Chapter 5. Noise in Thin-Film Disk Media Introduction. V rms. Frequency (MHz)

Transcription

1 5.1. Introduction Noise in continuous thin-film disk media is mainly due to the random fluctuations in the micro and macroscopic properties of the magnetic layer. Figure 5.1 illustrates the signal and noise spectra as measured with a spectrum analyser for a typical thin-film disk with coercivity 000 Oe ( kA/m), M r δ product of.1 memu/cm (0.01A) and coercive squareness, S *, of (µv) 0.04 (d) V rms 0.0 (c) (e) (b) 0 (a) Frequency (MHz) Figure 5. 1 Signal and noise spectra. (a) Electronic + head noise, (b) DC erase noise, (c) signal and transition noise at 47KFCI, (d) signal and transition noise at 94KFCI and (e) modulation noise shoulder. 146

2 The signal was replayed with a thin-film head of gap length 0.18µm, head-to-medium separation 0.041µm, velocity of 1.81 m/s, track width 4µm and pole length 3.5µm (symmetrical). The undulations in the spectrum are the consequence of the finite pole length of the replay head and their periodicity confirms the value of the pole length. The portion of the spectrum shown by Figure 5.1(a) represents the noise due to the channel electronics and the head. It was measured by unloading the head from the disk and measuring the spectrum to isolate this type of noise from that associated with the medium. The head and electronics noise is usually considered to be stationary white noise with Gaussian distribution (Tang, 1985; Carley and Moon, 1987; Williams, 199) and can be minimised with suitable averaging. Nevertheless, its influence must be accounted for at high packing densities as it affects the effective signal-to-noise ratio of the recording system (Arnoldussen, 1997). The DC erase noise spectrum, curve (b), was measured by saturating a track in one direction with a high write DC current and measuring the residual noise using a spectrum analyser in a particular bandwidth. The mechanism of this type of noise was taken to be the fluctuations in the head-to-medium separation yielding fluctuations in the recording field and, via the magnetisation curve, fluctuations in the medium magnetisation (Bertram, Hallamasek and Madrid, 1986). Coercivity fluctuations were also considered to be another source of noise (Bertram et al, 1986; Katti et al, 1988) causing fluctuations in medium magnetisation through the applied field. This effect is magnified in media with high squarenesses where small variations in the applied field produce minute transitions giving rise to demagnetising fields which induce a signal in the replay head. Eperimental evidence has indicated that surface roughness can also give rise to noise (Ogawa and Ogawa, 1979; Bertram et al, 1986). Reduction of this type of noise can be achieved with proper fabrication techniques to reduce the surface roughness and by using media with high orientation ratio (ratio of M r in the circumferential direction divided by M r in the radial direction) (Ogawa and Ogawa, 1979). Saturating a recorded track in one sense with a high constant write current and then writing in the opposite sense with small DC current increments and measuring the 147

3 noise spectrum, yields a distinct and reproducible peak in the integrated noise power. The reverse DC current value that gives the maimum noise corresponds to a write field value that approimately equals to the medium s remanance coercivity (Bertram et al, 1986; Aoi et al, 1986; Tarnopolsky et al, 1989). At this field value, the medium is at a state corresponding to M r = 0 on the major hysteresis loop. At this condition, large irregular magnetic domains are generated through the magnetic interactions between crystallites giving rise to a noise maimum (Aoi et al, 1986; Tarnopolsky et al, 1989). The reverse current that gives the maimum DC erased noise provides an in situ, non-destructive technique for characterising and monitoring media properties (Tarnopolsky et al, 1989). Comparison of the maimum reverse erased noise power spectra with maimum transition noise spectra (giving the maimum integrated transition noise power) for different media, showed a strong linear correlation between the two indicating that both noise sources are generated by related mechanisms (Aoi et al, 1986). Micromagnetic simulations along with simplified modelling have shown that the total reverse DC erase noise is directly proportional to the field derivative of the remanance hysteresis loop (dm(h)/dh) (Zhu, 199). This dependence implies that the dc reverse erase noise is additive in nature (Arnoldussen, 199). Small modulation noise is also present in the recorded square wave pattern manifested in the form of sidebands around the write frequency (shoulder in Figure 5.1(f)) (Belk et al, 1986). This is caused by the spatial fluctuations in the macroscopic parameters of the head and medium such as variations in flying height and in the local coercivity (Arnoldussen, 199). A wide range of eperimental and theoretical techniques (Yuan and Bertram, 199; Bertram and Che, 1993; Tang, 1985; Belk et al, 1985) have identified two major modes of transition noise; namely transition position jitter and variation in the width parameter. For modelling purposes, the magnetisation transition is usually assumed to a have a fied functional form (e.g. arctangent, tanh) and to be uniquely determined by two variables: the transition centre position and the width parameter. Transition noise is then modelled by allowing the variation of these two particular variables. 148

4 When writing magnetisation reversals at large bit separations on a DC saturated track, the measured spectrum includes the contribution of both quiet (DC saturated) and noisy transition regions. Increasing the packing density increases the proportion of the transition regions in the written track compared to the DC saturated regions and gives the voltage spectrum (c) in Figure 5.1, after subtracting the electronic and head noise. A doubling of the linear density increases the voltage spectrum to curve (d) as shown in Figure 5.1. Further increase in the packing density will essentially increase the voltage spectrum. Thus it was concluded that the noise is concentrated in the transition regions and the increase of the noise voltage spectrum with reduced bit spacing is due to the increase of the fluctuations in the transition regions where the transition free zones are reduced to a minimum (Baugh et al, 1983; Belk et al, 1985). This type of noise, known as transition noise, is the dominant source of medium noise in thin-film disks as illustrated in Figure 5.1. The physical basis of transition noise is the irregular zig-zag domain structures (Belk et al, 1985). Transition noise is an additive, non-stationary noise process (Tang, 1985; Bertram and Che, 1993). By additive it is meant that the noise is functionally unrelated to the signal, but yet its mean square value (variance) is related to the signal or its derivative (Arnoldussen, 199). Non-stationarity indicates that the noise varies with time and is different for different locations on the recorded track (Murdock, 199). This chapter will concentrate on the two transition noise modes; position jitter and width variation. Simplified epressions for the magnetisation fluctuation will be developed and the corresponding noise voltages will be determined. Epressions for the noise power spectral densities will then be evaluated by transforming the noise voltage epressions to the Fourier domain. The noise autocorrelation function and the total noise power as a function of packing density will then be derived. A Williams and Comstock (1971) type theory will be used to eplain the behaviour of the broadband noise power through the influence of the interaction fields. Finally, the subject of signal-to-media noise ratios will be studied. The following analyses are developed mainly for comparison with the spectral measurements in Chapter 7. The spectrum analyser essentially measures the time- 149

5 averaged signal power in a narrow bandwidth centred on a frequency that is swept over the range of interest (Murdock, 199). Since the recording disk rotates rapidly past the head, the spectrum analyser produces the average noise power of thousands of transitions and the magnetically saturated regions between transitions. This introduces another form of non-stationarity in the measured spectra manifested in the differing noise statistics between the noisy transition regions and the saturated regions. Therefore, for the spectral measurements to be meaningful (characterise transition noise) and to allow fittings to the theoretical models developed in this chapter (which ignore the noise from the saturated inter-transition regions), the recorded track must be filled with transitions and, ideally, written at a frequency which produces the maimum noise in the spectrum (Belk et al, 1985; Murdock, 199). In this case, the measured noise is the average non-stationary transition noise after averaging out all the stationary noise components. Time based noise measurements of isolated transitions (Xing and Bertram, 1997; Carley and Moon, 1987) and dibits (Ale et al, 1997) reveal the same noise statistics as that obtained from spectral measurements. The fluctuations in both position jitter and transition width are assumed to be very small compared to the separation between transitions and the transition width parameter. This allows the magnetisation and noise voltage epressions to be epanded into a first order Taylor series hence separating the noise from the original magnetisation and signal. Only the longitudinal component of transition noise is assumed present. The noise fluctuation distribution is assumed to be a Gaussian and within the width of an isolated replay pulse on either side of the transition centre. In the following tet, the power of a signal or a noise voltage is taken to refer to that dissipated into a 1Ω resistor, so for any voltage signal v(), the instantaneous power is: P = v() (V ) Definitions of Statistical Operators Noise is a random process, thus is it customary to use the epectation or variance of these random variables. The details of these operators can be found in Clarke and 150

6 Cooke (1980). A summary of the basic definitions of these operators will be given here. The mean or epected value, µ, of the continuous random variable X which has probability density function f() is: N µ = E[X] = f () d 1 E[X] = i N i= 1 The discrete version of the epectation operator for equal probability is shown in the brackets. The mean or epectation of two independent (uncorrelated) random variables X and Y is: E [XY] = E[X]E[Y] and if either of the variables has a zero mean, then the above reduces to zero. The variance,, of the continuous random variable X which has probability density function f() and mean µ is (discrete version for equal probability is shown in the brackets): N var[x] = = ( µ ) f () d 1 var[x] = ( i µ ) N i= 1 which can be written in terms of the epectation operator as: ( E[X ) = E[X ] ] where it can be seen that the variance of a zero mean random variable is simply the mean square value of the random variable: = E[X ] The standard deviation,, is the positive square root of the variance. The variance of the sum of a finite set of independent random variables is: var[a ± B ± C ± ê ] = var[a] + var[b] + var[c] +ê The correlation coefficient between two random variables X and Y is: ρ = E[(X E[X])(Y E[Y])] E[(X E[X]) ]E[(Y E[Y]) and is used to measure the strength of the linearity of the relationship between the two random variables. The numerator is the covariance between the random variables X and Y, and the denominator represents the square root of the product of the variance of 151 ]

7 the two random variables. If the two random variables have zero mean, then the above becomes: ρ = E[X E[XY] ]E[Y ] = E[XY] X Y Since transition noise is concentrated in the transition region, then the probability density function can be approimated by a Gaussian (Madrid and Wood, 1986, Arnoldussen, 199) with probability density function: in which: f () = 1 π ( µ) e E [X] = 0 and var[ X] = E[X ] = 5.. Modes of Transition Noise The cross-track average longitudinal written magnetisation transition for any of the functional shapes listed in Table 3.1, can be written in a normalised form as M (( o ) / a) where o defines the location of the transition centre position and a is the cross-track average transition width. Allowing variations in transition centre position and width, the noisy magnetic transition can be epressed as: o n M n() = M (5.1) a + an where n and a n are taken to be zero mean random variables representing the fluctuation in the transition position and width respectively. For small fluctuations, equation (5.1) can be epanded into a first order Taylor series giving: where: M M n () = M a o + n M (( o ) / a) M (( o ) / a) = ( ( ) / a) o (( ) / a) o + a n M (( a o ) / a) (( o ) / a) 1 M (( o ) / a) = a ( ( ) / a) o (5.) 15

8 and hence the derivative with respect to a in (5.) can be epressed as: M (( o ) / a) M (( o ) / a) ( ( o ) / a) = a ( ( o ) / a) a ( ) M (( ) / a) = Therefore, (5.) can be written in the form: a o M (( o ) / a) a n M n () = M + n ( o ) (5.3) a a Subtracting the average magnetisation M from M n in (5.3) yields the noise in the magnetisation transition. Squaring and taking the epectation yields the magnetisation noise variance variance, o M in terms of the average magnetisation gradient, the position jitter, and the transition width fluctuation variance, (( ) / a) o a, i.e.: M o ( o ) M = + a (5.4) a Figure 5. illustrates an arctangent transition along with the standard deviation (positive square root of variance) of the magnetisation position ( 0 ) and width ( = 0) variances evaluated using (5.4) for o = 0. a = By taking the Fourier transform of (5.3), it can be shown that for o = 0: a n M n () M km (k) j n (5.5) a a Equation (5.5) indicates that the two magnetisation transition noise modes are orthogonal having the same Fourier transform (km (k)) and are out of phase by π/ scaled only by n and a n. This is certainly true for any transition shape. Consequently, the transition position jitter and the width fluctuation noise distributions illustrated in Figure 5. are therefore a Hilbert transform pair. In the following tet, the values of and a will be eaggerated for demonstration purposes. 153

9 1.0 M / M r Position jitter Arbitrary Magnitude Width varation a = 0.1µm = a/ a = a/ /a Figure 5. Arctangent transition with the corresponding transition noise modes. If the transition position variance distribution in Figure 5. was added to the original magnetisation transition, a transition with the centre shifted from zero will be produced. Thus the position jitter variance can be considered to represent, physically, the difference between one transition and another perturbed by distance n. This in fact was the basic idea behind the shifted transition model proposed by Belk et al (1985) to eplain the transition noise phenomenon in disk media. By adding the transition width fluctuation distribution to the average arctangent transition, on the other hand, the slope of the modified transition will increase yielding a narrower transition. To determine the noise voltage profile for the two transition noise modes, the gradient of the noisy magnetisation transition must be found. Differentiating (5.3) with respect to yields: M n () M = + M (( ) / a) a M (( ) / a) (( ) / a) o o a n n a a n ( o o ) (5.6) 154

10 Subtracting the average magnetisation gradient from the noisy magnetisation gradient of (5.6), squaring and evaluating the epectation gives the variance of the total magnetisation gradient noise as: dm = a M ( ) ( ) / a M (( ) / a) a o + o + a o a (5.7) Considering initially an ideal lossless replay head where the head sensitivity function can be represented by a Delta function; the replay voltage in this case, via reciprocity, will be directly proportional to the magnetisation gradient. Accordingly, the noise voltage will be proportional to the difference between the average and noisy magnetisation gradients. From (5.6), the Fourier transform of the noise voltage was found to be: M n () M ( / a) εk n km a (k) + j a n M dm (k) (k) + k dk (5.8) where ε is a suitable constant. Equation (5.8) indicates that the two transition noise modes are no longer orthogonal and the corresponding noise voltage spectra for both modes will be different and dependent on the shape of the written transition. The eception only occurs when the transition is described by a Fourier transform of the form: ±ck e M(k) α k c constant as eemplified by an arctangent transition (Table 3.1), in which case the noise voltage components will be orthogonal and their spectra will be identical as indicated in Fung et al (1997). It can be seen from (5.8), however, that the transition width noise voltage transform (imaginary part) can effectively be identified as the derivative with respect to k of the noise voltage transform due to position jitter. For a semi-infinite gapped head, the isolated replay pulse for an arctangent transition computed using (3.8) is shown in Figure 5.3 along with the noise voltage due to a = position jitter ( 0 ) and transition width variation ( 0 ) evaluated using (5.6). = 155

11 Normalised Voltage Replay pulse Position jitter Width variation a = 0.1µm a n = a/ n = a/ g = 0.µm d = a δ = g/ Figure 5. 3 Isolated replay pulse due to an arctangent transition and the noise voltages due to position jitter and transition width variation. /g The position jitter noise voltage represents the difference between two displaced isolated pulses (Belk et al, 1985). Adding the width variation noise voltage to the replay pulse in Figure 5.3 yields a narrower isolated pulse with slightly higher amplitude as epected due to the reduction of the transition width (Figure 5.). Fluctuations in the transition width parameter causes broadening of the isolated replay pulse according to (3.6). Since the area underneath an isolated pulse is constant, variation in the transition width also causes fluctuations in the amplitude of the replay pulse. The position jitter and width variation noise voltage distributions (or their sum) illustrated in Figure 5.3 were identified by a number of workers using different eperimental methods. Yuan and Bertram (199) applied a technique of decomposing the standard deviation of the total noise voltage of eperimentally captured isolated pulses into weighted components of the main noise sources. The dominant noise sources, with the highest weights compared with the other noise components, were the position jitter and width variation having similar profiles as the noise voltages shown in Figure 5.3. The driving force of that technique was the assumption that the 156

12 dominant noise sources in the transition are orthogonal. In addition to noise decomposition techniques (Yuan and Bertram, 199), theoretical noise autocorrelation analyses (Slutsky and Bertram, 1994) have shown that the contributed weight of position jitter noise was always greater than that due to transition width noise and this contribution increases with reduced transition width. The scale of the contribution of each noise mode, however, was found to have a strong dependence on the head field parameters such as the read gap length and the flying height (Slutsky and Bertram, 1994). Another important observation was that the transition position and width fluctuations were found to be statistically uncorrelated (Slutsky and Bertram, 1994) as found from temporal noise measurements (Carley and Moon, 1987). The spatial noise distribution similar to that shown in Figure 5.3 was also observed from MFM images of recorded dibits (Ale, Scott and Arnoldussen, 1997). Eperimental and theoretical signal noise autocorrelation calculations also yielded the same noise distributions for the two modes of transition noise shown in Figure 5.3 (Tang, 1985; Bertram and Che, 1993; Xing and Bertram, 1997). Fitting to spectral noise measurements for a number of thin-film disk media has indicated that the position jitter variance, the width fluctuation variance and the resultant transition noise all decrease with decreasing M r δ and increasing H c (Xing and Bertram, 1997) Noise Power Spectral Density (PSD) In order to perform comparisons and fittings to spectral noise measurements obtained using a spectrum analyser, the noise power spectral density (PSD) must be determined. This is achieved by first deriving an epression for the Fourier transform of the noise voltage. As described earlier, the noise voltage can be obtained by substituting the transition noise derivative (given by subtracting (5.6) from the average transition gradient) into the reciprocity formula of equation (3.30). Taking the Fourier transform of the correlation integral and integrating through the medium thickness, it can be shown using equation (5.8) that the Fourier transform of the noise voltage is given by: 157

13 kδ * 1 e E n (k) = o s kδ (5.9) a n a n dm (k) M (k) nk jk jk a a dk kd ( µ vwnηδ) [ H (k)] [ e ] The first three terms in square brackets are the gap-loss, spacing loss and thickness loss terms respectively. The last term in square brackets is the spectral transition noise term containing the contributions of position jitter and transition width fluctuation. The derivative of the magnetisation transform with respect to k in the last square bracket in (5.9) can be written in terms of the magnetisation transform times a transition dependent term, i.e.: dm (k) = M (k) F(k, a) (5.10) dk where F(k,a) is a general function dependent on the transition shape and is defined in Table 5.1 for the arctangent, tanh and error function magnetisation transitions (for positive transition transforms). The Fourier transform of the noise voltage can therefore be written in terms of the Fourier transform of an isolated replay pulse times the contribution of the transition noise terms: a n E n (k) = E (k) j nk + ( 1+ kf(k, a) ) (5.11) a where E (k) is the Fourier transform of the isolated replay signal as given by equation (3.33). Transition Arctangent Tanh Error Function F(k,a) 1+ ka k π a π ka ctnh πk a 1 + k Table 5. 1 Transition dependent function used for the evaluation of the Fourier transform of the noise voltage. 158

14 The one-sided power spectral density (the average power measured by a spectrum analyser or equivalently the variance of the noise voltage transform (Thurlings, 1980)) of an isolated transition, neglecting correlations between adjacent transitions, is given by (Belk et al, 1985): { E[Re E (k) ] E[Im E (k) ]} Pn (k) = n + n (5.1) where E[ ] is the epectation operator. Substituting (5.11) into (5.1) yields the power spectral density of the transition noise as: 1+ kf(k, a) P n (k) = P (k) k + a (V ) (5.13) a where P (k) is the isolated replay pulse power spectral density (i.e. (k) E (k) P = ), and a are the transition centre jitter and width fluctuation variances. The root mean square (rms) noise voltage spectrum as displayed on a spectrum analyser is found by evaluating the positive square root of the power spectral density (Belk et al, 1985), i.e.: n E rms (k) = Pn (k) (V) (5.14) Figure 5.4 illustrates the noise power for a tanh transition normalised by twice the signal power as computed from (5.13). P n (k)/ P (k) Position jitter Width variation Position + Width a = 0.1µm = a = a/ Wavenumber, k (10 6 m -1 ) (a) 159

15 P n (k) / P (k) Position jitter Width variation Position + Width a = 0.1µm = a/ a = a/ Wavenumber, k (10 6 m -1 ) (b) Figure 5. 4 Position jitter and width fluctuation noise power spectra for a tanh transition for (a) equal and (b) unequal contributions of noise modes. From Figures 5.4(a) and 5.4(b) and equation (5.13), it can be seen that the position jitter noise power spectrum increases quadratically with wavelength, while the transition width fluctuation noise spectrum increases at a higher rate. For an error function transition, the noise PSD, using Table 5.1, is given by: P (k) = P n (k) π a 4 where it can be seen that the transition width noise has a fourth order dependence on wavenumber. Therefore, for the error function magnetisation transition (and similarly for the tanh function), the transition width noise increases at a fourth order rate compared to the second order increase in transition jitter noise PSD (in agreement with the approimations of Xing and Bertram (1997)). At long wavelengths, position jitter noise dominates the spectrum. At short wavelengths, the effect of width variation becomes more prominent and clearly dominates the total noise power spectrum in the case where the contribution of both noise modes is equal (Figure 5.4(a)). This is also true at short wavelengths even in the case where the contribution of transition width noise is less than that due to position jitter as indicated in Figure 5.4(b). Therefore, it can be concluded that limiting the noise theory to position jitter will not be sufficient 160 k + a k 4

16 to predict accurately the eperimental noise spectral shape and the contribution of transition width noise must be included. Figure 5.5 illustrates the position jitter, width variation and total noise power spectral densities for a tanh transition, including head losses, assuming equal weights of position and width fluctuations. P n (k) / C Position jitter Width variation Position + Width a = d = 0.1µm g = 0.µm δ = g/6 = a = a/ Wavenumber, k (10 6 m -1 ) Figure 5. 5 Normalised noise PSD for a tanh transition where C = µ o vwn ηm δ. From Figure 5.5, it can be seen that the position jitter noise power spectral density dominates the long wavelength response of the total noise PSD. The transition width noise spectrum, on the other hand, determines the short wavelength noise response. The magnitude of each noise spectrum will be scaled by variance of the corresponding jitter variable. As a result, the measured noise PSD can give an indication of the dominant noise mode; an increase in the long wavelength noise at the epense of the short wavelength noise with increased linear density would imply that position jitter is the largest contributor to the total noise in the medium (Madrid and Wood, 1986). Attenuation of the long wavelength noise and enhancement of the short wavelength noise component, on the other hand, would imply that transition width noise is the dominant noise source in the medium. 161

17 The effect of the functional form of the written transition on the shape of the noise power spectrum is demonstrated in Figure 5.6. Shown in this figure, are the total noise PSDs (position and width fluctuation) for the arctangent, tanh and error function transitions as calculated using (5.13) P n (k) / C Arctangent Tanh Error function a = d = 0.1µm g = 0.µm δ = g/6 = a = a/ Wavenumber, k (10 6 m -1 ) Figure 5. 6 Net noise PSD for different transition distributions for equal weights of transition position jitter and width variation where C = µ o vwn ηm δ. As illustrated in Figure 5.6, the error function and tanh transitions produce noise spectra with higher amplitudes and with faster roll-offs with increased wavenumber when compared to the arctangent noise spectrum. Therefore, the characteristics of the transition noise spectrum will be influenced by the shape of the written transition. Actual recorded transitions are not fied to a single functional form such as the arctangent or error function. Inverse filtering has indicated that the actual recorded transitions are in fact asymmetrical and can be represented by a combination of an arctangent and tanh transitions (Wells, 1985; Aziz et al, 1999b). Noise measurements favour the error and tanh transition functions (Arnoldussen and Tong, 1986; Xing and Bertram, 1997; Aziz et al, 1999a). 16

18 Figure 5.7 illustrates the total noise PSDs for the arctangent and tanh transitions evaluated for different values of the transition width parameter a. P n (k) / C a = 0.08µm a = 0.06µm a = 0.04µm a = 0.01µm d = 0.1µm g = 0.µm δ = g/6 = a = d/ Wavenumber, k (10 6 m -1 ) (a) P n (k) / C a = 0.08µm a = 0.06µm a = 0.04µm a = 0.01µm d = 0.1µm g = 0.µm δ = g/6 = a = d/ Wavenumber, k (10 6 m -1 ) Figure 5. 7 Total noise PSD for (a) an arctangent and (b) a tanh transition for different transition widths. (b) 163

19 While the noise power due to the arctangent transition varies as the Fourier transform of its isolated replay pulse scaled only by the position jitter and width variances (noise modes for an arctangent transition are orthogonal with similar power spectra), the noise power due to a tanh transition varies differently with transition width as illustrated in Figure 5.7(b). For a tanh transition, reduction in transition width parameter causes a small increase in the noise power magnitude followed by a decrease in the noise PSD. In addition, the slopes of the noise power spectra for the tanh transition decrease with reduced transition width at short wavelengths, without any noticeable change at long wavelengths. This implies that transition width noise is more sensitive to the change in transition width than position jitter noise for the tanh and error function transitions. When the spectrum of the replay signal is measured with a thin-film head, the spectrum will ehibit undulations or head bumps, characteristic of the finite pole lengths of the replay head as illustrated in Figure 5.1. This can be simulated using the thin-film head gap-loss function given by Lindholm (1975) (Chapter, equation (.56)) in the noise power formula of (5.13). The result for a tanh transition is shown in Figure Thin-film head 0.16 Karlqvist head P n (k) / C a = d = 0.1µm g = 0.µm D = g/6 = a = a/ P 1 = P = 10g Wavenumber, k (10 6 m -1 ) Figure 5. 8 Effect of finite pole width on the transition noise spectrum for an a tanh transition. The power is normalised to C where C = µ o vwn ηmrδ. 164

20 Good agreement has been obtained with eperiment when using the thin-film gap loss epressions of Lindholm (1975) and Bertero et al (1993) (Aziz et al, 1999a) Integrated Noise Power At large bit spacings, the total rms noise power, obtained by integrating the area underneath the measured power spectral density, was observed to increase linearly with packing density (Baugh et al, 1983; Belk et al, 1985). This is a consequence of the nature of the spectrum analyser measurement which produces the average noise power of the transition regions along with the quiet regions. The linear increase of the average transition power with packing density is due to the increase of the transition regions in proportion to the quiet regions. This linear dependency continues up to a bit spacing λ / πa representing the onset of overlapping between transitions. Beyond o that point, the integrated noise power value starts to increase at a supralinear rate with linear density reaching a peak value where the track is filled with overlapping transitions (Belk et al, 1985). The bit spacing at which the broadband noise power maimises (on-track correlation length) is thus epected to be proportional to the transition width parameter (Arnoldussen, 199). Eperimental studies showed that the on-track correlation length is related to the Williams and Comstock (1971) transition width parameter as a. With further increase in the linear density, the total noise decreases due to reduction of net moment caused by incomplete saturation and domain wall bridging effects (Arnoldussen and Tong, 1986). At etremely high densities, the field variation from the head is spatially so rapid during recording that the medium is ac erased leading to a diminishing of net moment and loss of signal and noise. Assuming that the replay signal due to a recorded square wave pattern can be constructed by the superposition of alternating isolated pulses, the noisy replay signal due to small fluctuations in position and width at each pulse can be written using (3.35) as: ( n= n ( nb,a + a ) e () = 1) e (5.15) ns n n 165

21 where e is the replay voltage of an isolated pulse, n and a n are the position jitter and transition width fluctuations in the nth transition and are considered to be random variables of zero mean with variances and a. If it is assumed that n << B and a n << a, then epanding (5.15) to a first order Taylor series and subtracting (3.35) yields the noise voltage as: n e ( nb,a) e ( nb,a) e n () = ( 1) n + a n = (5.16) n a The first term yields the noise voltage due to position jitter while the second term accounts for the contribution of the transition width noise voltage. Squaring (5.16) and taking the epectation considering only nearest neighbour noise voltage correlations yields the noise voltage variance (spatial noise power spectral density). The corresponding total noise power (written as the square of the rms noise voltage) is then given by (Belk et al, 1985): V RMS = = 1 B 1 B = E = [( e + e ) ] n 1 [ ] d + E[ e e ] E e n n B d = n n 1 d (5.17) Consider first the noise voltage due to position jitter as given by (5.16) and substituting in (5.17) yields the total position jitter noise power as (Bertram, 1994): n n 1 = e () E[ ] e () e ( B) RMS d + d V (5.18) B B = = The first integral in (5.18) effectively represents the autocorrelation function with zero displacement while the second integral is the autocorrelation function with displacement B. Hence, in general, the total noise power in (5.17) can be written in terms of the autocorrelation function R in the form (Belk et al, 1985): 1 VRMS = [ R(0) + R(B) ] (5.19) B Following the Wiener-Khinchin relations (Brigham, 1988), the noise voltage autocorrelation function is also the inverse Fourier transform of the noise power spectral density, i.e.: 166

22 Pn = 1 jkb R (B) = (k) e dk (5.0) π k where P n is the noise PSD as defined in (5.13) for position jitter and transition width variation. Since the autocorrelation function is always even and its Fourier transform is purely real and even, then (5.0) can be written in terms of the Fourier cosine transform: Pn = 1 R (B) = (k) cos(kb) dk (5.1) π k Substituting equation (5.13) for the noise power spectral density in (5.1) and evaluation of the selfcorrelation, R(0), and autocorrelation, R(B), functions yields the total broadband noise power for position jitter and transition width variation as (Belk et al, 1985; Barany and Bertram, 1987b): V RMS = πb a + πb k= k= P (k) k ( 1+ ρ cos(kb) ) 1+ kf(k, a) P (k) a dk ( 1+ ρ cos(kb) ) a dk (5.) where ρ E[ = n n 1 ] and ρ a E[a = a n n 1 a ] are the position jitter and transition width fluctuation correlation coefficients signifying the noise correlation between two adjacent transitions. These vary between 1 and +1 depending on the degree and direction of interactions between noisy transitions. The noise power spectral density for square wave recorded patterns including adjacent transition correlations is given by the integrand of (5.) since (Bertram, 1994): = 1 V RMS = PSD(k) dk (5.3) π k and the form in (5.) is in agreement with the PSD derived by Madrid and Wood (1986) and that obtained by Moon et al (1988) for a Lorentzian replay pulse. With ρ = 0 in (5.), the uncorrelated noise power spectral density is obtained (equation (5.13)) where the noise power increases linearly with packing density (Belk 167

23 et al, 1985; Baugh et al, 1983). In the presence of correlation between noise voltages, it is the second terms in (5.) that are thought to give rise to the supralinear region of the broadband noise power curve. The total noise power for an arctangent transition replayed with a Karlqvist head can be integrated eactly in (5.) to give the total noise power for a thin-film medium as (Barany and Bertram, 1987a): where V RMS C = C + i ((a + d) πb i + ρ πb + B i a 1 (a + d)[(a + d) + i a ρ (a + d) / 4)((a a [(a + d) 3B / 4 + g / 4] + d) + (B g) + (g / ) ] / 4)((a + d) + (B + g) / 4) (5.4) C = µ ovwnηm rδ. The noise variances and correlations for the two noise modes simply add in (5.4) since the position jitter and transition width fluctuation noise spectral shapes are identical for an arctangent transition. Equation (5.4) is plotted in Figure 5.9 assuming negative correlations to demonstrate the supralinear increase in the total noise power / C V RMS a = d = 0.1µm g = 0.µm = a = a/ ρ = ρ a = /B (KFCI) Figure 5. 9 Total noise power for an arctangent transition assuming negative noise correlations for both noise modes. 168

24 With the assumption of negative correlations, equation (5.4) fails to predict the packing density at which the onset of the anomalous noise increase would occur (for a = 0.1µm in Figure 5.9, the onset of the supralinear curve should be at 1 / B 1/ πa = 8 KFCI ) and the bit spacing at which the total noise power maimise ( 1 / B 1/ a = 18 KFCI ). For zero correlation between adjacent noise voltages, the total rms power increases linearly with packing density as indicated in Figure 5.9. By considering position jitter only, the supralinear increase of noise beyond λ / πa was attributed to the interference between zig-zag fluctuations causing negative correlation between adjacent transition noise voltages (Belk et al, 1985). In this tet, positive correlation is taken when transitions tend to jitter in opposite direction, while negative correlation occur when transitions jitter in the same direction as illustrated in Figure 5.10 in accordance with Belk et al (1985) and Madrid and Wood (1986). o 169

25 Positive correlation - jitter in opposite direction Positive correlation - jitter in opposite direction Negative correlation - jitter in sam e direction Figure Positive and negative correlation between position jitter noise voltages. Solid line is the isolated replay pulse, the dashed line is the shifted isolated pulse and the dotted line is the noise voltage. Madrid and Wood (1986) have performed a similar analysis to that of Belk et al (1985) and obtained similar epressions for the transition noise power spectral density. They eplained negative correlations by either variation of disk velocity due to head flutter, fluctuation of head-to-medium separation due to surface roughness, or by local variations in the coercivity of the medium. These effects give rise to noise that is concentrated around the square wave frequency in the measured spectrum (shoulders of the signal spikes - modulation noise). Positive correlations, on the other hand, were attributed to the demagnetisation fields of neighbouring transitions during record and relaation causing transitions to shift towards previously written transitions as 170

26 demonstrated in the discussion on non-linear interactions in Chapter 4. Based on the observation that considerable enhancement of the low frequency response of transition noise spectra at the epense of the higher frequencies with increased packing density, Madrid and Wood (1986) argued that positive, rather than negative, jitter correlations occur at higher packing densities. However, the theory of Belk et al (1985) does not accommodate this argument as positive correlations would effectively reduce the total noise power as suggested by Figure Hence, Madrid and Wood (1986) concluded that the anomalous supralinear increase of the broadband noise power is due to the increase in the intrinsic noise per transition as the transitions are packed more closely. From studying Lorentz images of the zig-zag patterns of recorded transitions at different packing densities, Arnoldussen and Tong (1986) argued that the negative correlation between noisy transitions is an improbable high energy state. Furthermore, jitter in the same direction indicated by Figure 5.10, as proposed by Belk et al (1985), causes a block shift of all bits giving rise to phase shift rather than noise. At moderate and high packing densities, Arnoldussen and Tong (1986) have observed amplitude modulation noise (introducing even harmonics in the noise spectrum) along with the positive correlation bit-shift noise (alternate transitions spaced closer and farther apart) as indicated by the Lorentz images. Their conclusion was that at high packing densities, domain bridging (percolation) occurs to lower both the wall and magnetostatic energies. This domain bridging causes an increase in the zig-zag variance (increase in the zig-zag dimensions). Because they occur in the tails of transitions, variance increases contribute strongly to the amplitude modulation noise causing the supralinear increase with bit density as can be seen from equation (5.). Further increase in the linear density reduces both signal and noise due to the ac erasure action of the record head Barany and Bertram (1987a, 1987b) Model To account for interaction fields and to remedy the limitations imposed by the negative correlations proposed by Belk et al (1985), a theoretical model was developed by Barany and Bertram (1987a, 1987b). Their theory allowed the noise variances and 171

27 correlations to change as a function of packing density through the coupling of the interaction fields between two recorded transitions. For position jitter, variation in the medium s local coercivity was proposed as the physical mechanism by which transitions are shifted from their nominal write position. While fluctuations in the medium s magnetisation in the neighbourhood of the transition centre was taken to model the change in transition width parameter. In Barany and Bertram (1987b), the two noise sources were taken to be correlated. In this tet, the two noise sources will be assumed independent to simplify the modelling. This assumption is supported by eperimental evidence of a number of workers using time series analysis (Carley and Moon, 1987; Yuan and Bertram, 199; Slutsky and Bertram, 1994). y 1 o B n n-1 M dh d s (B) d 0 d dh s (1) d H Figure Recorded dibit with fluctuations in transition centre positions. Figure 5.11 illustrates a previously recorded transition with centre at o separated a distance B from a newly written transition with centre at 1. At the centre of the newly written transition, the total fields in the system must equal the medium coercivity H c, i.e.: d s H = H (, y) H (B, y) (5.5) c 1 + where H ( 1,y) is the write field at the centre of the new transition and H d s (B, y) is the net demagnetising field (including image fields) of the previous written transition with 17

28 centre o evaluated at 1. All the fields are evaluated in the medium centre line with y = d + δ /. Transition position jitter was assumed to be caused by local variations in the medium coercivity or by variations in the head-to-medium spacing. These effects can be modelled by introducing a zero mean random field H r at the centre of each recorded transition thus causing the transitions to be shifted from their nominal write position (Barany and Bertram, 1987a). Thus by adding H r to the coercivity and epressing the position jitter in the new and previous transitions as n and n-1 as shown in Figure 5.11, then (5.5) becomes: d ( c r 1 n s n 1 n H + H ) = H ( +, y) + H (B + ( ), y) (5.6) Assuming that the jitter is much smaller than the bit separation, then epanding (5.6) to a first order Taylor series and subtracting (5.5) yields the jitter in the new transition as: n = H H ' r + H d' s d' s (, y) + H 1 n 1 (B, y) (B, y) (5.7) where H d s ' (B, y) is the net demagnetising field gradient of the previous transition evaluated at the centre of the new transition and ' (, y) is the maimum head field H 1 gradient. Squaring n and evaluating the epectation, assuming linearity n 1 ] = E[ n ] E[ = and that H r and n-1 are uncorrelated (i.e. E[ n 1Hr ] = 0 ), yields the position jitter variance as: ' E[H r ]/ H (1, y) = (5.8) d' 1+ H s (B, y) / H (, y) The numerator in (5.8) effectively represents the variance of position jitter due to the variation in the head field alone at the centre of the newly written transition and can be written using (4.4) as: r c ' 1 E[H ]y H = (5.9) Q H 173

29 To determine the position jitter correlation coefficient, both sides of (5.7) were multiplied by n-1 and the epectation was evaluated. Using the above assumptions, the transition position jitter correlation coefficient was found to be: d' H s (B, y) / H (1, y) ρ = (5.30) d' ' [1 + H (B, y) / H (, y)] s Barany and Bertram (1987a) define positive and negative correlations to be in opposite direction to that given by Belk et al (1985). Therefore, a negative sign was applied to (5.30) to maintain consistency with the definition of positive and negative correlations postulated by Belk et al (1985) and Madrid and Wood (1986) as illustrated in Figure ' 1 Assuming that the transition width parameter is not affected by the adjacent transition demagnetising field and to introduce the influence of the medium parameters in the noise statistics, the head field gradient in (5.8) and (5.30) can be epressed in terms of the demagnetising field gradient of the newly written transition. Using the differential equation of Williams and Comstock (4.8), it can be shown that: H ' 1 (, y) 1 * 1 y(1 S ) = 1 d' H (, y) Qπ s 1 a d' where s (, y) is the net demagnetising field gradient at the centre of the newly H 1 written transition. Substituting for the maimum head field gradient, the position jitter variance and correlation coefficient can be written in the form: H = (5.31) * d' y(1 S ) H + s (B, y) 1 1 d' Qπa H s (1, y) * d' y(1 S ) H s (B, y) 1 d' Qπa H s (1, y) ρ = (5.3) * d' y(1 S ) H (B, y) 1 + s 1 d' Qπa H s (1, y) From equations (5.31) and (5.3), it can be seen that the position jitter variance and correlation coefficient are functions of the ratio of the demagnetising field gradient of the previous transition to the demagnetising field gradient of the new transition. 174

30 Therefore, it is epected that the jitter variance and correlation would increase with increasing packing density as the effect of the previous transition demagnetising field becomes more significant. The transition position jitter variance and correlation coefficient for an arctangent transition evaluated using (5.31) and (5.3) are plotted in Figure 5.1(a). The corresponding integrated noise power is computed by substituting (5.31) and (5.3) into (5.4). Position jitter variance / H ρ Hr = H c / /B (KFCI) (a) Position jitter correlation coefficient 175

31 Total noise power (normalised) B = πa B = a Hr = H c / /B (KFCI) (b) Figure 5. 1 (a) Position jitter variance and correlation coefficient for an arctangent transition, (b) corresponding total noise power (solid) compared with total power calculated using only negative correlations (dashed). Recording parameters: a=0.067µm, d=0.1µm, g=0.µm, δ=g/6, S * =0.9, H c =160kA/m, Mr/H c =3 and Q=0.93. In Figure 5.1, it is assumed that the flying height is the same in recording and replay. The transition width parameter was calculated using the write limit epression of Williams and Comstock (1971) as given by (4.6). Figure 5.1(b) compares the total noise power calculated with changing variance and correlation coefficient due to interactions with that using only negative correlation as suggested by Belk et al (1985). It can be seen clearly that with negative correlations, the total power does not conform with the eperimental observations that correlate the bit spacing at which the noise power is a maimum with twice transition width parameter (Belk et al, 1985; 1986). The jitter noise variance indicated in Figure 5.1(a) is shown to increase with packing density reaching a peak at a transition spacing that approimately equals to twice the transition width parameter. This peak is also observed on the jitter correlation coefficient curve. At small packing densities, the jitter correlation coefficient is zero where the transitions are far apart. When the bit spacing approaches πa, the interaction fields cause positive correlations between transitions producing the supralinear portion of the total noise power curve. 176

32 Positive correlations can be eplained by eamining the effect of displacement of the previous transition and its demagnetising field on the newly written transition. If the previous transition was displaced in the direction of increasing (Figure 5.11), then the newly written transition will eperience a reduced value of the previous transition demagnetising field and the two transitions will appear as if they were repelled. When the previous transition jitters in the direction of reducing closer to the new transition, then the effect of its demagnetising field on the new written transition will increase and the new transition will shift closer to the previous transition as discussed in Chapter 4. Therefore, the two transitions will appear as if they attract each other. The displacements of neighbouring transitions in opposite directions give rise to the positive correlations. The variation in the transition width parameter was eplained by fluctuations in the remanent magnetisation that is concentrated at the transition centre (Barany and Bertram, 1987b). To model the change in the transition width, consider the magnetisation gradient at the newly written transition centre which from Williams and Comstock (1971) can be written, for a square hysteresis loop (i.e. ' d' d' 0 1 s r 1 + s r dm / dh ) as: = H (, y) + H (M,a, ) H (M,a, B) (5.33) Ignoring position jitter and including variation in the a parameter due to fluctuations in the remanent magnetisation, (5.33) becomes: ' d' d' 0 1 s r n n 1 s r n 1 = H (, y) + H (M + M,a + a, ) + H (M,a + a, B) (5.34) where a n and a n-1 are the variation in the transition width parameter for the new and previously written transitions and are considered to be zero mean random variables. M n is a zero mean random variable signifying the variation in the remanent magnetisation. Epanding (5.34) into a first order Taylor series assuming small fluctuations and subtracting (5.33) yields: d' d' H s (M r,a, 1)M n / M r + a n 1 H s (M r,a,b) / a a n = (5.35) d' H s (M r,a, 1) / a Squaring and taking the epectation noting that [M a ] 0 (uncorrelated) and assuming that variance gives: E[a n ] E[a n 1 ] a E n n 1 = = =, then solving for the transition width fluctuation 177