Journal of Applied Physics 61, 1580 (1987); https://doi.org/10.1063/1.338094 61, 1580

© 1987 American Institute of Physics.

Magnetization reversal, coercivity, and theprocess of thermomagnetic recording inthin films of amorphous rare earth–transitionmetal alloysCite as: Journal of Applied Physics 61, 1580 (1987); https://doi.org/10.1063/1.338094Submitted: 28 August 1986 . Accepted: 10 October 1986 . Published Online: 04 June 1998

M. Mansuripur

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Magnetization reversal, coercivity, and the process of thermomagnetic recording in thin films of amorphous rare earth-transition metal aUoys

M. Mansuripur Boston University. College of Engineering, Boston, Massachusetts 02215

(Received 28 August 1986; accepted for publication 10 October 1986)

A model is proposed for the mechanism of magnetization reversal in thin films of amorphous alloys with perpendicular magnetic anisotropy. Examples of these alloys are ThFe, GdCo, DyFe, and GdThFeCo, which are currently under investigation as storage media for erasable optical recording applications. The model exhibits the observed behavior of the media such as nucleation and growth of reverse-magnetized domains under external magnetic fields; square hysteresis loops; temperature dependence of coercivity; formation and stability of domains under conditions of thermomagnetic recording; and incomplete erasure with insufficient applied fields.

I. INTRODUCTION

The process of magnetization reversal in thin amor­phous films ofrare earth-transition metal (RE~TM) alloys is of considerable practical interest because of its role in ther­momagnetic recording. I

-5 Amorphous films of GdCo,

GdFe, ThFe, GdTbFe, DyFe, GdTbFeCo, etc., have been fabricated in recent years and their magnetic, optical, and thermal properties have been studied extensively. These films are typically deposited on glass or plastic substrates using one of several sputtering or evaporation techniques. They range in thickness from a few tens to a few thousand angstroms and are usually overcoated with some metallic or dielectric layer to protect them against the formation of un­desirable oxide layers at the surface.6 Structural and compo­sitional analyses have shown that properly fabricated films are pure, uniform, and without microcrystalline structure (no long-range order has been found down to the 15-A reso­lution limit of the diffractometers used). Magnetically, these alloys are ferrimagnets with the rare-earth and transition­metal subnetworks forming antiparallel moments perpen~ dicular to the film surface. Perpendicular magnetic anisotro­py is probably due to some sort of short-range order on the nearest-neighbor atomic scale,7 although no direct observa­tion of this structural anisotropy has been reported to date. The net magnetization M" is the difference between the mag­netizations of the RE and TM subnetworks and, when the magnetic anisotropy energy constant (Ku ) is larger than the demagnetizing energy (21l'Ms 2), the net magnetization is perpendicular to the film surface. By proper choice of com­position, it is possible to make the two subnetwork magnetiz­ations equal at some desired temperature, in which case the alloy is said to have a compensation point temperature. To date, most of the compositions studied for thermomagnetic recording/magneto-optic readout applications have a com~ pensation point in the vicinity of the ambient temperature. Although this choice results in highly stable recorded do­mains, it is by no means essential to confine attention to such a narrow range of compositions. Shieh and Kryder,8 for in­stance, have recently shown that films with compensation points well above room temperature have potential for direct overwrite.

In observations of the magnetization reversal process under polarized light it has been noted that the reversal is a nucleation and growth process. The nucleation sites are usually intrinsic "weak points" of the film which initiate the reversal process at every repetition of the experiment. Some of these nuclei are active in both directions while others ap~ pear in only one direction (i.e., reversal from up to down only). The presence or absence of a given nucleus may be a function of the history of magnetization of the sample. In general, it is possible to reduce the number of nuclei by ap­plying a large premagnetizing field. 9

,10 The nuclei that re~ main after saturation in a strong field are usually due to pinholes or other structural (as opposed to magnetic) de­fects.

The reverse field required for initiating the magnetiza­tion reversal process is known as the nucleation field. For reasons that were mentioned earlier, the value of this field is a function of the state of premagnetization. At the nuclea­tion field it is observed that the nuclei grow in time, albeit very slowly. Sometimes a nucleus stops growing and its wall appears to be stuck at one or several points; in such cases increasing the external field (usually by a small amount) allows the domain wall to overcome the trap and continue its growth. The growth at this larger field is now faster every~ where. Trapping seems to prevent the growth only when the domains are small (say, about a micron), with larger do­mains growing indefinitely until the entire sample is re­versed. The hysteresis loops are thus highly square and the coercivity is equal to the nucleation field.

Coercivity is a function of temperature and diverges at the compensation point 1 i Above the compensation point the coercivity drops continuously until it vanishes at the Cu­rie temperature. Below the compensation point, the behav­ior is not wen understood; in some samples the coercivity drops monotonically with decreasing temperature while in others it drops first and then increases; in yet other samples, it is too high to be measured.

In this paper we present a model for the magnetization reversal process that can explain the main features of the observed phenomena. The model is described in Sec. II and its predictions by computer simulation are presented in Sec.

1500 J. Appl. Phys. 61 (4). 15 February 1987 0021-8979/87/041580-08$02.40 @ 1987 American Institute of Physics 1580

III. Section III also contains a discussion of the thermomag­netic recording process. Some final remarks and conclusions are induded in Sec. IV.

II. ANALYSIS BY COMPUTER SIMULATION

In order to understand the observed behavior of thin amorphous films ofthe RE-TM alloys under various combi­nations of external field and temperature distribution, we propose the foHowing model for the magnetization reversal process. The film is considered to be a mosaic of square tiles, as in Fig. 1, with each tile having dimensions A X A X tf .

Here tf is the film thickness and A is a parameter of the model for which the choice of numerical values will be dis­cussed later. There is no physical reason for the particular tile geometry chosen here and, in fact, irregular tiles would have been preferable if computational complexity were not a concern. A more realistic and yet computationally feasible model should have hexagonal or higher-order polygonal tiles. We do not delve into this matter further since the objec­tive is merely to show the feasibility of the concept with the least complicated model and to explain the gross features of the phenomena observed in real films.

To each tile we assign a magnetization vector Ms and restrict it to be perpendicular to the film surface. Thus for the magnetization direction there are only two possibilities which will be referred to as up and down. M., then becomes a scalar that is positive when the magnetization points up and negative when it points down. The magnitude of M~s varies randomly from tile to tile according to the following distri­bution:

(1)

M.o is a function of temperature only and represents the spatial average of magnetization. x is a random variable with uniform distribution in the interval [ - 1,1] and is chosen for each tile independently. a, a parameter of the model, is a number in the interval [0,1 J which represents the extent of variations of magnetization across the film. With high-reso-

1-.1.-1 j=

T b. 1

.1 2

3

Jmax

i= 1 2 3 I max

FIG. 1. Array of square tiles with 1m .. columns and Jm .. rows. Each tile represents a A X f:t. area cfthe magnetic film surface. The film thickness (not shown here) is tf' The algorithm described in the text scans the array ill a checkerboard fashion, scanning the shaded tiles before the unshaded tiles.

1581 J. AppL Phys., Vol. 61, No.4, 15 February 1987

lution magnetometry it should be possible to obtain realistic estimates for the spatial distribution of M" the origin of which is the random structure ofthe amorphous state. In the absence of experimental observations, however, we set a = 0.25 throughout the paper. The uniform distribution of x is also chosen for convenience but if the true distribution were known i.t could be used i.nstead. The assumption that x for a given tile is independent of x for other tiles needs elabo­ration. The spatial variations of magnetization on a real sam­ple are correlated and, if the correlation length happens to be larger than the dimension A of the tiles, this dependence must be taken into consideration. Of course, one can always choose t:. large enough to eliminate such dependencies, but as will be shown later, A is not quite arbitrary and certain compromises are involved in its selection. Therefore, our as­sumption concerning independent values of x for different tiles is purely for convenience and must be avoided in a more realistic model.

The values of a, fl., and x for the array of tiles are chosen at the outset and remain unchanged throughout the compu­tations. The average saturation magnetization M sO' how~

ever, is a function of temperature that can be measured by standard methods of magnetometry. Figure 2 shows the measured values of M,{) versus temperature for a Tb21 Fe79

alloy.2 The solid curve is a mean~field match to the experi­mental data. 12 Also shown are the iron and terbium sub­network magnetizations as calculated from the mean-field model.

Next we assign a domain wall energy density (O'w) to each wall of length fl. that separates neighboring tiles. The value of 0'11.) varies from wall to waH and is given by

O'w = O'wQ (1 + /3y). (2)

(7 wQ' the average domain wall energy density, is a function of temperature and has the same value for all the walls at any given temperature. The average domain wall energy density is a function of the anisotropy energy constant Ku and the

1000 MTb

1100

~ 600 ......

:::I E C!.I

:E 400

200 M§

0 0 100 200 300 4011 500

T 0<)

FIG. 2. Circles show magnetization YS temperature as measured by VSM for a rf-sputtered Tb21Fe79 thin film sample (Ref. 2). The solid curves are obtained from a mean-field model matched to the data (Ref. 12).

M. Mansuripur 1581

1,00 5X10 ·-7 Ku (erg J em l )

OM] 0,60 i

,

0.40 106 A,(erg/cm)

10-10",. (erg I cm2 )

0.20

10' 3 "",(A)

0.00 0 100 200 300 400 5011

T (K)

FIG, 3. Anisotropy energy constant Ku. exchange stiffness coefficient A" domain-wall energy density 17 w' and domain-wall thick..'1ess Aw vs tempera­ture. The curves correspond to Tb2 ,Fe79 composition and are obtained from the mean-field model under certain reasonable assumptions concerning the nature of perpendicular magnetic anisotropy (Ref. 12).

exchange stiffness coefficient Ax. Both these parameters as well as UuAJ are plotted versus temperature in Fig. 3. The composition of the assumed sample is Th21Fe79 and the func­tions are derived from the mean-field model under certain reasonable assumptions. i2 Also shown in Fig. 3 is the do­mai.n-wall thickness AU) versus temperature. Notice that a physical constraint on the model is b.;>Aw' In Eq. (2) f3 is the extent of variations of O'w across the film and, for simpli­city, has been set equal to 0.25 throughout the paper. y, an independent, uniformly distributed random variable in [ - 1,1], is chosen for convenience, but must be replaced with an experimentally obtained distribution for more realis~ tic simulations. Given that the tile size f1 is constrained by factors other than the correlation length of the spatial varia­tions of 0' w' this more realistic distribution of y must neces­sarily include some sort of spatial correlation.

Associated with each tile are three forms of energy as described below.

a. External field energy. In the presence of an external magnetic field H ext applied perpendicular to the sample, each tile has an energy equal to the product of the net mag­netic moment and the field. This energy is positive when Ms andHext are antiparallel and is negative otherwise. Using the convention that positive values of H ext andMs correspond to "up" and negative values to "down," the energy difference between the two states of magnetization is

(3)

Depending on the initial direction of magnetization with re­spect to the field, f1E i could be positive or negative, In any event, this energy difference is only a function of the moment of the given tile and does not depend on other tiles,

b. Domain-wall energy. Each tile has four boundaries which it shares with its four nearest neighbors. When the moments of two neighboring tiles are antiparal1el the bound-

1582 J. Appl. Phys., Vol. 61, No.4, 15 February 1987

ary between them represents a domain wall with associated energy flt,l7w; when the moments are parallel there is no domain waH and the energy is zero. (There is an exception to this rule that applies to situations where the two neighboring domains are at different temperatures, one above and the other below the compensation point. Here the domain wall exists when the net moments are parallel.) Thus the energy difference between the two states of magnetization for a giv­en tile, while the neighboring moments are fixed, is

!1E2 = ( ± O'wl ± O'w2 ± O'w3 ± O'w4 )1ltf . (4)

The proper sign for each term (plus or minus) depends on the mutual orientation of the corresponding moments.

c. Self- or demagnetizing energy. This contribution to energy arises from dipole-dipole interactions and can be ac­curately described in terms of a double integral on the entire film volume,4 The computation of the double integral, how­ever, is time consuming and, in order to avoid it, we have introduced the following approximations: (a) Each tile is replacedbyasingledipoleofstrengthm = b.2tfMs; (b) only a finite number of interactions are considered.

The interaction energy £ of a pair of dipoles with respec­tive strengths m I and m 2 , separated a distance r, and located in a plane to which both dipoles are perpendicular is given by

(5)

The demagnetizing energy density of a uniformly magne­tized, saturated thin film in the above approximation is

(1/2) N N (fl2tf M s)2

bhf i=2;.N j~2: N (i2 + l) 3/2t:.?'

j2 + l=l=o, (6a)

or, equivalently,

(

N N ) 21TM/Ctf lf1) 4~ i~L N j=~N (P+/)--3/2 ,

j2 +/=1=0. (6b)

The factor 1/2 in Eq. (6a) is needed to avoid counting each pair of dipoles twice. The double sum is the energy of interac­tion of a single dipole, located at the center of a (2N + 1) X (2N + 1) lattice of dipoles, with all other di­poles in the lattice, and normalization by fllt f gives the ener­gy per unit volume.

Figure 4 is a plot of the bracketed term in Eq. (6b) versus N. The limiting value for large N is 0.715' . '. Notice that in order for Eq. (6b) to properly approximate the de­magnetizing energy density of a uniform thin film (21TM;), Il must be roughly equal to tf . This is to be expected on physical grounds considering the nature of the demagnetiz­ing interactions. It must be emphasized, however, that in no way does this restriction on Il limit the generality of the modeL As mentioned earlier in relation to Eqs. (1) and (2), the model can be made consistent by inclusion of the correla­tion lengths of Ms and 0' w through the appropriate choice of the random variables x and y. Also notice in Fig. 4 that at N = 5 the function has achieved almost 90% of its asympto­tic value. For practical computation of the demagnetizing energy in this paper we will thus use the value of N = 5 in Eq. (6b) and related equations that follow.

M. Mansuripur 1582

1.00

0.80

S I

0.60

zr ... l f 0 1"1 Ii '1l

z .-

Z~,~ ~~ 0.40

~I:

0.20

0.00 -j----,-----r---r---,.----, (I 10 20 30 40 50

N

FiG. 4. Plot of a function appearing in the expression fOl' the demagnetizing energy density [Eq. (6b)].

For a given distribution of moments across the sample, one can calculate the difference in demagnetizing energy that arises from the switching of a single moment at location (ioJo). This energy difference is given by

io+ N jo+N (+ )2m .. m .. !.W3= L L - IJ ~o,

t= io --- N j=-jo - ,:v jrij ~ rio},,!

(7a)

where, depending on the mutual orientation of the moments at (ioJo) and (fJ), either the plus or the minus sign applies. Replacing for m and r in Eq. (7a) one obtains

io+N jot N

f:.E3 = 2b.t/Ms (io,jo) 2: L i-=io--Nj=jo N

( ± )M, (iJ)

x [(i _ iO)2 + (j _ jO)2p12'

(i - to )2 + U - jO)2#O. e7b) The total change in the energy of the magnetic system

upon reversal of a single tile is now given by

f:.E = b.E1 + tlE2 + b.E3• (8)

Depending on the magnitude of b.E the reversal mayor may not take place. The actual decision algorithm used in the simulations is the so-called Metropolis algorithm with the following features:

(a) Periodic boundary conditions: The array oftiles in Fig. 1 repeats itself in the two~dimensional space so that the end of a row/column is connected to the beginning of that row/column,

(b) Metropolis decision rule: If f:.E for a given tile is negative its magnetization will be reversed. If t::.E is positive the reversal occurs with probability P = exp( - f:.E /kT} where k is the Boltzmann constant and T is the absolute temperature at the center of the tile under consideration.

(c) Checker board scanning: Each scan of the array consists of two half-scans. In the first half-scan the shaded tiles of Fig. 1 are analyzed and those that must be reversed are marked. The reversal of the marked moments, however,

1583 J, Appl. Phys., Vol. 61, No.4, 15 February 1987

does not take place until the end of this half-scan. In the second half-scan the same procedure is applied to the re­maining tiles.

We are now ready to describe the results of simulations. In all of these studies the arrays used are 40 X 40 and the film thickness and tile size are tf = 1l. = 500 A. III. RESUL. TS AND DISCUSSION

To study bulk magnetization reversal and coercivity, we started with a saturated sample at room temperature ( T = 300 K) and applied a reverse field in the perpendicular direction. As long as the field was below 20.4 kOe no rever­sals occurred. At H ext = 20.4 kOe, however, the sequence of Fig. 5 was observed: first a single tile is reversed as shown in Fig. 5 (a), then in successive iterations as in Figs. 5 (b), 5 (c), and 5 (d) the reversed nucleus grows and eventually covers the entire array. The diamondlike structure of the domain is due to the particular choice of tile geometry (square) in these simulations and has no significance other than show­ing regular growth from the initial nucleus. Regular growth is a feature of samples in which domain-wall energy domi­nates over the demagnetizing effect. When the demagnetiz­ing effect is dominant, the domain boundaries tend to be jagged and irregular.

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FIG. 5. Bulk magnetization reversal starting from the saturated state, The sample has II unifonn temperature of T = 300 K and the applied reverse field isH.x, = 20.4 kOe. Patterns (a)-(d) correspond to 1,4,7, and 10 itera­tions.

M. Mansuripur 1563

30

Rnoc = 0

o:=Il~G.25

25 . 20 A=50()A,

T~300K

t f ~ 500'\ ... 20

0 ~ 10

i ~ 0 ::g:

" 2500 5000

:J: 0

Rn.«A)

T(K)

FIG. 6. Coercivity vs temperature as obtained from simulations with no initial nucleus. The inset shows the dependence of coercivity at room tem­perature on the radius of an initial nucleus (see Fig. 8 and the related text for further explanations).

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FIG. 7. Same as Fig. 5 except for the value of H ext which is now 21 kOe. Patterns (a}-(d) correspond to 1, 3, 6, and 9 iterations.

1584 J. AppL Phys., Vol. 61, No.4, 15 February 1987

We define coercivity He as the smallest field at which complete reversal occurs. Thus for the above sample at room temperature the coercivity is He = 20.4 kOe. A plot of He versus temperature is shown in Fig. 6. Notice that this curve has the general characteristics of the experimentally ob­tained curves for media with compensation temperature. 1 I

When the applied field is larger than coercivity, several nucleation centers appear simultaneously and the reversal proceeds faster. Figure 7 corresponds to an applied field of H ext = 21 kOe, with conditions otherwise similar to Fig. 5. It should be noted that in these simulations the coercivity is dominated by the nucleation process: once a nucleus is formed, its growth proceeds uninhibited. On the other hand, if we allow nucleation centers (i.e., structural or magnetic defects) to exist in the saturated state, then the growth pro­cess becomes the determining factor. Figure 8(a) shows a nucleus (diameter = 5000 A) in the otherwise saturated sample. Under an applied field of H ext = 11.5 kOe this nu­cleus grows to Fig. 8(b) and then to Fig. 8(c) but remains trapped in this last position. The rectangular shape of the domain in Fig. 8 (c) is due to the chosen geometry of tiles and has no physical significance. If the field is now raised to H ext = 12 kOe the growth starts again, proceeding very slowly until the entire array is covered. Some of the patterns during this growth process are shown in Figs. Sed), 8(e), and 8 (f). Since the growth is now dependent on the random variations of Ms and u w' the patterns are no longer symmet­ric (as in the diamondlike patterns in Figs. 5 and 7) and the growth proceeds along paths of minimum resistance [like the path starting at the upper left comer in Fig. 8(d)]. For the same reason, the number of iterations it takes to com­pletely reverse the array is much larger in this case. Notice, however, that the coercivity of growth from existing nuclei is less than the coercivity ofnucIeation-dominated reversal. In the example of Fig. 8 the coercivity is only 12 kOe. The coercivity is somewhat dependent on the size of the initial nucleus and the inset in Fig. 6 shows this dependence at room temperature; a similar behavior is observed at other temperatures as well. When the nucleus is sufficiently large, the probability of a minimum resistance path starting on the boundary of the nucleus approaches unity. Consequently, the coercivity reaches a limiting value for large nuclei.

Some of the experimental results on amorphous TbFe and DyFe thin films indicate that below the compensation point the coercivity tends to decrease and then increase again with decreasing temperature. This behavior, which is not observed in our model, may be due to the sharp increase in the stress-induced anisotropy at low temperatures,

We now tum to the process of thermomagnetic record­ing and assume that a 50 ns laser pulse with Gaussian inten­sity distribution (e- I radius = 5000 A) illuminates the cen­ter of the array of tiles. The resulting temperature distribution is shown in Fig. 9 where plots of temperature versus time are given at various radii. 13 Note that the maxi­mum temperature is about 500 K which is attained at the center of the beam at the end ofthe pulse. The region within a circle of approximate radius r = 5000 A is above the Curie temperature at this time (Tc = 418 K). We apply a reverse field of H ext = 1 kOe and observe the behavior of the system

M. Mansuripur 1584

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FIG. 9. Temperature rise vs time in It soo-A film of TbFe alloy on glass substrate. The beam is Gaussian with e-' radius of 5000 A and pulse dura­tion of 50 ns. The profiles are calculateil numerically, taking into account the effects of nonuniform light absorption and three-dimensional heat diffu­sion. The incident power is 5 mW and the beam is stationary. Different curves correspond to different distances from the beam center.

1585 J. AppL Phys., Vol. 61, No.4, 15 February 1987

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FIG. 8. Bulk magnetization reversal from an existing nucleus. The sam­ple has a uniform temperature of T = 300 K and the radius of the ini­tial nucleus in (a) is R rroc = 2500 A. An external field H ext = 11.5 kOe is applied and the patterns of (b) and (c) are obtained after 5 and 10 itera­tions, respectively. The pattern in (c) does not cha.'"]ge with further iterations. When HeAt is raised to 12 kOe the patterns in (d), (e),and (0 are obtained after 1, 30, and 40 itera­tions. Further iterations cause the entire array to reverse.

as time evolves. Prior to t = 8 ns no reversals occur. At this point, however, a reverse magnetized domain appears at the center of the heated region, as shown in Fig. 10 (a). (Since the dynamics of the process are not incorporated in the mod­el, a sufficient number of iterations are performed to reach steady state at each time, assuming, in effect, that the nuclea­tion and growth processes have subnanosecond time con­stants.) Figure 10(0) shows the domain at t = 20 ns. The domain is now larger, because the heated region is larger, and most of it is above the Curie temperature. This is why the reversals within the domain are more or less random. Figure lOee) corresponds to t = 50 ns, the end of the heating peri~ od. At this point the laser is turned off and the temperatures begin to decline. Since the cooling is quite rapid, we in~ creased the time in I-ns steps for several nanoseconds after t = 50 and allowed the domain to achieve steady state at each step. The patterns aU = 51, 53, and 55 ns are shown in Figs. lO(d), Wee), and 10(0, respectively. The cooling pro­ceeds from the boundary towards the center. The pattern does not change after t = 55 ns.

M. Mansuripur 1585

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To study the stability of the thermomagnetically record­ed domain, we applied a reverse field to the domain in Fig. 1O(f), now at room temperature, and allowed the system to relax, Figures 11 (a), 11 (b), and 11 (c) correspond to H ext

= 0, - 1 kOe, and - 2.5 kOe, respectively. It is seen that under relatively small fields, the domain shrinks a bit but does not collapse. However, when a - 5 kOe field was ap-

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1586 J. Appl. Phys., Vol. 61, No.4, 15 February 1987

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FIG. 10. Thermomagnetic record­ing with the temperature profiles of Fig. 9 and an external reverse field of 1 kOe. Patterns (a)-(f) correspond to t = 8, 20, 50, 51, 53, and 55 ns. Iterations are performed at each stage until the steady state is achieved. The regions with random distribution of magnetization are above the Curie temperature .

plied, the domain collapsed towards the center and, in a few iterations, totally disappeared.

Finally, we turn to the thermomagnetic erasure process. The laser pulse used is the same as that in the recording process and the temperature profiles of Fig. 9 still apply. The magnetic field, however, was turned off this time (Hext = 0). Figure 12 represents the erasure process; succes-

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FIG. 11. Stability of recorded do> mains in the presence of an erasing field. The domain is recorded with a 50-ns pulse and a recording field of 1 kOe as in Fig. 10. When the tempera­ture returns to normal (T = 300 K) and the external field is turned off (H"", = 0) the stable pattern in (a) is obtained. With an erasbg field of - 1 kOe the domain shrinks to (b)

but remains stable. (c) Corresponds to He« = - 2.5 kOe; although this domain is substantially smaller than the original domain it is still stable. (Raising the field to - 5 kOe, how­ever, causes the domain to collapse in a few iterations. )

M. Mansuripur 1586

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. .,) ...... .

......................• "'''' .............. , ••••••••••• ••••• , .... CIiIllC ••••••••••••••

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sive patterns correspond to t = 0, 10,20, 53, 54, and 55 ns. Notice in Fig. 12(b) that the domain is disturbed by two mechanisms: nucleation from within and wall collapse from without. Also notice that the erasure is incomplete due to insufficient external field. We repeated the simulation with an external field of HeKt = - 1 kOe and found that complete erasure occurred in this case.

IV. CONCLUDING REMARKS

The model presented in this paper can explain the ob­served features of magnetization reversal and the thermo­magnetic recording process in thin amorphous films of the RE-TM alloys. It is argued that fluctuations in Ms and (I", are responsible for the observed behavior of these films. The implications for different behavior ofS-state and non-S-state RE alloys, particularly with regard to the temperature de­pendence of coercivity, are that fluctuations will occur only through compositional fluctuations in, e.g., Gdeo alloys but will derive in addition from the sperimagnetism of, e.g., Tbeo alloys. These two cases might therefore be modeled quite well by different values of a and /3.

The model is based on certain assumptions that need experimental verification. These include the assumptions concerning random spatial variations in magnetic properties and their statistical distributions. Also the dynamics of do-

1587 J. Appl. Phys., Vol. 61 , No.4, 15 February i 987

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FIG. 12. ThemlOmagnetic erasure in the absence of external field (lIe>[ ~~ 0). (a) is the recorded do­

main created in Fig. 10. A 50-ns laser pulse is applied and the steady state patterns corresponding to t = 10, 20, 53,54 and 55 ns are shown in (b)­(f). The last pattern remains even after the temperature returns to nor­mal. Incomplete erasure is caused by lack of external field. When the sim­l.i1ation was repeated with Hex! = - 1 kOe, erasure was complete .

main-waH motion have been ignored in the analysis. The model, however, can be improved in this respect if one re­lates the number of iterations at each stage to the temporal evolution of the growth process, an undertaking that re­quires a better understanding of the domain-wall dynamics at various temperatures.

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M. Mansuripur 1587