compared with conventional current-in-plane (CIP) giant magneto-resistive (GMR) memory, whose signal level is inversely proportional to the square root of the storage density, these current-perpendicular-to-plane (CPP) STram elements provide an excellent readout property in that their signal level is independent of their cross-section area. This result is so attractive that the density of STram should not be limited by signal degradation. Moreover, a magnetic flux closure design was found to reduce the crossfeed by about a factor of five, compared with conventional keeperless design, which is the most favored approach for achieving [10.sup.9] bits/[cm.sup.2] areal density. Although the storage mechanism described in this paper is made of STrata, the flux-closure design could be generally applicable to other magnetic solid state memories. Index Terms - Creep, current-perpendicular-to-plane giant magnetoresistance effect, readout, selective property, solid state memory, spin-tunneling effect.

A previously published two-dimensional conformal mapping solution for a shielded magnetoresistive (MR) sensor is used to explore the effect that recessing the shields has on replay from a vertically oriented medium in the presence of a soft magnetic underlayer. The horizontal and vertical head-field components, spectral response function, linear dibit shift, and roll-off curve of each of two heads, where the sensor tip and the shield faces are coplanar, are compared with those of the same heads with recessed shields. By recessing the shields, the solution may be applied to the case of an unshielded MR sensor. Index Terms - Digital recording, fields, magnetic recording/reading heads, magnetoresistive materials/devices, perpendicular magnetic recording, Schwarz-Christoffel mapping.

Three-dimensional finite element models have been utilized to simulate the crossfeed problems in read-while-write (RWW) tape heads, and the results have been compared with experimental measurements. The most significant contributors, such as shield structure, intershield spacing, and reproductive mode, have been identified. A new thin-film inductive/dual-magnetoresistive tape head design was found to reduce the crossfeed by about a factor of 30, compared with the conventional shared-pole design, which is the most favored approach for achieving simultaneous operation of the READ head and WRITE head. Index Terms - Dual magnetoresistive head, magnetic recording, magnetic tape head, read-while-write operation.

A new theoretical analysis of an asymmetrically shielded magnetoresistive sensor in the presence of a high-permeability underlayer is obtained by conformal mapping. The symmetrically shielded head is a special case. The solution also covers recessed sensors, and can be extended to the no underlayer situation. Head fields, spectral response functions, phase, dibit shift, and roll-off results for various head configurations are provided and discussed.

This paper discusses the spectral response functions of various head geometries in perpendicular magnetic readback from a double layer medium. New analytic expressions, based on idealized two-dimensional models, for the spectral response of a symmetrically shielded constant potential pole and a thin film head with either constant or linearly varying pole potential are presented. The spectral response function for a single pole with an arbitrary magnetic potential distribution, varying through the pole thickness only, is also given. Specific examples are discussed in conjunction with related linear dibit shift results. The analysis indicates that certain graded magnetization single poles may have superior readback capabilities in comparison with conventional ring, single pole, or thin film heads, especially for 'in contact' recording. It appears that a single pole with a cosine-squared magnetization may have almost identical replay characteristics to a symmetrically shielded pole.

The magnetic field of a graded magnetization single pole head in the presence of a highly permeable magnetic underlayer is analyzed. Vertical field amplitudes and gradients in the region close to the trailing edge of the pole are shown to be higher than those produced by a conventional single pole. For a perpendicularly magnetized medium, output spectral response functions, roll-off curves and peak shift computations indicate improved replay performance. The exact theoretical solution is obtained in the form of an infinite series using Fourier analysis but it is shown that the leading term alone can lead to a simple and accurate approximation. New theoretical results are also presented for a constant potential pole, including a simple approximation to the field and explicit exact formulae for certain output calculations.

A new and exact solution for the magnetic field of a thin film recording head in the presence of a high permeability magnetic underlayer is presented. Typical Fourier coefficients and corresponding fields are given. The leading term alone of the Fourier solution is shown to be more accurate than other previously published approximations when used in conjunction with an integral representation. A novel proposal of variable pole potential is analyzed in a similar manner and significant improvements in peak field strength, field gradients and the absence of secondary field peaks are demonstrated and quantified.

The new solution of an idealized mathematical model of the field due to a symmetrically shielded magnetic pole suitable for perpendicular recording is presented. Accurate Fourier coefficients and corresponding magnetic fields are obtained for a representative pole dimension and a range of pole/shield separation distances and comparisons are made with the field of an unshielded isolated pole. It is found that the presence of a shield may significantly increase both the peak field strength and the field gradient. The highest field gradients presented here are predicted for in-contact recording with certain thin-film pole/shield geometries.

Wage changes in a sample (1979-86) of Canadian wage contracts are analyzed. The sample is split into cost-of-living allowance (COLA) and non-COLA contracts. Econometric estimation corrects for sample selectivity bias. In non-COLA contracts, the unemployment rate and a proxy for expected inflation are significant. Expected industry price changes and productivity changes exert smaller but significant effects. In COLA contracts, ex ante inflation coverage ranges between 60% and 100% of expected inflation. Catch-up for previous uncompensated inflation, the unemployment rate, and expected inflation are also significant. A common wage structure across COLA and non-COLA contracts is rejected by the data.

The size, shape, and distribution of islands in a patterned medium depends on the patterning process adopted. The off-track performance in this case is mainly dominated by inter-track interference due to the neighboring islands. In this paper, the effect that the island distribution has on the off-track performance is investigated with respect to read channel bit-error-rate performance. Index Terms--Patterned media, PRML, track misregistration.

The effects of gap length, head-underlayer separation, and return pole potential on the magnitude and direction of the field of a gapped head are investigated using an analytic two-dimensional model. Criteria derived from a micromagnetic model of a perpendicular recording medium are used to optimize the head geometry for various media parameters. A head with a gap length which is at most 2.5 times the head-medium separation can write on higher coercivity media than a single pole. Index Terms--Digital recording, magnetic-head fields, magnetic recording/reading heads, perpendicular recording.

A dynamic hierarchical micromagnetic model has been used to study the effect of off-axis (in-plane) fields upon the recording properties of perpendicular media. The transient and high-frequency switching response are improved. On longer timescales, the susceptibility and squareness are significantly increased. However, transitions written in the medium with a pole head change only slightly when the in-plane field is included. This is because the transition gets written at a point where the head field gradient is lower and because in-plane demagnetizing fields act to reduce the in-plane head field. Index Terms--Magnetic recording, modeling, perpendicular magnetic recording, transient response.

The key technique in applying keepered media system in a reproduction is how to 'open' the keeper layer. In this paper we propose a new concept of keepered-media-reproduction with Dual Magnetoresistive (DMR) heads. The keeper layer may be biased by the stray flux from DMR sensors. The shunting-releasing ratio between reproduced signals without and with a sense current is found to be - 26 dB. This media system is completely stable against thermal demagnetization. It is believed that keepered media may offer a similar advantage with MR heads and potentially allow magnetics to break the 10 Gb/[in.sup.2] areal density barrier with the aid of the high performance gains of MR heads. Index Terms: keepered media, Dual Magnetoresistive head, magnetic recording

A new concept of Dual Magnetoresistive reproductive mode is applied to improve the design and operation of high-performance multi-track Read-While-Write (RWW) heads for tape and/or potential disk applications. RWW operation exists in some magnetic recording units in which verifying what was recorded is necessary. A new multi-track Inductive/Dual Magnetoresistive (DMR) head design is found to reduce the crossfeed noise radiation by about a factor of 20, compared with the conventional Soft-Adjacent-Layer (SAL) Magnetoresistive Heads design, which is the most favoured approach for achieving simultaneous read and write operations, especially in a multi-track head with ultra-narrow trackwidth. To verify these analysis results, measurements with 2.9 [[micro]meter] track width inductive/MR and inductive/DMR heads were done. Index Terms: Read-While-Write operation, Dual Magnetoresistive head, magnetic recording, multi-track recording

The paper describes write fields and replay characteristics of gapped heads with magnetically recessed leading poles for high-frequency digital video. It describes an investigation by finite-element analysis and compares computed results for laminated Sendust and [Co.sub.91][Nb.sub.6][Zr.sub.3] heads and discusses the effects due to depositing a layer of copper inside the leading gap edge. Recessing the leading pole yields an asymmetric field, while eddy currents in the copper layer cause an advantageous redistribution of flux in the head. Predicted increases in both maximum horizontal field and horizontal field gradient by at least 130% and 17% are predicted for Sendust and [Co.sub.91][Nb.sub.6][Zr.sub.3] heads, respectively, at 100 MHz. Hence, such a head can write on higher coercivity media than a conventional gapped head. The playback performance is also improved by an extension of the frequency response. The paper gives a linear model for the phase of the spectral response, valid over a wide range of frequencies, for each of two leading pole recessions. Index Terms--Digital magnetic recording, eddy currents, finite element methods, magnetic recording/reading heads, video recording.

The paper derives expressions for the magnetization distributions that result in one-sided fluxes in infinitely long cylinders whose cross sections are confocal ellipses. It discusses their relationship to the expressions for a circular cylinder, and gives examples of the flux distributions due to specific magnetizations. Index Terms--Elliptical cylinders, Laplace equations, magnetic fields, permanent magnets.

This paper describes an investigation, based on finite-element models, of the effect of throat height reduction in gapped heads constructed from realistic, saturable materials. It considers materials and two gap region geometries. It shows that throat height reduction in a gapped head improves both the maximum horizontal field and the maximum horizontal field gradient provided that saturation does not partially block the flux path toward the gap. Saturation of planar heads limits both the maximum horizontal field and the maximum horizontal field gradient at higher throat heights than it does in conventional gapped heads. The paper shows that the optimum write field of a conventional gapped head is both stronger and sharper than that of a planar one. Index Terms--Digital magnetic recording, finite element methods, magnetic recording/reading heads.

Integrals arising in two-dimensional models of shielded magnetoresistive heads are expressed either in terms of special functions or as rapidly convergent infinite series. In each case, form of expression provides an alternative and normally more efficient means of evaluation than direct numerical integration. Index Terms--Fourier integrals, magnetic fields, magnetic recording, magnetoresistive heads.

Secondary gaps between the ferrite bulk and the Sendust (AlFeSi alloy) shim may occur in metal-in-gap (MIG) heads. Finite-element models are used to explore the effect that these gaps have on the read/write characteristics of single-and double-sided MIG heads. The presence of a secondary gap is shown to cause an additional peak in the writing field and oscillations in the magnitude of the response during replay. The field and response of a double-sided MIG head is affected more than those of the corresponding single-sided MIG head. Index Terms--Finite-element methods, magnetic recording heads, magnetic reading heads.

In planar structures, magnetization patterns exist that have the surprising property that all the external fringing field of flux emerges from one side of the structure, with precisely none appearing on the other side. In the last 25 years, this fact has led to a number of useful applications. More recently, the same concept has been found to hold for both cylindrical and spherical hollow structures, where all the fringing flux emerges either inside or outside the structure, with precisely none appearing on the other side. Again, this has led to a number of important practical applications. While many of the concepts have been published previously, this paper contains, for the first time, a uniform mathematical analysis of these one-sided flux configurations. Additionally, the paper reviews the principal applications of one-sided flux structures. Index Terms--Brushless rotating machines, cylinders, focusing, Laplace equations, magnetic fields, permanent magnets, spheres.

New analytic solutions are presented for idealized mathematical models of double-element shielded magnetoresistive (MR) magnetic recording heads. Two types of heads are considered, represented by poles of either equal or opposite magnetic potentials, each in the presence of an infinitely permeable underlayer. The solutions for heads without underlayers are special cases of the more general results. Typical field plots and spectral response functions are given and comparisons are made with published approximate solutions. Index Terms--Digital recording, magnetic fields, magnetic recording, magnetoresistive heads, mathematics.

An improved method of calculating the Fourier coefficients for a single unshielded magnetic pole head is presented. The method is based on the knowledge of the exact Fourier coefficients for a magnetic ring head. Index Terms--Magnetic recording, single pole heads.

The magnetic field of a ring head has been analyzed by Westmijze, using a conformal mapping, and by Fan, using Fourier techniques. Here these methods are reexamined and combined to give, for the first time, an explicit analytic expression for the harmonic coefficients in the Fan solution. Index Terms - Heads, frequency response, magnetic fields, mathematics, recording.