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24ISSN 2072-5981
Volume 17, Issue 2 Paper No 15201, 1-6 pages
2015
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In Kazan University the Electron Paramagnetic Resonance (EPR) was discovered by Zavoisky E.K. in 1944.
*
Computer simulation of growth and magnetic properties of quasi 2D magnetic cluster^
N.A. Balakirev, V.A. Zhikharev* Kazan National Research Technological University, K. Marx 68, 420015 Kazan, Russia
* E-mail: [email protected]
(Received December 15, 2015; accepted December 24, 2015)
The computer simulation of the formation of 2D magnetic cluster is performed. It is shown that the magnetic interaction between growing cluster and randomly walking atoms results in the cluster elongation in the cluster magnetization direction. The features of microwave field absorption (FMR or SPR signals) in thin granular film consisting of elongated clusters are discussed. Magnetic resonance measurements can provide the detailed information on shape anisotropy of the particles and give an evidence for considered model of magnetic cluster growth.
PACS: 75.30.Gw, 75.50.Bb,75.70.Ak
Keywords: silicide, ferromagnetic clusters, implantation, granular film, diffusion limited aggregation, ferromagnetic resonance
1. Introduction
The method of ion-beam synthesis is a very effective way to obtain thin films of new chemical phases within the surface layer of any target. In particular, high-dose implantation of iron ions into the silicon results in the formation of thin films of different silicides FeSi, Fe2Si, Fe3Si, depending on the implantation dose [1]. It is assumed that the synthesized films are not continuous and consist of clusters of the silicide phase. Of particular interest are thin films of magnetic silicide Fe3Si as a promising material for microelectronics and spintronics. Moreover, when ion-beam synthesis is carried out in presence of external magnetic field parallel to the target surface, the synthesized films reveal noticeable magnetic in-plane anisotropy [2]. It was suggested that shape anisotropy of created ferromagnetic Fe3Si clusters constituting the film may be considered as one of the reasons for such effect. In this paper the process of magnetic cluster growth in external magnetic field and the features of magnetic resonance in the film, consisting of the clusters, are discussed.
The formation of magnetic Fe3Si silicide clusters begins with a creation of stable grain of this new phase. During ion bombardment along with the creation of the grains the process of their destruction due to atomic collisions takes place. Hence, an initial randomly created grain of the Fe3Si phase should reach a certain critical size to become stable and serve as a nucleus of the Fe3 Si cluster growth. It can be assumed that the process has the noticeable probability only within thin layer, where the concentration of implanted Fe atoms is maximal. Therefor the computer simulation of cluster growth in 2D system can be used to study geometrical properties of created clusters. For this purpose the Witten-Sander model of the particle-cluster diffusion-limited aggregation (DLA) [3] is widely used. There are numerous modifications "sticking" of moving particle to the growing cluster (the different probabilities of the process).
^This paper material was selected at XVIII International Youth Scientific School "Actual problems of magnetic resonance and its application", Kazan, 26 - 30 October 2015. The paper was recommended to publication in our journal and it is published after additional MRSej reviewing.
a)
b)
Figure 1. Clusters generated according to classical DLA model (a) and in frame of the model considered in present paper (b). The number of Fe atoms included in cluster in both cases is the same.
In the case of the formation of magnetic (ferromagnetic) clusters from diffusing magnetic particles it is necessary to take into account the influence of the magnetic field created by the cluster on the motion of particles near the cluster surface. Such type of the anisotropic diffusion of magnetic particles was considered in the description of the growth of electrodeposited ferromagnetic aggregates in the applied magnetic field [4] and the shape transformation of iron precipitates in alloys under the strong magnetic field [5]. The modified multi-center DLA model, taking into account the effect of the dipole-dipole magnetic interaction between the cluster and randomly walking atoms was developed in [6]. It was shown that the interaction results in cluster elongation in the direction of the cluster magnetization. Thus, if growing in the external magnetic field cluster elongates along the field.
In all mentioned modifications of DLA model the basis of the original DLA approach remains: the cluster grows by taking the diffusing particles up one by one and each of the particles becomes a new unit of the cluster. However, the atoms embedded into the solid react with the host atoms and form a new chemical compound. Such a situation occurs during the ion-beam synthesis of FeaSi silicide when not one but several Fe atoms must come near the cluster boundary at a time to add a new unit (with Fe3Si structure) to the cluster. To take into account this process a modified model was developed [7]. In the model a new unit is added to the cluster in the site of the cluster perimeter near which the local concentration of randomly walking Fe atoms becomes sufficient to provide the FeaSi phase composition. The examples of clusters generated in frame of original DLA approach and according to the suggested model are presented in Fig. 1. The figure shows that in case B) generated cluster has a much denser structure.
In the considered model the influence of the magnetic field created by the cluster of magnetic silicide
Fe3 Si on the motion of iron atoms near the cluster also Figure 2. Cluster simulated in the pres-
take place. Accounting of the magnetic interaction re- ence of the external magnetic field di-
sults in cluster elongation (see Fig. 2). The simulation rected al°ng the OX axis.
of the cluster growth was performed on square lattice (400 x 400). The model critical size of the stable "grain" is taken to be 3a (a is the lattice constant). Each of 10000 free iron atoms makes 10 steps during each stage of the simulation. It was assumed that the saturation magnetization Ms of the growing ferromagnetic cluster is equal to that for silicide FesSi (Ms = 900 emu/cm3). All calculations were performed using the C++ code compiled under Linux OS.
Of course, in a real situation each cluster within the film is not two-dimensional but has a certain size in the direction perpendicular to the plane of the implanted layer. However, it is conceivable that found above effect of cluster elongation along the applied magnetic field will remain. Thus, ion-beam synthesized magnetic film can be considered as a planar system of elongated ellipsoids, the long axes of which lie in the implanted layer plane.
2. Magnetic resonance
Magnetic resonance method (ferromagnetic resonance (FMR), superparamagnetic resonance (SPR)) is usually used to obtain the information about the shapes of particles constituting the granular film, their size distributions and inter-particle interactions. In the scientific literature there are a great number of theoretical and experimental works devoted to the study of the FMR in granular systems (see the detailed overview [8] and references in it). The theoretical description of microwave absorption in real granular magnetic film is very complex, multi-parameter problem, but if the particle density in the film is sufficiently low the inter-particle interactions can be neglected and one can easily obtain the microwave absorption for a single particle. Then, the expression for absorption power must be averaged on the particle parameter distributions over the film to obtain the experimentally observed FMR signal. Below the simple calculation of FMR for planar system of elongated and non-interacting ellipsoids is made and several features of calculated signals connecting with the shape anisotropy of particles are discussed.
The microwave (at the frequency u) absorption P(uj) in the film can be represented as the sum of absorption signals Pi(uj) for i-th ellipsoid:
P(u) = '£Pi(u). (1)
i
According to Kittel approach [9] the magnetization Mi of ¿-th ellipsoid in external magnetic fields is obeyed the equation:
^L=1(MixHi)+Ri, (2)
where Hi = Ho — AnNiMi + h(t), Ho is the static magnetic field, h(t) is the microwave fields, Ni is the demagnetization tensor and Ri is the term describing magnetization relaxation. In the principle axes of ellipsoid demagnetization tensor is written as
(Nix 0 0 \
Ni = 0 Niy 0
I 0 0 Niz)
where NiX < Niy < NiZ (the elongation is supposed along a>axis ). Regarding the relaxation term Ri in the equations (2) it should be noted the following. The differences in the demagnetization factors (the presence of the easy magnetization axis of the ellipsoid) cause a predetermined orientation of the equilibrium magnetization Mio in the absence of a microwave field. The energy dissipation processes will seek to bring the magnetization to this direction. It is convenient to
write the relaxation terms and to solve the equations (2) in the coordinate system in which one of the axes coincides with the equilibrium direction for the magnetization.
The FMR absorption lines are usually observed in two static field Ho orientations: perpendicular one, when Ho is normal to the film plane and h(t) lies in the plane (case A) and parallel orientation, when Ho lies in the film plane and h(t) is normal to the plane (case B). Below both of these cases are briefly considered.
Case A. When Ms^°_N. ) > 1 the magnetization relaxes toward direction of static field Ho (along z-axis) and equations (2) in usual stationary resonance approximation can be written as:
9Mi» 1U 7 • ,, Mix
= ui\Miy — ^jh(t) sin aMs —
dt iy ' v 7 s T '
dMiy M uu\ ** Miy (4)
- = LOi2Mix - 7h(t) cos aMs - -j^-, y >
Miz w Ms,
where a is the angle betwenn h(t) and longest axis of ellipsoid. In (4) the Bloch-type relaxation
_ Mix, ix,y — t1
terms RiXy = — M™'y are introduced. The frequencies wn and wi2 are determined as follow:
wn =7^o-7 (Niz - Niy) Ms,
(5)
C012 = jHo-J (Ntz-Ntx) Ms.
Equations (4) lead to the FMR absorption for i-th ellipsoid:
27r(ryh)2Ms[uJn cos2 a + wi2 sin2 a]
r> f UL-T W12 Olii Ltj , s
^W) - mu. .0-—-. 1 X9 . 4^11 TJIPT- W
Case B. In this case static magnetic field Ho lays in the (xy) plane, where «-axis coincides with longest axis of ellipsoid, microwave field h(t) being directed along z-axis. In coordinate system (x', y', z') in which rc'-axis is directed along equilibrium magnetization Mjo the equations (2) can be written as:
Mix, « Ms,
-gj- = U2iMiz, - jh(t)Ms - -^f, (7)
dMiz, MiZi - u^Miy'--
where
W21 = 7[iio(sin (f> sin 9 + cos (f> cos 9) + (NiZ — Niy sin2 9 — NiX cos2 9)Ms], w22 = 7[iio(sin (f>sin9 + cos (f>cos 9) + (Niy — NiX cos2 9 — NiX sin2 6)Ms]■
(8)
In (8) (f) is the angle between static magnetic field Ho and longest axis of ellipsoid and the angle 9 is determined from the condition
Ho (cos (f) sin 9 — sin <f) cos 9) = (NiX — Niy) Ms cos 9 sin 9. (9)
The solving of (7) leads to the microwave absorption:
2ir(^h)2Msuj22 T[(w2 - W2lW22 + + ^22]
- -:-:-. 1 A2 , 4co21CQ—-
The expressions (6) and (10) need to be averaged on parameters of ellipsoids to obtain the experimentally observed FMR signals. To make some numerical estimations of the FMR lines (6) and (10) it can be supposed that all ellipsoids in the film have the same shape, but the orientations of ellipsoid axes in film plane may be different. Let the dimensionless axes of the ellipsoid a, b and c are supposed to be equal to: a = 200, b = 180 and c = 20. These values correspond to a thin cluster with in-plane elongation in a>axis direction. These parameters of the cluster lead to the following values for demagnetization coefficient [10]: Nx = 0.095, Ny = 0.105, Nz = 0.8. The value of the magnetization Ms is taken equal to 1000G and the frequency uj of microwave field is set equal to 9.5 GHz (X-band).
Microwave absorption in case A (Eq. (6)) is described by single resonance line with the maximum at H= ^y'wnwii and the width, determined by relaxation parameter 1/T. Note, that the resonance frequency uj & y/wnw2i is not dependent on the angle a. Thus, the averaging on orientation of the ellipsoid in the film plane only leads to a change in amplitude of FMR signal, but not to its broadening. Consequently, in case A the FMR signal is the same in ordered or disordered arrangements of ellipsoids and at above introduced numerical parameters the signal is the resonance line with maximum at = 12900 Oe.
Contrary to the case A, in case B the resonance frequency uj & y/W2iW22 depends on the angle (f). If the arrangement of ellipsoid is ordered i.e. the longest axes of all ellipsoids have the same direction in the plane (taken clS 2>clXlS of coordinate system), the absorption maximum will be observed at H^esl = 880 Oe when Ho lies along a>axis and at ii|'es2 = 1320 Oe, when Ho lies in plane and perpendicularly to «-axis. This case was considered in [11]. The numerical values were obtained using introduced above ellipsoid parameters, the width of each signal being determined by relaxation time T. If the orientation of the longest axes in the film plane is random the FMR absorption is described by inhomogeneously broadened line, practically absorption "band", extended from 800 Oe to 1200 Oe.
3. Conclusion
Finally, a simple model of the formation of FesSi clusters during high-dose Fe ion implantation into silicon is considered. The magnetic interaction between growing cluster and randomly walking Fe atom results in the cluster elongation in the cluster magnetization direction. This can explain the origin of the uniaxial in-plane magnetic anisotropy in films ion-beam synthesized in the presence of an external magnetic field. The features of microwave field absorption (FMR, SPR signals) in thin granular film consisting of elongated clusters are discussed. The observation of discussed above differences between resonance lines in perpendicular (Ho -L film plane) and parallel (Ho 11 film plane) orientations of static magnetic field provides the useful information on shape anisotropy of particles constituting the magnetic granular film and can give evidence for the considered model of cluster growth.
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