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Submitted on 1 Jan 1988
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RANDOM SURFACE ANISOTROPY AND THE
MAGNETIZATION OF EPITAXIALLY GROWN
THIN FILMS
J. Cullen, K. Hathaway
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, Supplement au no 12, Tome 49, d6cembre 1988
RANDOM SURFACE ANISOTROPY AND THE MAGNETIZATION OF EPITAXIALLY GROWN THIN FILMS1
J. R. Cullen and K. B. Hathaway
Naval Surface Warfare Center White Oak, Silver Spring, MD 20903-5000, U.S.A.
Abstract. - We introduce a model of a thin magnetic film which includes coherent bulk and surface anisotropies as
well as an incoherent, random surface anisotropy. A phase diagram showing the stable magnetization states in thickness, surface anisotropy space is presented. Modifications of the diagram due to the random anisotropy are calculated. The case of aFe on GaAs is discussed
1. I n t r o d u c t i o n
Experimental studies of the magnetic properties of single-crystal aFe grown on GaAs by molecular-beam epitaxy have revealed anomalous decreases in the mag- netization as a function of film thickness, for both (100) and (110) faces [I, 21. The bulk magnetization is only recovered in films of a thousand
A.
This anomaly in the magnetization, in particular the lack of satura, tion for thick films, was recently attributed to the pres- ence of random surface anisotropy at the GaAs face [3]. The Fe films grown on the (110) face have another in- teresting feature, i.e. the easy direction of magnetiza- tion switches from [OOl] to[lIO]
as the thickness of the films is decreased [4, 51. The latter phenomenon is at- tributed to the pesence of uniaxial surface anisotropy. Assuming that the sign of the surface anisotropy is such as t o favor the [ l i O ] direction, in a film of a given thickness, the surface and bulk anisotropies compete in determining the net magnetization direction. In thick films, the bulk aisotropy wins, and [001] is easy. In thin films, however, the surface anisotropy per volume, which varies as the inverse of the thickness, prevails, and the [IiO] is the easy direction. However, these considerations leave open the question of the nature of the transition or transitions and the effect of the random anisotropy on them. In this paper we dis- cuss some of the consequences of an extended version of the model of random surface anisotropy which in- cludes a coherent (i.e., non-random) uniaxial surface anisotropy. The results are summarized by means of a phase diagram with film thickness and strength of the coherent surface anisoropy as coordinates. [OOl],
[ o ~ o ] and [uiiv] regions are separated by 1st or 2nd order transition lines. In this [utiv] region the net spin direc- tion varies with distance from the surface. The effect of the random surface anisotropy is significant in two regions of this thickness-surface anisotropy space; the region where the coherent surface anisotropy is small (for any thickness) and the region where the surface anisotropy energy per area is roughly equal to the bulk domain wall energy per area (and the thickness is in the vicinity of the wall width). The latter region is centered along the 2nd order line separating [001] from [uiiv].
The case of aFe on (110) GaAs is discussed on the basis of these results.The existence of a uniaxial, in-plane surface anisotropy for low-symmetry surfaces (such as (110)) was predicted by NBel[6]. Rado [7] showed how surface anisotropy influences high-field magnetization and fer- romagnetic resonance frequencies. NBel estimated K,.
t o be between 0.3 and 3.0 ergs cm-2 for aFe. Grad- mann et al. [8] inferred from conversion electron Moss- bauer spectroscopy a value of 0.04 for (110) Fe on W. The latter authors also re-examined the (110) Fe on GaAs results and obtained
K,
equal to 0.047. Their analysis is based on the "homogeneous magnetization approximation",
which is identical to our model in the limits of ultra-thin films and no random surface anisotropy.2. The m o d e l and the phase d i a g r a m
We consider a model of ferromagnetically coupled spins lying in the (110) plane subjected to a fourth order crystalline anisotropy and a unixial surface anisotropy. The total energy F is a functional of the angle variable O which the spins make with respect to the [001] direction
K1
F =
/
d3x{A
2 ( u B ) ~-
- E 8 ( e ) )+
. -(28
-
27 (x)) (1) Here, A is exchange stiffness (= 2 x Rado's value),Kl the usual fourth order anisotropy constant, K,
the strength of the coherent surface anisotropy, K, the strength of the random component of surface anisotropy and r] is a random function of position x
on the reacted GaAs face. 1 and 2 refer t o the two faces at z = 0 and z = L respectively. The function
3
E (O) is given by E (63) = -COS 4 8
+
cos2@. It is the4
last term in equation (1) that distinguishes the present work from that of reference [5]. The requirement that
F be an extremum leads to the following equations for
e
v2e
-
(z)
sin 2 e (3 cos 2 8+
1) =o
(2)and
'supported in part by the Materials Division of the Office of Naval Research and in part by Naval Surface Warfare Center's Independent Research Funds.
JOURNAL DE PHYSIQUE
Consider first the case for which
K,
= 0. The above equations then predict the existence of three stable phases; [OOl] for small K, and large L, [IiO] for smallL and [utiv] for large K, and large L. In the [uiiv] phase the magnetization direction ( 8 ) is a function of z, de- pending on the parameters K, and L. The first-order boundary separating the [OOl] and [OiO] phases is given- by the formula L1 = 4K.
/
K1 (Eq. (5)). These results are summarized in figure 1. We use dimensionless pa- rameters for thickness and surface anisotropy strength;@ =
K,
/ ( ~ A K I ) " ~ and 7 = L / (2 (2A/ ~ 1 ) " ~ ).
Fig. 1.
-
The phase diagram for the magnetic states of the (110) face of a cubic ferromagnet. Bold solid lines show first-order boundaries; the thin solid line indicates a secon-order boundary. The [uiEv] phase is characterized by a non-uniform magnetization. The dotted line represents the [loo] t o [lIO] boundary as modified by random surface anisotropy. Note that this line intersects the ordinate a t a non-zero value. r and p are dimensionless measues of thick- ness and coherent surface anisotropy, respectively, defined in the text.3. The effect of r a n d o m surface anisotropy
The random surface anisotropy at z = 0 gives rise t o a fluctuating magnetization which is greatest near this surface. We expect the fluctuations t o be largest along the second-order line and at and near
K,
= 0. We estimate the size of the fluctuations from the [001] state by calculating the mean square value of Q at z = 0. In so doing we assume that correlations between then (x) are finite only over distances of the order of the lattice parameter ao. We find thatQ2
( ~ ? d
/
A2)f
(P) In (Af (P)l
KI~:), (6)f
= 1/
((1-
P ) ~ - (1+
P ) ~ exp ( - 8 ~ ) ) (7)f - I vanishes along the second ordr line, so the fluc-
tuatons diverge there and this perturbative approach breaks down. However, we can estimate the boundary of the region in which there is a large reduction in the magnitude of the magnetization by setting Q2 B 0.2,
say, and solving for T for fixed P. Using K, w K,,
the prefactor in equation (9) is of the order of Expanding f for T near 7 2 , its value on the critical line,
and setting /3 =
P
= 1.0, AT w 0.03. Thus only for thicknesses very close to the critical thickness would we expect a significant magnetization reduction. Note however, that whatever the size of the reduction, it penetrates a distance of the order of the domain wall with into the bulk of the specimen [3].4. Discussion
We come now to the question of Fe films on (110) GaAs. Where do they fit on the phase diagram? All indications are that the easy-axis transition takes place at 100
A;
from equation (8) for the first-order line using Kl = 5 xlo5
erg cm-3 we find K. = 0.125 erg cmd2 This puts aFe in the lowP
portion of the phase dia- gram, where the (virtual) second-order line gets close to the first-order line, indicating that significant fluc- tuations in magnetization are expected. In fact, using equation (9) with A = 2 x erg cm-l, the value of KI given above, and a0 = 2x
cm, we findthe surface random anisotropy distorts the ,first-order line, pushing it to higher T for a given
P,
as shown schematically by the dotted line in figure 1. Note that the shifted line intersects the T axis at a finite value, proportional toK,.
Thus it is that random anisotropy induces a[loo]
to [ l i O ] transition, even in the absence of coherent surface aniosotropy. We estimate thatTaking
K,
= 1.0 erg-cm--2, this translates into a re- duction of K, of about ten percent, thereby placingKs
close to its value for metal interfaces 151.
To summarize, we have extended the model of ran- dom surface anisotropy to include a coherent uniaxial term. We chose the latter to be in competition with the bulk anisotropy in order to study the nature of the transition, as a function of film thickness, from [001] to [110]. As depicted in the phase diagram, two different first-order transitions are expected, depending on the surface anisotropy strengths. The second order line is expected t o be significantly altered by the presence of the random surface anisotropy, and so is the first- order line near K, = 0. The latter modification is used to calculate a new first-order line and thus determine a reduced value of K, for Fe on GaAs.
[I] Prinz, G. A. and Krebs, J. J., Appl. Phys. Lett. 39 (1981) 397.
[2] Prinz, G. A., Phys. Rev. Lett. 54 (1986) 1051. [3] Cullen, J. R., Hathaway, K. B. and Coey, J. M. D.,
J. Appl. Phys. 63 (1988) 3649.
[4] Prinz, G. A., Rado, G. T. and Krebs, J. J., J.
Appl. Phys. 53 (1982) 2087. [5] NCel, L., C. R. 237 (1953) 1468.
[6j Rado, G. T., Phys. Rev. B 26 (1982) 295. [7] Gradmann, U., Korecki, J. and Waller, G., Appl.
Phys. A 39 (1986) 101;
Gradmann, U., J. Magn. Magn. Mater. 5457
