IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 6, NOVEMBER 2004
3545
A Simple Slope Model for Oriented and Isotropic Longitudinal Media Hans Jürgen Richter, Senior Member, IEEE, Gene M. Sandler, Member, IEEE, and Erol Girt
Abstract—A simple model to predict the effect of medium orientation of longitudinal media on the transition sharpness is reported. It is shown that a traditional slope model can be suitably modified by multiplying the total field gradient with the medium squareness to reflect the loss in effective field gradient due to reversible magnetization rotations. The results agree well with experimental data. Index Terms—Longitudinal recording media, oriented media, signal-to-noise ratio (SNR), slope model.
I. INTRODUCTION
I
T IS generally agreed that media for high-density longitudinal recording should have grains with axes parallel to the film plane. Today, most of the longitudinal media used in hard disk drives are of the oriented type, i.e., these media not only have axes in the film plane, but also show an anisotropy of the magnetization in the film plane [1]. Hence, the magnetization is preferentially oriented along the circumference, i.e., in the same direction as the record head passes over the disk. Most theoretical work has assumed some degree of -axis alignment to explain the recording behavior of oriented media (e.g., [2] and [3]). It is generally agreed that media with an isotropic distribution of axes in the plane (“2-D-isotropic” media) have poorer recording performance than oriented ones. To our knowledge, no simple model has been reported which successfully predicts the effect of orientation on transition sharpness and jitter. This paper reports on the development of a slope model for oriented longitudinal media. The paper is organized as follows. First, conventional slope models are reviewed and it is outlined that these fail to predict the orientation effect. Then, suitable modifications are suggested. Finally, the results are compared with literature and experimental data. II. CONVENTIONAL SLOPE MODELS This paper uses a slope model formulation according to [4] and [5] rather than the original formulation [6]. As it stands, the model as outlined in [5] was intended for recording on two-dimensional (2-D)-isotropic longitudinal media. The standard for-
mulation for the one-dimensional (1-D) slope model for longitudinal recording is [6] (1) where is the component of the magnetization, is the component for the head field, and is the demagnetizing near cofield. The slope of the magnetization curve ercivity is determined by the intrinsic switching field distribution (SFD), and the degree of -axis orientation. It is emphasized that the definition of the SFD used here is concerned with the anisotropy field distribution and excludes angle effects. Now, consider an oriented medium and a 2-D-isotropic medium having otherwise identical properties (notably and saturation magnetization ). medium thickness ( , Due to the orientation, the squareness remanent magnetization) of the oriented medium is greater than that of the 2-D-isotropic one and, consequently, of the oriented medium is greater. Hence, the oriented medium experiences a penalty due to the higher demagnetization at recording and should record wider transitions. This contradicts general experience as well as micromagnetic analysis [2]. as meaIt is generally observed that the loop slope sured in a magnetometer is greater for oriented media. However, it should be kept in mind that the magnetization loops are measured with a field applied along the axis, whereas the recording process occurs in a rotating field. It has been worked out in [5] that the additional spread of recording locations for a 2-D-isotropic medium due to the -axis dispersion is very small and, hence, the loop slope (of the remanence curve) entering in (1) is mainly controlled by the anisotropy field dispersion. Conis very similar for oriented and sequently, the value isotropic media and using the correct loop slope does not resolve the problem. III. SLOPE MODEL MODIFICATIONS For the general vectorial case, the slope condition has to be written as (2)
Manuscript received October 16, 2003. This paper was presented at The Ninth Joint Magnetism and Magnetic Materials–International Magnetics Conference, Anaheim, CA, January 5–9, 2004. See IEEE Trans. Magn., vol. 40, July 2004, Part II. H. J. Richter and E. Girt are with Seagate Recording Media, Fremont, CA 94538 USA (e-mail:
[email protected];
[email protected]). G. M. Sandler is with Seagate Technology, Longmont, CO 80503 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TMAG.2004.829292
where denotes the perpendicular components. In (2), the gradient of the perpendicular demagnetizing field has been neglected. The last term considers the magnetization change due to a change of the perpendicular component of the head field and, thus, considers the variation of the head field direction at the writing point. (It is noted that the formulation used in [4], [5] is erroneous because it misses the feedback of the
0018-9464/04$20.00 © 2004 IEEE
3546
IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 6, NOVEMBER 2004
=
=
Fig. 1. Sketch of the recording geometry. x down-track, y transverse; z perpendicular direction. The dashed line is the (mean) easy axis.
=
Fig. 2. Magnetization along the x direction for various angles '. The center of the transition is at x x . In a typical recording system, a 0.1g (g record gap length). The inset illustrates reversible and irreversible switching at the hysteresis loop.
=
perpendicular head field on the magnetization slope). The last term can be identified with
(3)
=
Equation (2) implies that the magnitude of magnetization is identical in either state, switched or nonswitched. It can be shown that the magnetization projection along the axis is
where is the switching field. Using Fig. 1, the angle between the easy axis and head field can be derived (7) (4) is the angle between the head field and the axis where and is the angle between the easy axis and the axis. In the is defined as the location following, the freezing location furthest downstream where the (total) head field exceeds the switching field. If each grain switches according to the model of Stoner–Wohlfarth [7], the angle dependencies of the head and to increase monotonically with inthe switching field cause creasing . For a 2-D-isotropic medium, the center of the transi. To simplify the further analysis, tion corresponds to the magnetic behavior of the entire medium is approximated by . The distribution of axes is that of one grain oriented at assumed to follow the modified Gaussian distribution suggested by Bertram [8]
where is the Stoner–Wohlfarth equilibrium angle in the plane formed by the easy axis and the field direction. Equation (7) is as a function of the down-track plotted in Fig. 2 for various coordinate . As can be seen the magnetization change 2 is less than 2 and, hence, all terms containing have . Fig. 2 neglects both magnetostatic and to be multiplied by exchange interaction effects. For a simplified model, it is reasonby the medium squareness able to approximate the average ) so that interaction effects are taken into consid( eration at the expense of catching all the details of the reversible magnetization changes. A simple heuristic interpretation for the multiplication of the total field gradient with is that some portion of the field gradient is “wasted” for reversible magnetization rotations rather than for irreversible switching of the grains. An approximation for the medium squareness is obtained from
(5) (8) is the angle between the particles’ easy axis and the where preferred axis, and is an orientation parameter ( correto perfectly oriented media). sponds to isotropic and It is straightforward to find the required values . Assuming Stoner–Wohlfarth switching, the last term is
(6) Using (4) and (6), it is readily found that the last term in (2) can be neglected. This confirms a finding reported in [9], where good agreement between micromagnetics and a modified slope model was obtained if the term containing was , (6) omitted. With suitable variable permutations and can also be used for the analysis of oriented perpendicular media.
is the medium squareness as it would occur if there where is the number of nearest were no magnetic interaction, where is the intergranular exneighbors, change constant, the anisotropy constant and the grain diameter, , and is the demagnetizing factor of the (cylindrical assumed) grains perpendicular to the film plane. Equation (8) uses again the concept of the representative angle as well as the approximation that the magnetization depends on for small field . the applied field as IV. RESULTS AND CONCLUSION A. Comparison With Micromagnetic Data This comparison is carried out using the results given in [2]. Using the parameters given in [2], the orientation parameter was adjusted to match the orientation ratio for the remanent
RICHTER et al.: SIMPLE SLOPE MODEL FOR ORIENTED AND ISOTROPIC LONGITUDINAL MEDIA
magnetization to 1.5. For both cases (with and without exchange), the transition parameter for the isotropic medium is found to be 6% larger in the present modeling, which agrees well with the result of 7% reported in [2].
3547
TABLE I MEASURED AND MODEL PARAMETERS
B. Comparison With Measured Data To verify the theory experimentally, three different samples orientation ratios of 1, 1.3, and 1.6 were prepared. with The orientation ratio of the media was varied by different oxidation of the seedlayers; all other properties of the media (alloys, etc.) were kept identical. The only exception was the thickness constant. of the magnetic layer, which was varied to keep Table I summarizes some properties of the media. For the modand the remanent coercivity at eling, the orientation ratio, 1 ms were matched by adjusting the distribution function pa. rameter , the medium thickness and the anisotropy field Only very minor changes of the anisotropy field were required to match the data. All other parameters were kept constant. With the transition parameter from the slope model and the cross-track correlation length obtained using the equations given in [5], the transition jitter can be determined. To compare the theory with the data, the equalized signal-to-noise ratio (SNR) definition from [10] was used, which uses a pseudorandom sequence. The determination of the medium SNR (SMNR) requires a background subtraction. If this background subtraction is accomplished by lifting the head, the resulting SMNR number is smaller than the “true” transition SMNR (SMNR ) because the noise contains the transition noise and the dc noise, which is located in the bits. If the SMNR is obtained by subtracting the noise associated with a dc erased track, the dc noise is subtracted in the bits and at the transitions and the obtained number SMNR is greater than SMNR (because the dc noise is subtracted twice for the transitions). Since the average transition spacing is 2 in a pseudorandom sequence, the dc-erased fraction of the track is estimated to 2 , where is used as an estimate for the be 1 transition width. Hence, the true SMNR can be estimated from the two measurements SMNR (9) SMNR SMNR SMNR For the samples investigated here, the two values for SMNR and SMNR differ by about 1.5–2.5 dB at a density of 23.6 kfc/mm (600 kfci) so that (9) does not introduce significant corrections. Fig. 3 shows the SMNR for both the experimental data as well as the theoretical data for the three samples. Note that the theoretical value for the transition parameter is needed to obtain SMNR . Fig. 3 demonstrates that the orientation effects on the transition sharpness and SMNR are reasonably well described with the simplified model.
Measured orientation ratio (M =M ), remanent coercivity at 1 ms and M . The medium thickness and the anisotropy fields were adjusted to match M and H . All other parameters are common: head medium separation
=
17 nm, grain size 9.5 nm, grain size distribution =D = 0:25, anisotropy distribution H =H = 0:12, M = 250 kA=m, intergranular exchange A = 0:05 2 10 J=m, gap length 120 nm, reader width = 250 nm.
Fig. 3. Comparison of experimental and theoretical SMNR (transition contribution only) for media with different orientation ratio OR.
REFERENCES [1] A. Anjan and I. Okamoto, “Crystallographic orientation of Cr in longitudinal recording media and its relation to magnetic anisotropy,” Appl. Phys. Lett., vol. 81, pp. 1465–1467, Aug. 2002. [2] H. Zhou and H. N. Bertram, “Micromagnetic analysis of anisotropy orientation in high density longitudinal magnetic recording,” J. Appl. Phys., vol. 87, pp. 5675–5677, May 2000. [3] M. Plumer and J. van Ek, “Micromagnetic study of track edge and medium orientation effects in high areal density recording,” IEEE Trans. Magn., vol. 36, pp. 2225–2231, Sept. 2000. [4] H. J. Richter, “An approach to recording on tilted media,” IEEE Trans. Magn., vol. 29, pp. 2258–2265, Sept. 1993. [5] H. J. Richter, E. C. Champion, and Q. Peng, “Theoretical analysis of longitudinal and perpendicular recording potential,” IEEE Trans. Magn., vol. 39, pp. 697–703, Mar. 2003. [6] M. L. Williams and R. L. Comstock, “An analytical model of the write process in digital magnetic recording,” in Proc. 17th Annu. AIP Conf., vol. 5, 1972, pp. 738–742. [7] E. C. Stoner and E. P. Wohlfarth, “A mechanism of magnetic hysteresis in heterogeneous alloys,” Phil. Trans. R. Soc. A, vol. 240, pp. 599–642, May 1948. [8] H. N. Bertram, “Geometric effects in the magnetic recording process,” IEEE Trans. Magn., vol. MAG-20, pp. 468–478, Nov. 1984. [9] L. Zhong, Y. Zhou, J. Giusti, and J. Fernandez-de-Castro, “Head field angle-dependent writing in longitudinal recording,” IEEE Trans. Magn., vol. 39, pp. 1851–1854, May 2003. [10] N. H. Yeh, D. Wachenschwanz, and L. Mei, “Optimal head design and characterization from a media perspective,” IEEE Trans. Magn., vol. 35, pp. 776–781, Mar. 1999.
