## Jetplet24_00iskhakovlo

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*JETP Letters, Vol. 72, No. 12, 2000, pp. 603–607. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 72, No. 12, 2000, pp. 872–878.*

Original Russian Text Copyright 2000 by Iskhakov, Komogortsev, Moroz, Shalygina.
**Characteristics of the Magnetic Microstructure**
**of Amorphous and Nanocrystalline Ferromagnets**
**with a Random Anisotropy: Theoretical Estimates**
**and Experiment**
**R. S. Iskhakov*, S. V. Komogortsev*, Zh. M. Moroz**, and E. E. Shalygina*****
** Kirenskiœ Institute of Physics, Siberian Division, Russian Academy of Sciences,*
*Akademgorodok, Krasnoyarsk, 660036 Russia*
*** Siberian University of Routes of Communication, Krasnoyarsk Branch, Krasnoyarsk, 660028 Russia*
**** Moscow State University, Vorob'evy gory, Moscow, 119899 Russia*
An experimental determination (both direct and indirect) of the characteristics of the magnetic microstructure,namely, the ferromagnetic correlation radius

*R*
*f* and the rms fluctuation of the mean anisotropy

*D*
formed for amorphous and nanocrystalline ferromagnets with a random anisotropy characterized by the quan-tities

*R*
*Ha*, respectively. The magnetization curves of amorphous and nanocrystalline ferromagnets are
found to exhibit a dependence on

*H* that is caused by the alignment of the magnetizations of individual magneticblocks with the field.

* 2000 MAIK “Nauka/Interperiodica”.*
PACS numbers: 75.30.Gw; 75.50.Kj; 75.60.-d

**1.** Amorphous and nanocrystalline ferromagnetic

anisotropy field

*D*1/2

*Ha*, where

*D* is the symmetry factor
alloys can be represented as an ensemble of clusters or
equal to 1/15 for a uniaxial anisotropy [4]. In the case
grains of size 2

*Rc*, which are bound together by the
of large grains satisfying the inequality
exchange interaction and have randomly oriented easy
axes. In the approximation of a continuous medium,

*R *>

*D *1/4(

*A*/

*K *)1/2,
such a system is described by the internal energy den-sity
the correlation properties of the inhomogeneous state

of the orientation of

**M** always coincide with the corre-

lation properties of the local anisotropy fluctuations. In

*U *= - α(∇

**M**)2

- β(

**Ml**)2

this case, the approximation of crystallites withoutexchange interaction between them is valid. Beginning
where the magnetization

*M* is characterized by a con-
from the publications [11, 12], this approximation was
used for calculating the law of magnetization approach
2

*A*/

*M*2 is determined by the exchange interaction con-
stant

*A*, the parameter β =

*H*
∆

*M*/

*M *= (

*DH */

*H*)2,

*H *>

*D*1/2

*H *.

*a*/

*Ms* = 2

*K*/

*M * is deter-
mined by the local anisotropy constant

*K*,

**l** is the unit

For amorphous and nanocrystalline ferromagnets, the
vector of the easy axis of this anisotropy, and

*H* is the
It is known that, in a ferromagnet, an orientational
irregularity of the magnetic anisotropy of any origin
If this inequality is satisfied, the correlation properties
(crystallographic, elastic, or other) gives rise to the for-
of the SMS fundamentally differ (in the fields

*H* <

*Hex*)
mation of an inhomogeneous state of the magnetic
from those of the local anisotropy: the deviations of the
moment

**M**(

**x**) [1–10]. This state is called a stochastic

magnetization

**M**(

**x**) from the direction of the external

magnetic structure (SMS). The parameters of the SMS
field are correlated in space and form a static wave with
are determined by the relations between three charac-
the characteristic wavelength

*R * = (2

*A*/

*MH*)1/2. The
teristic fields: the external field

*H*, the exchange field
changes occurring in the correlation properties of the

*ex* = 2

*A*/

*Ms R *, and the rms fluctuation of the local
SMS in the vicinity of the field

*Hex* lead to a change in
0021-3640/00/7212-0603$20.00 2000 MAIK “Nauka/Interperiodica”

* * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * *
such a material can be described by an ensemble ofweakly coupled magnetic blocks (Fig. 1). The blocksize is 2

*R*
*f*, the mean anisotropy in the block is

*K * =
, and the unit vector

**n** of this anisot-

ropy is randomly oriented. In the approximation of acontinuous medium, such a system can be described bythe internal energy density represented in the form

**U **= –- β

*e*(

**Mn**) –

**HM**,

where the parameter β

*e* = 〈

*H *〉

*a */

*M* = 2

*K */

*M * is deter-

**Fig. 1.** Schematic representation of a ferromagnet with a

mined by the constant 〈

*K*〉 characterizing the mean
random anisotropy. The small arrows indicate random ori-
anisotropy in the magnetic block. In zero field, the
entation of the local magnetic anisotropy

**l**(

**x**), and the large

magnetization of a magnetic block is oriented along the
arrows show random orientation of the mean anisotropy of
unit vector

**n**. Therefore, in this case, the correlation

a magnetic block

**n**(

**x**).

properties of the irregular orientation of

**M**(

**x**) com-

pletely reproduce the correlation properties of the fluc-

tuations of mean anisotropy 〈

*K*〉. This means that the

magnetization curve in low magnetic fields should be

described by dependence (3) modified as follows:

∆

*M*/

*M *= (

*D*1/2 〈

*H *〉/

*H*)2 = (

*R */

*R *)4,

*H *>

*D*1/2 〈

*H *〉 or

*R *<

*R *.

Simultaneously, using the definition of the characteris-tics of the system of magnetic blocks (

*R*
in terms of the characteristics of the grain system (

*Rc*
**Fig. 2.** Theoretical dependences of the variance of magnetic

and

*D*1/2

*Ha*), Eq. (5) can be represented in the form
moment on the external magnetic field in the

*D*1/2〈

*H *〉
the solid curve corresponds to Eq. (11) and the dashed curveto Eq. (14).

∆

*M*/

*M *= (

*D*1/2 〈

*H *〉/

*H*)1/2 =

*R */

*R *,

*R *<

*R * Ӷ

*R *.

the law of magnetization approach to saturation. For
The aim of our work is the experimental study of the
three-dimensional and isotropic inhomogeneities of
aforementioned effects. Its significance is determined
anisotropy, the following relationship was obtained
by the fact that the experimental measurement of
dependence (3') is a direct, rather than indirect [seeEq. (5')], proof of the existence of magnetic blocks and
(

*D*1/2

*H */

*H *)2(

*H */

*H*)1/2
allows one to directly measure the characteristics of the
magnetic microstructure (the quantities 〈

*H *〉
= (

*D*1/2

*H */

*H *)2(

*R */

*R *),

*f*) of amorphous and nanocrystalline ferromagnets.

**2.** Let us theoretically estimate the correlation prop-

erties of the irregular orientation of

**M**(

**x**). The main

For anisotropic and low-dimensional inhomogeneities,
characteristic of these properties is the correlation func-
the dependence on

*H* may be different [6, 7, 13–16].

tion

*Km*(

**r**) or the spectral density

*Sm*(

**k**) related to the

correlation function through the Fourier transform:

As the magnetic field further decreases (

*H* Ӷ

*Hex*),
the situation changes. In low magnetic fields, the mag-
〈

**m**⊥

**x**
( +

**r**)〉 =

*K ***r**
netic system of amorphous and nanocrystalline ferro-magnets exhibits the well-known Imry–Ma effect [17].

( )

**m**⊥*

**x**
( +

**r**) =

*S ***k**
( )δ(

**k **–

**k**');

This effect consists in the instability of the ferromag-netic state with respect to the randomly oriented local

*K ***r**
*S ***k**
( )

*ei***kr k**
magnetic anisotropy. In this case, the ferromagneticorder is characterized by the correlation length

*Rf* =
where

**m**⊥(

**x**) are the transverse components of the unit

*Ha*) [5, 10], so that the magnetic structure of
vector of magnetization

**m**(

**x**) =

**M**(

**x**)

**/M**. The magneti-

CHARACTERISTICS OF THE MAGNETIC MICROSTRUCTURE
zation curve is related to

*Km*(

**r**) and

*Sm*(

**k**) by the stan-

dard relationships

∆

*M*/

*M *≡

*d *(

*H*) =

*K ***r**
*S ***k**
The general expression for

*Sm*(

**k**) through the arbitrary

spectral density

*S*(

**k**) of the fluctuations of local anisot-

ropy axis has the form [3, 4, 17]

2

*S ***k**
*S ***k**
where

*kH* = 1/

*RH* is the wave number of exchange cor-relations. If we model the stochastic properties of theorientational irregularity of magnetic anisotropy by thesimplest correlation function

**Fig. 3.** High-field portions of the magnetization curves

*M*(

*H*): (

*1*) Fe73.5Cu1Nb3Si13.5B9 and (

*2*) Co80Zr10 amor-
tape; Co90P10 amorphous films with

*t* = (

*4*) 500 and(

*5*) 2000 Å; and (

*6*) a Co90P10 amorphous coating with

*t* =
where

*kc* = 1/

*Rc*, we obtain a symmetric expression for

*Sm*(

**k**):

for the variance of the random deviations of 〈

**m**(

**x**)〉 :

In this case, the expression for the variance

*d*
*d*' (

*H*) = --------------------- ---
Here, the rms fluctuation of anisotropy in a magnetic
One can see that, for

*H* ӷ

*Hex*, Eq. (11) yields expres-
block is

*D*1/2〈

*H *〉 =

*D*1/2

*H*
sion (3) for both inequalities (2) and (4) and, for

*H* Ӷ

*a*/(

*Rc*/

*Rf*)3/2 and the field

*Hf* is

*ex*, Eq. (11) yields expression (5).

We now consider a random function 〈

**m**(

**x**)〉 that

is obtained by averaging the random function

**m**(

**x**) over

is reduced to Eq. (3'). Figure 2 shows dependences (11)
the space interval (

**x** – |

*R *|

*f *,

**x** + |

*Rf *):

and (14) as functions of magnetic field in

*D*1/2〈

*H *〉
One can see that, up to

*H* ≈ 10

*D*1/2〈

*Ha *, the variations of
the experimental error). This means that the magnetiza-
tion of amorphous and nanocrystalline ferromagnets
The stochastic properties of the orientational irregu-
occurs through the alignment of the mean magnetiza-
larities of the mean anisotropy 〈

*K*〉 of magnetic blocks
tions of the blocks with the field [according to Eqs. (3')
are determined by the rules described in [18]. In the
and (14)] and only after that the decrease in the ampli-
case

*Rf* ӷ

*Rc*, these properties are described by the cor-
tude of

**m**⊥(

**x**) [described by Eqs. (5), (5'), and (11)]

relation function and the spectral density of the follow-

**3.** Figure 3 presents the high-field portions of the

magnetization curves

*M*(

*H*) for films and foils of amor-
phous and nanocrystalline alloys produced by different
techniques (fast quenching of the melt or chemical dep-
osition). The magnetization curves were obtained using
where

*kf* = 1/

*Rf*. Substituting Eqs. (13) into Eq. (8) and
vibrating-coil magnetometers with an electromagnet
then into Eq. (7), we obtain the following expression
for fields of up to 15 kOe and with a superconducting
solenoid for fields of up to 30 kOe. One can see that, in
inside the sample do not coincide with the external
magnetic field strength at

*H* ~

*H*
*s*,

*H*–1/2) coordinates, the experimental curves
can be described by linear dependences (of the type

*y* =
This difficulty can be overcome with the use of an

*ax*) determined by Eqs. (5) and (5'). This means that
experimental technique that allows one to exclude the
inequality (4) is valid for the amorphous and nanocrys-
effect of magnetostatic mechanisms. Such a technique
talline alloys under study. The experimental curves also
is realized by a magneto-optic micromagnetometer
suggest that the numerical value of the coefficient of
[19] allowing the measurements of the local magnetiza-
linear dependence is determined by the characteristics
tion curves by using the equatorial Kerr effect δ(

*H*) ~
of the random anisotropy,

*Rc* and

*D*1/2

*Ha*. To calculate

*M*(

*H*) with the light spot diameter of 1 to 30 µm in mag-
the latter, it is necessary to measure the magnetization
netic fields of up to 200 Oe. In this technique, the exter-
curve in the fields

*H* >

*Hex*. By recording the part of the
nal magnetic field is applied along the sample surface
curve described by dependence (3), one can determine
normally to the plane of light incidence. Before mea-
suring the local magnetization curves

*M*(

*H*), the system

*a*. The revealed crossover of

*M*(

*H*) (the transition
from ∆

*M* ~

*H*–1/2 to ∆

*M* ~

*H*–2) provides the possibility
is tuned so as to depart from the significant magneto-
static sources. For this purpose, it is necessary to con-

*ex*. Substituting the exchange interaction
constant

*A*, which is calculated from the low-tempera-
struct the distribution curves δ(

*L*)/δ

*s* ~

*M*(

*L*)/

*Ms* at a
ture thermomagnetic curves ∆

*M* ~ (

*T*/

*A*)3/2, into

*H*
constant external magnetic field

*H* (much lower than

*c*. However, in many cases this pro-
scanning the light spot over the sample surface along an
gram cannot be implemented. Many amorphous and
arbitrarily chosen direction. (The scan length

*L* is cho-
nanocrystalline ferromagnetic alloys are characterized
sen so as to exceed the spot diameter by two to three
by the values of

*Hex* that exceed the maximal fields used
orders of magnitude). Typical distribution curves can
in the experimental measurements of

*M*(

*H*) (see, e.g.,
be found in [20]. They exhibit irregular deviations of

*M*
[16]). Then, the grain size (or the cluster size) 2

*Rc* can
from the mean magnetization 〈

*M*〉. As the field is
be determined by direct methods such as the X-ray
increased and the scanning along

*L* is repeated, the
structural analysis or transmission electron microscopy
value of 〈

*M*〉 increases and the deviations decrease, but
and the values of

*Hex* and

*D*1/2

*Ha* can be calculated (the
the spatial scale of deviations is retained. The field

*Hs*
latter is obtained from the linear dependence ∆

*M* ~

*H*–1/2
is determined as the one at which the amplitude of the
deviations is of the order of experimental error. For oursamples, the following deviations were observed:
For the characteristics of the magnetic microstruc-

*f* and

*D*1/2〈

*Ha *) the situation is entirely different.

For their indirect determination, it is sufficient to record
the linear dependences shown in Fig. 3. The measure-
ment of the coefficient of linear dependence allows one
nanocrystalline tape. The local magnetization curves
to calculate the rms fluctuation of the anisotropy field in

*M*(

*H*) were obtained from the light spot, 20–30 µm indiameter, set at the center of a soft magnetic region. It
a magnetic block,

*D*1/2〈

*H *〉

*a * ≡

*Hf*. The substitution of

*A*
was believed that the deviations of

**M**(

**x**) in this region

into

*Hf* provides the value of

*Rf*. For the amorphous and
are caused by the scattering due to the chaotic orienta-
nanocrystalline alloys presented in Fig. 3, the following
tion of the axis

**n**(

**x**) of a magnetic block.

*a * and

*Rf* were obtained: 340 Oe and
Figure 4 shows the local magnetization curves

*M*(

*H*)
130 Å, respectively, for a Fe73.5Cu1Nb3Si13.5B9 amor-
phous tape 20 µm thick; 25 Oe and 640 Å for a
73.5Cu1Nb3Si13.5B9 amorphous foil and a Co90P10
amorphous film; the curves were measured for different
Co80Zr10V10 amorphous tape 30 µm thick; 100 Oe and
parts of the samples. One can see that, in the (∆

*M*/

*M*
*H*–2) coordinates, the experimental magnetization
20 µm thick; 120 Oe and 380 Å for a Co90P10 amor-
curves are described by linear dependences of the type
phous film 500 Å thick; 9 Oe and 1430 Å for a Co90P10
of Eq. (3'). The slopes of these dependences character-
amorphous film 2000 Å thick; and 3 Oe and 2460 Å for
ize the values of the rms fluctuation of the mean anisot-
a Co90P10 amorphous coating 30 µm thick. To directly
ropy field in a magnetic block,

*D*1/2〈

*H *〉

*f* and

*D*1/2〈

*Ha *, it is necessary
presented in Fig. 4, we obtained:

*D*1/2〈

*H *〉 ≈ 3–6 Oe and
to record the part of the dependence

*M*(

*H*) that is
described by Eqs. (3') and (14) rather than by Eqs. (5),

*f* ≈ 970–1300 Å for a Fe73.5Cu1Nb3Si13.5B9 foil and

*D*1/2〈

*H *〉 ≈ 60–70 Oe and

*R*
(5'), and (11). With a vibrating-coil magnetometer, such
a measurement is apparently impossible, because real
film (500 Å thick). One can see that the difference
amorphous and nanocrystalline alloys contain microc-
between the values of

*D*1/2〈

*H *〉
racks, pores, and inclusions of a second phase, giving
the Co90P10 amorphous film by the direct [Eq. (3')] and
rise to magnetostatic mechanisms of scattering for the
indirect [Eq. (5')] methods does not exceed 50%. The
magnetic moment [these mechanisms are not included
greater difference between the corresponding values
in Eqs. (1) and (1')]. As a result the true values of

*H*
obtained for a Fe73.5Cu1Nb3Si13.5B9 tape is no surprise.

CHARACTERISTICS OF THE MAGNETIC MICROSTRUCTURE

**Fig. 4.** Local magnetization curves

*M*(

*H*) measured for different parts of (a) a Fe73.5Cu1Nb3Si13.5B9 amorphous tape (

*t* = 20 µm)

and (b) a Co90P10 amorphous film (

*t* = 500 Å).

The point is that the local signal δ(

*H*) ~

*M*(

*H*) is
6. H. Kronmüller, IEEE Trans. Magn.

**15**, 1218 (1979).

obtained from a thin surface layer ~200 Å thick. There-
7. A. P. Malozemoff, IEEE Trans. Magn.

**19**, 1520 (1983).

fore, a coincidence between the “surface” signal

*M*(

*H*)
8. E. M. Chudnovsky, W. M. Saslow, and R. A. Serota,
and the integral value of

*M*(

*H*) should be expected only
Phys. Rev. B

**33**, 251 (1986).

for films whose thickness is comparable to the penetra-
9. W. M. Saslow, Phys. Rev. B

**35**, 3454 (1987).

tion depth in the magneto-optic technique. The lattercondition is fulfilled for the 500 Å-thick Co
10. G. Herzer, IEEE Trans. Magn.

**26**, 1397 (1990).

phous film and does not hold for the foils showing
11. N. S. Akulov, Z. Phys.

**69**, 278 (1931).

noticeable differences between the values of

*D*1/2〈

*H *〉
12. W. F. Brown, Jr., Phys. Rev.

**58**, 736 (1940).

and

*Rf* obtained for the bulk of the samples and for the
13. E. M. Chudnovsky, J. Magn. Magn. Mater.

**40**, 21

14. A. G. Chernykh, P. P. D’yachuk, and V. B. Kruglov, Izv.

V.P. Ovcharov for providing the films and foils and to
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*et al.*,
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Source: http://natoarw.kirensky.ru/zdoc/00_2je.pdf

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