IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 5, SEPTEMBER 2002 2081 3-D Analysis of the Playback Signal in Perpendicular Recording for an Off-Centered GMR Element Bogdan Valcu and H. Neal Bertram, Fellow, IEEE Abstract—The playback signal characteristics in perpendicular recording is analyzed using three-dimensional geometry. Analyt- ical expressions are deduced for both the width of the pulse and the asymptotic value taken by the signal when the sensor is far away from the transition. Index Terms—Fourier transform, perpendicular recording, reci- procity. I. INTRODUCTION T HE general form of the single pulse in perpendicular recording is well understood. Here a quantitative analysis of the signal dependence on specific recording parameters is given. To obtain the correct form of the pulse shape we consider the complete three-dimensional (3-D) geometry of the reading process [1] (see Fig. 1 for a description of all parameters). Due to the asymmetry in the construction of the head, the sensor is off-centered, with one gap (shield-to-element) distance typically double the other gap distance . The soft underlayer (of permeability ), providing the returning path for the magnetic flux, is considered an integral part of the head. Only the case of a centered read track will be considered. II. SIGNAL IN THE FREQUENCY DOMAIN By reciprocity, [2] the signal is the convolution of the mag- netic charge distribution at the two surfaces of the perpen- dicular recording medium ( and ) and the mag- netic potential along these surfaces. The problem of solving the potential can be easily handled in Fourier space. Conse- quently, the total flux through the sensor at a distance from the center of transition is (1) The potential satisfies Laplace’s equation at any point under the head. In Fourier space the solution is a superposition of waves propagating with wave vectors in the directions. The mixing coefficients are determined by im- posing the appropriate boundary conditions for the potential and they are proportional to the Fourier transform of the potential at the surface of the head . In the cross-track direc- tion, we consider the simplest dependence of , namely, Manuscript received February 11, 2002; revised June 1, 2002. This work was supported by an IBM Partnership. The authors are with the Center for Magnetic Recording Research, University of California-San Diego, La Jolla, CA 92037 USA. Digital Object Identifier 10.1109/TMAG.2002.801838. Fig. 1. Geometry of the reading process. The insert shows the sensor positioned in the middle of the track. and are the widths of the read and written tracks, respectively. The total distance ABS to SUL is . The SUL has thickness . a box function [3], while the dependence along the direction is approximated by a numerical two-dimensional (2-D) finite-element method, using the MAXWELL software package [4]. For the written transition a tanh shape in the downtrack direc- tion is considered. In the cross-track direction we neglect pos- sible curvature. Thus the transition does not vary along the cross track direction and vanishes abruptly at the track edge. The complete form for the Fourier transform of the signal is given by (2) where with (3) (4) It is worthwhile to note that in the 2-D analysis the function in (2) is replaced by . For a symmetrically positioned sensor( ) the Fourier transform is a sine transform (plotted in Fig. 2). It becomes apparent why it is important to use the 3-D analysis to ob- tain the pulse shape for large . The low frequencies determine the signal value when is large. In the 2-D case, the Fourier transform vanishes, inducing a decay to zero of the signal. The 0018-9464/02$17.00 © 2002 IEEE