©2002 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.Analytical expressions of the replay spectra from single- and multiple-recorded bits in patterned media have been derived. The expressions were extended to include position jitter noise and bit-size variations

Aziz, Mustafa M.

Du, Huan

Middleton, Barry K.

Nutter, Paul

Wright, C. David

IEEE Transactions on Magnetics

Open Research Exeter

1964 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 5, SEPTEMBER 2002Signal and Noise Characteristics of Patterned MediaMustafa M. Aziz, Member, IEEE, C. David Wright, Member, IEEE, Barry K. Middleton, Member, IEEE, Huan Du,and Paul NutterAbstract—Analytical expressions of the replay spectra fromsingle- and multiple-recorded bits in patterned media have beenderived. The expressions were extended to include position jitternoise and bit-size variations.Index Terms—Noise, patterned media, signal.I. INTRODUCTIONTHE AIM of this paper is to provide simple analyticalexpressions of the replay spectra from isolated and singlerows of rectangular magnetic islands in recorded patternedmedia. Furthermore, noise arising from bit-position jitter andbit-size variations is also modeled.The theoretical development of signal and noise in this paperfollows the methods used in magnetic recording theory on con-tinuous media [1]. There are, however, differences between thetwo systems since the replay transducers in patterned media areexpected to have dimensions comparable to the magnetic bits[2]. In addition, the nonmagnetic spacings between the magneticbits will introduce additional nulls in replay spectra at wave-lengths comparable to the bit length.II. HEAD FIELDSThe generality of the spectral theory presented in this paperallows field functions for any transducer type or shape to beused. This includes inductive, magneto-resistive [3] and mag-netic force type transducers [4]. In the following analysis, thesingle pole head will be utilized as the replay transducer as-suming that the track width is greater than the bit width. Forsimplicity, the far-field approximations to the head field are usedhere with components defined as [1](1a)(1b)where is the current in the replay head of coil turns andefficiency . As indicated in Fig. 1(a), is the pole length andis the distance from the head surface. The Fourier transforms ofthe and components of the head field are given, respectively,Manuscript received February 15, 2002; revised May 20, 2002. This work wassupported by the Engineering and Physical Sciences Research Council underGrants GR/M46648 and GR/L83462.M. M. Aziz, C. D. Wright, and H. Du are with the School of Engineeringand Computer Science, University of Exeter, Exeter EX4 4QF, U.K. (e-mail:M.M.Aziz@exeter.ac.uk; David.Wright@exeter.ac.uk; H.Du@exeter.ac.uk).B. K. Middleton and P. Nutter is with the Department of Computer Science,University of Manchester, Oxford Road, Manchester M13 9PL, UK. Tel.: +44(0)161 2754551 (e-mail: B.K.Middleton@cs.man.ac.uk).Digital Object Identifier 10.1109/TMAG.2002.802787.by and , whereis the wavenumber and is the wavelength.III. SIGNALA. Single BitsIndividual magnetic bits are assumed to have a rectangulargeometry with length , thickness , and width as indicatedin Fig. 1(a). The replay flux in the head coil is found from reci-procity [5](2)where is the permeability of free space. In two dimensions,the head field is given by (1), and the magnetization insideeach individual bit is given by withmagnitude and orientation , as shown in Fig. 1(b). andare unit vectors in the and directions, respectively. Thus, (2)reduces to(3)where to indicate the direction of magnetization.Taking the Fourier transform of (3) with as the operator yields(4)In the absence of a soft underlayer, integration over the volumeof the magnetic bit yields the replay flux assgn(5)where is the Fourier transform of the head-field functionat the head surface ( ). The term in (5)indicates a null in the replay spectrum at a wavelength equalto the bit length . The replay voltage is given by the spatialderivative of (3) with respect to and, hence, in the Fourierdomain is simply(6)where is the head-to-medium speed.0018-9464/02$17.00 © 2002 IEEEAuthorized licensed use limited to: University of Exeter Exeter University Trial User. Downloaded on December 19, 2008 at 12:00 from IEEE Xplore.  Restrictions apply.AZIZ et al.: SIGNAL AND NOISE CHARACTERISTICS OF PATTERNED MEDIA 1965Fig. 1. (a) Coordinate system and magnetic bit size. (b) Magnetization vectorof individual magnetic bits.B. Multiple BitsFor a single infinite row of recorded islands magnetized in thesame direction and separated by a distance from their centersas shown in Fig. 1(a), the replay voltage can be written as thesuperposition of the voltages from isolated magnetic islands inthe form(7)Taking the Fourier transform of (7) yields(8)where . Similarly for an alternating sequence of mag-netized bits (with wavelength 2 ), the Fourier transform of thereplay voltage was found to be(9)IV. NOISENoise in patterned media is modeled as jitter in the bit posi-tions and variations in the bit sizes. Including position jitter, ,the replay signal for an “all ones” sequence can be written aswhich for small perturbations in , can be expanded to a first-order Taylor series(10)The spatial autocorrelation function of (10) is given by(11)where is the averaging length with periods. Eval-uating the autocorrelation function assuming that there are nonoise correlations between neighboring bits and that is azero-mean random variable yields(12)Taking the Fourier transform of (12), and writing the summationas an average, yields the replay signal power spectral density(13)where is given by (6) and the overbar indicates spatialaverages.Variations in the bit sizes include variations in the thick-nesses of magnetic bits. These variations cause fluctuations inthe head-to-medium spacing. To account for this variation, thelower limit of integration with respect to in (4) is changed tothe random variable which has a mean value of , while theupper integration limit is maintained at (i.e., changes inare counteracted by changes in in the opposite direction, thuskeeping the sum fixed). Therefore, (13) becomes(14)In (14), and it was assumed that the magnitude of themagnetization of individual islands is not changing.To evaluate the averages in (14), a distribution of bit sizesfrom a patterned medium sample is required. Fig. 2 shows aSEM image of the PtCo patterned medium used in this analysis,which was produced using laser interference lithography [6],[7]. The magnetic islands produced have an average circulardiameter of 190 nm, thickness 35 nm, and bit spacing of 590nm yielding an areal density of 1.9 GB/in .Using the SEM image, a distribution of bit diameters was ob-tained as shown in Fig. 2. For comparison, the Gaussian distri-bution, calculated from(15)using the measured average bit diameter, and standard devi-ation , is also plotted in Fig. 2. It can be seen that the Gaussiandistribution fits reasonably well with the measured distribution.Using the measured distribution of bit diameters as a guide, theGaussian function will be assumed to describe the variations inbit sizes (length, width, and thickness) in the following analysis.Evaluation of the averages in (14) according toAuthorized licensed use limited to: University of Exeter Exeter University Trial User. Downloaded on December 19, 2008 at 12:00 from IEEE Xplore.  Restrictions apply.1966 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 5, SEPTEMBER 2002(a)(b)Fig. 2. SEM image of a patterned magnetic sample and measured distributionof bit diameters. Average bit diameter (length) = 190 nm with standarddeviation 17 nm and bit spacing (center-to-center of 590 nm).using the distribution in (15) yields the replay signal power spec-tral density, including noise, as(16)where , , , and are the standard deviations of the bitwidth, length, position jitter, and head-to-medium spacing, re-spectively.Using the bit dimensions and spacings of Fig. 2, the replayspectra including bit size variations and jitter noise are plotted,respectively, in Figs. 3 and 4 using a head-to-medium spacingof 30 nm.Fig. 3 illustrates the effect of jitter noise alone on the replayspectrum, with all the other noise variances equal to zero. It canbe seen that jitter noise affects the short wavelength componentsof the replay spectrum.Bit-size variations, on the other hand, cause the disappear-ance of the spectral null corresponding to the bit length and aslight increase in the magnitude of the replay spectrum at longwavelengths, as indicated in Fig. 4.Fig. 3. Replay spectrum for increasing jitter noise variance (no bit-sizevariations).Fig. 4. Replay spectrum for increasing bit size variance (no position jitter).V. CONCLUSIONGeneral expressions for the replay signal and noise fromrecorded patterned media were derived. Jitter noise was foundto dominate the short wavelength response of the replay spec-trum. Variations in bit size and head-to-medium modulationwere found to affect the long wavelength part of the replayspectrum and to result in the disappearance of the bit-lengthnull in the spectrum. The theoretical development of signal andnoise in this paper assumed inductive type replay transducers.REFERENCES[1] B. K. Middleton, “The recording and reproducing processes,” inMagnetic Recording Technology, 2nd ed, C. D. Mee and E. D. Daniel,Eds. New York: McGraw-Hill, 1996, ch. 2, pp. 2.1–2.72.[2] G. F. Hughes, “Read channels for patterned media,” IEEE Trans. Magn.,vol. 35, pp. 2310–2312, Sept. 1999.[3] S. Shtrikman and D. R. Smith, “Analytical formulas for the unshieldedmagnetoresistive head,” IEEE Trans. Magn., vol. 32, pp. 1987–1994,May 1996.[4] C. D. Wright and E. W. Hill, “A reciprocity-based approach to under-standing magnetic force microscopy,” IEEE Trans. Magn., vol. 32, pp.4144–4146, Sept. 1996.[5] H. N. Bertram, Theory of Magnetic Recording. Cambridge, U.K.:Cambridge Univ. Press, 1994.[6] J. P. Spallas, A. M. Hawryluk, and D. R. Kania, “Field emitter array maskpatterning using laser interference lithography,” J. Vac. Sci. Technol. B,vol. 13, pp. 1973–1978, 1995.[7] A. Fernandez, P. J. Bedrossian, S. L. Baker, S. P. Vernon, and D. R.Kania, “Magnetic force microscopy of single-domain cobalt dots pat-terned using interference lithography,” IEEE Trans. Magn., vol. 32, pp.4472–4474, Sept. 1996.Authorized licensed use limited to: University of Exeter Exeter University Trial User. Downloaded on December 19, 2008 at 12:00 from IEEE Xplore.  Restrictions apply.

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