Grain Flipping Probability Model

Solving the LLG equation numerically in a micromagnetic channel model gives an accurate prediction of the magnetization phenomenon associated with the writing and reading of information bits. The reproduction of read-back signals from this process is however slow and computationally intensive. The GFP channel model aims to reproduce read-back signals via statistical means instead to ensure a significant amount of time savings and therefore practicality.

The GFP channel model consists of a multi-dimensional LUT of probabilities that represent the likelihood of a grain flipping given a set of circumstances per bit interval. The areal density of the characterized GFP model is determined by the parameters chosen in the micromagnetic simulations that are used in populating the GFP’s LUT in the training phase. In the GFP model, a region of interest (ROI) is first defined for each nominal bit location. The ROI is selected such that it is wide enough to capture grains within the vicinity of the magnetic head field that have

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a non-zero probability of flipping in a given bit interval. Every grain in the ROI has a probability of flipping conditioned on four parameters.

The first parameter for inclusion in the LUT is the anisotropy constant, Hk, of the magnetic grains. The second and third parameters are the position of the center of the grain, (x,y), relative to the write head. The last parameter involves the surrounding bit patterns. The surrounding bit patterns are defined as the current bit, the previous bit written on the current track and two other bits written on the previous track. This is illustrated in Figure 3.5. The current bit’s magnetization largely influences the flipping of grains in the ROI. The remaining bit patterns determine the magnetization of grains within the vicinity of the ROI and affect the likelihood of grains in the ROI flipping through exchange coupling and demagnetization field. In order for the GFP model to provide reliable statistics, these four parameters of the LUT need to be filled with sufficient data.

Figure 3.5 Inclusion of surrounding bit patterns in the GFP LUT.

The LUT of the GFP channel model is stored as two four-dimensional (4D) arrays, known as the numerator and denominator array. The denominator array counts the number of grains that have the capability to flip at every index position in the same bit interval and under the same set of circumstances. This is essentially a count of the grains within the ROI that are opposite in magnetization to the applied field. The numerator array, on the other hand, counts the number of grains that flipped under the influence of the magnetic field (as well as thermal field for a HAMR system) at every index position in a given bit interval. The grain flipping probabilities at each position on the Voronoi media can be obtained from the division of the elements of the numerator array by that of the denominator array.

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As it is not possible to plot a figure of 4D arrays, the numerator and denominator arrays can be represented as a lower dimensional probability tables as a function of fewer parameters. For example, the arrays can be summed over the bit pattern and (x,y) bins, and the resulting arrays divided to arrive at a grain flipping probability as a function of Hk. A one-dimensional (1D) plot of the grain flipping probability as a function of Hk is shown in Figure 3.6. Similarly, the arrays can be summed over the dimensions of Hk and bit pattern, resulting in a grain flipping probability as a function of (x,y). This gives a probability distribution as a function of the grain position within the ROI. Figure 3.7 shows the numerator and denominator arrays in this case and the resulting probability footprint. The probability footprint takes on the shape of the head field with grains in the middle region having a higher tendency to flip than those further out.

The footprint of Figure 3.7(c) was obtained with Mason William’s head field for a HAMR channel (Figure 3.2(a)). It can be noticed that there is a slight reduction in the probability of grains flipping in the center of the footprint. This phenomenon is observed because the angle of the applied field is coming down almost vertically near the center of the footprint, while it is coming down at an angle towards the edge. Grains that experience fields 180° to their easy axis, have a harder time flipping than grains that experience fields at a slight angle. This phenomenon is captured in the micromagnetic simulations and characterized in the GFP LUT.

Figure 3.6 Grain flipping probability as a function of Hk.

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(b)

(c)

Figure 3.7: A (a) denominator array and a (b) numerator array that has been summed over the dimensions of Hk and bit patterns, resulting in a (c) GFP footprint as a function of (x,y).

With a characterized LUT, a random number generator can be used to flip grains in a short interval of time. Table 3.1 compares the computation time required between a micromagnetic and GFP model, for a SMR and HAMR channel under a fixed set of circumstances. A time savings of approximately 1400 times and 2000 times is reaped for the HAMR and SMR channel respectively with the use of the GFP model. In Table 3.1, the number of bits written per frame varies from 15 to 45. In actual error performance studies, the number of bits written typically

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range between 512 to 4096 bits per frame, and the number of frames simulated vary from 100 to 50 000.

The ability of the GFP model to mass produce grain magnetizations and therefore read-back signals accurately and in a reasonable amount of time makes it a good model for error performance studies of a fully integrated system. The remaining sections of this chapter present a novel system level study of SMR and HAMR. In these studies, the proposed GFP model is applied to each channel to generate reliable read-back waveforms that are processed with conventional signal processing techniques.

Table 3.1: Comparison of computation time for a micromagnetic and GFP channel model Channel No. of bits written Time taken (s)

Micromagnetic GFP

HAMR 1 x 15 1561 1.130

1 x 25 2532 1.776

SMR 3 x 15 5841 3.777

3 x 25 8601 4.238