**UDC 538.1; 548 **

**THE TEMPERATURE DEPENDENCE OF THE FARADAY EFFECT IN TERBIUM **
**GARNET FERRITE IS INVESTIGATED NEAR ITS MAGNETIC-COMPENSATION **

**TEMPERATURE **

Sharipov M. Z.

Bukhara Engineering - Technologies Institute, 200100, Bukhara, Uzbekistan, [email protected]

*The temperature dependence of the Faraday Effect in terbium garnet ferrite, Tb**3**Fe**5**O**12**, is investigated *
*near its magnetic-compensation temperature, Т**с **= 249 K. A non-monotonous variation in the value of the *
*Faraday rotation angle Ф is observed in a weak magnetic field as the temperature approaches Т**с**: the *
*temperature plot of the Faraday rotation angle has two local maxima observed left and right of the magnetic *
*compensation point. A theoretical model is proposed, which follows from the phenomenological theory of *
*domain-boundary displacement under the action of a magnetic field, offering an unambiguous description of the *
*principles of domain-structure influence on the Faraday Effect in Tb**3**Fe**5**O**12** near Т**с**. *

* Keywords: *magnetic compensation temperature, domain structure, Faraday effect.

**Introduction **

It is well known that many properties (thermal capacity, Faraday rotation, Young modulus,
etc.) of rare-earth garnet ferrites (REGFs) undergo an abrupt change. By far the majority of
experimental studies dealing with this problem report the results of their investigation performed
under conditions of uniform magnetization of the crystal (in a single-domain state of the specimen)
[1].In a weak magnetic field, however, where there is a certain domain structure (DS) in the crystal,
variation of the physical properties of REGFs at Т → Тс can be much more complex. Within recent
years, an interest in the investigation of variations in DS and the related magnetic characteristics of
REGFs near *Т*с is prompted by the prospects of developing promising materials, based on these
ferrimagnetics, for the elemental base of devices of the new-generation magnetic microelectronics,
whose operating principle relies on the flexomagnetoelectric effect [2].

To this end, we performed an experimental investigation of the influence of DS on the Faraday Effect (FE) in garnet ferrite, Tb3Fe5O12, in the temperature interval close to the point of its magnetic compensation, Тс = 249 K [3], and present its results in what follows.

**1. Samples and experimental procedure **

It is well known that in Tb3Fe5O12 the easy-magnetization axes are the crystallographic
directions of the 〈111〈 type [1], i.e., the crystal has four easy magnetization axes along which we
could orient vector **M*** s*, which significantly complicates the resulting DS pattern. In order to
simplify the problem, in our experiments we used a single crystal Tb3Fe5O12 specimen shaped as a
parallel plate (measuring ≈100 µm in thickness) cut so as to let its [111] axis make an angle of ≈10°

with the normal to its plane, and one of the 〈111〈axes was directed to have a smaller tilt to the plane compared to the two others. This crystal orientation gave us a specimen with the simplest (from the point of interpretation of the experimental results) banded 180° DS of the Cotton type [4].

For the sake of illustration, Fig. 1 presents a number of photos of the specimen taken in its demagnetized state during heating from the lowest temperature attainable in the experiment Т= 85

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ferrimagnetic plate [5, 6]). The variation of DS with temperature occurs as a process of domain- boundary (DBs) hopping from one metastable position into another, the average width of the resulting domains remaining practically unvaried from case to case. Note that the positions of DBs on the specimen’s surface are somewhat different. In addition, we observe a marked irreversibility of DS in the course of heating-cooling of the specimen (more detail in [4]).

**Fig. 1. Photographs of domain structure of the specimen taken at different **
temperatures: 224(a), 245(b), 250 (c), 254 (d) and 295 K (e).

In the experiment we investigated the evolution of DS of the specimen during variations in the
temperature and magnetization of the external magnetic field and the resulting FE changes caused
by these variations. Within the framework of a two-sublattice model of REGF^{1} for the case of
nomal incidence of light on the specimen’s surface the spontaneous Faraday effect can be given by
the following [7]:

φ = ± (а MFe cosθ1 + b MR cosθ2) l, (1) where a and b are the magnetooptical coefficients depending on the light frequency, θ1, θ2 are the angles formed by vectors MFe and MR with the normal to the specimen’s plane (in our case θ1, θ2

≈ 80^{0}), l is the specimen’s thickness; the positive sign before the brackets refers to the temperatures
higher than Тс and the negative sign – to those lower than this temperature.

It is clear that in the case of a 180-degree DS, when we expose to light a specimen’s surface, whose area exceeds that of an individual domain, the FE will be given by

Ф= φ (S+– S–) / S ∝ φх, (2)

1 In this approximation Мs= MFe + MR, where MFe, MR are the spontaneous magnetic moments of the iron and RE
- sublattices, respectively (in a co-linear magnetic phase of the crystal vectors * M*Fe and

*R are antiparallel); in the external magnetic field for Т= Тс vectors MFe and MR turn through an angle of 180*

**M**^{0 }[1].

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where S is the exposed area of the specimen, S+ and S– are the areas of domains with opposite orientations of vectors Мs, and х is the average DB displacement under the action of Н with respect to the equilibrium position.

The DS of a specimen was visually examined using a ‘peek-a-boo’ method in a polarization
microscope by a conventional Faraday rotation technique [4]. The Faraday angle Ф was measured
by the method of a continuously spinning analyzer [8]. In order to increase the projection of vector
* М*s on the direction of light propagation (i.e., increasing φ), the specimen was rotated around the
axis perpendicular to the average direction of DBs, to make the angle of light incidence onto its
plane measure ≈ 15°. Magnetization of the specimen was performed by the Н ≤ 60 Oe magnetic
field oriented in its plane along the DB. A light source used was a halogen glow lamp. The main
focus of the investigations was on the temperature interval (200–295) K, including Tс of the crystal.

**2. Experimental results **

Figure 2 presents the temperature dependence of the field strength *Н*с, for which the
specimen transits into a single-domain (uniformly magnetized) state. Every point in Fig. 2 was
obtained by direct visual observation of the disappearance of DBs from the image in the course of
varying *Н and Т: in our experiment the specimen was cooled down to a certain temperature, *
following which we applied a magnetic field (vector Н was oriented along the middle direction of
DBs in the specimen’s plane) and its strength was gradually increased from zero to a value at which
no DBs were left in the image; further the specimen was warmed up to room temperature,
demagnetized in the alternating-sign magnetic field with an amplitude decreasing down to zero, and
then magnetized again at a different value of Т^{1}. The *Н**с**(Т) plot suggests that within the entire *
temperature range of interest 200 ≤ Т ≤ 295 K, except for a short (Тс ± 1.5) K (in this temperature
range we failed to maintain uniform magnetization state of the specimen up to the field strength
*Н = 60 Oe used in this experiment) a single-domain state of the specimen is formed at Н > 15 Oe, *
with the field Нс non-monotonously changing with temperature both on the right and left of Тс: for
*Т → Т*с the value of Нс first reaches its minimum and then sharply increases.

**Fig. 2. Temperature dependence of the strength of an external magnetic field under which the specimen **
transits into a single-domain (uniformly magnetized) state: calculation via Eq. (4) (solid line) and

experiment (filled circles).

It is evident from Fig. 3, which presents the FE temperature plots constructed for different magnetization field strengths, that under conditions of technological magnetization saturation the

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Ф(Т) curve has a classical form in accordance with Eq. (1), where the FE sign changes near Тс1. The investigations performed demonstrated that the behavior of the Ф(Т) curves in the region of Н < 20 Oe is most complicated: the observed dependences Ф(Т) are asymmetrical with respect to Тс; for Т

→ *Т*с the value of FE changes in a non-monotonous manner; the amplitude of the Ф(Т) curve
increases with *Н, with the maximum angle Ф approximating the value corresponding to the *
magnetic saturation condition; in contrast to the Ф(Т) curve constructed at Н= 35 Oe, from which
follows that that inversion (reorientation) of vectors MFe and MR at Т = Тс occurs abruptly, while in
weaker magnetic fields the process of reorientation of vectors * M*Fe and

*R proceeds in a certain temperature interval and is accompanied by a significant temperature hysteresis.*

**M**It is evident that at *Н< 20 Oe the shape of the Ф(Т) curve is controlled by the DS realized in *
the specimen. In particular, a decrease in the FE at the ends of the temperature interval is due to the
division of the specimen into domains of different areas with an inversely oriented projection of
vector Мs in the adjacent domains onto the direction of propagating light; the stepwise character of
the behavior of parameter Ф is controlled by the Barkhausen jumps; straight-line portions of the
Ф(Т) curve, observed at Н> 4 Oe in some temperature intervals right and left of Тс, indicate single-
domain states of the specimen formed in these temperature intervals.

**Fig. 3. Temperature dependences of the Faraday effect in the specimen under study, obtained at **
different field strength values, Н: 35(a), 15 (b), 10 (c), 3 (d) and 2 Oe (e). Arrows show the direction of

temperature variation. Horizontal axis corresponds to the zero Faraday rotation angle; the bar on the horizontal axis marks the temperature Т= 295 K.

The presence of local maxima in the Ф(Т) curve at Н< 20 Oe left and right of Tс and the non-
monotonous variation of the field *Н*с (Fig. 2) in the temperature range in question appears
somewhat unexpected, since the spontaneous magnetic moment in Tb3Fe5O12 will be approximately
linearly decreasing down to zero as a function of Т [3], as the temperature approaches the point of
compensation, and hence the pressure Р= 2Мs*Н acting on the DB will gradually decrease for Т → *
*T*с.

1 In the temperature interval in question, to a good approximation we can assume that quantities *M*Fe and *M*R

depend on Т in a linear fashion [3], which controls the linear temperature dependence of angle │Ф│observed at Н = 35 Oe (in the state of technological magnetization saturation of the specimen).

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**3. Discussion of experimental results **

In order to interpret our results, let us turn to the thermodynamic theory of a banded 180-degree
DS, since today there is no complete microscopic theory of DB mobility in REGFs [9]. It should be
noted at this point that the presence of DS at Т ≈ Tс in [4] was attributed to the transition of a crystal
from a collinear magnetic phase into an angular one, in which the angle between vectors * M*Fe and

*R is different from 180*

**M**^{01}. In particular, the latter fact results in a more complicated domain configuration of the specimen in the immediate vicinity to

*T*с: against a background of “light”

(“dark”) domains there are “dark” (“light”) wedge-like domains (see Fig. 1), whose length and
width is varied with variation in *Т. Thus, the model proposed below, which describes the effect of *
DS on FE in Tb3Fe5O12, wholly corresponds to the situation developing in the temperature range
where a banded 180-degree domain structure is realized.

It is well known that the position of DB in the external field Н in a perfect (defect-free) crystal is determined by the balance between its Zeeman and magnetostatic energies and depends on the geometry of the specimen under study [10]. In this case, the mean displacement of a DB from its equilibrium position at Н, which is directed along the easy magnetization axis (along the DB), could be derived from the following condition [10]:

2Мs*Н= kх, (3) *
where *k is the constant of a restoring force (elastic coefficient) generated in a finite-size *
specimen due to the demagnetization fields.

On the other hand, in a real specimen of a considerably large size the main reason controlling the position of DBs is the spatial distribution of the pinning centers (dislocations, non-magnetic impurity inclusions, etc.) in its crystal lattice: in terms of energy it is favorable for the DB plane to contain pinning centers [10]. It is evident that pinning of DBs plays a critical role in the processes of technological magnetization: specifically, the stepwise variation in the value of FE with temperature (see Fig. 3c-e) in weak magnetic fields is associated with the DB pinning on crystal- lattice defects. Thus, in order to initiate the motion of DBs as a single whole the external magnetic field has to exceed a certain threshold value determined by that of the pinning field Нр. Taking into account the latter consideration and relying on Eq. (3), the field of technological magnetization saturation can be given by

*Н*с = Нр+ kd/4Мs, (4)
where *d is the width of the domain (х= d/2 is the maximum DB displacement possible), and *
Eq. (2) can be rewritten as follows: where А is the proportionality factor.

*F = Ф/φ= 2АМ*s (Н – Нр)/k, (5)
If we assume the microscopic collections of non-magnetic impurities present in the crystal to
be the DB pinning centers, then Нр= γМs, where γ is a coefficient depending on the number, shape
and size of defects [10]. The temperature dependence of Мs for Tb3Fe5O12 is well known (see, e.g.,
[3]), which allows us to compare Eqs. (4) and (5) with the experimental *Н*с(Т) and Ф(Т) plots,
respectively.

The calculations via Eqs. (4) and (5) were performed using a standard likelihood-function- search routine, which ensured the least deviation of the curve from the experimental points, and the

*М*(Т) plot from [3]. The fitting parameters used were the γ, *kd and А/k, γ *

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pairs, respectively. It was assumed that coefficients γ and k do not depend on *Т, and the relatively *
weak (compared to the 1/Мs(Т) plot) variation in the value of *d within the range 200 ≤ Т ≤295 K *
(see Fig. 1) was neglected. Since it is difficult to analytically describe the variations in *М*s of the
garnet under study within the entire temperature range of interest (see [3]), we calculated the Нс(Т)
and *F(Т) curves in two stages. First, we fitted the functions (4) and (5) to the experimental plots *
*Н*с(Мs) and Ф/φ(Мs), respectively, then digitized the calculated *Н*с(Мs) and Ф/φ(Мs) curves, and
then automatically ascribed the respective temperature to every value *М*s, following which and
using an additional routine, we transformed (via interpolation) the digitized *Н*с(Мs) and Ф/φ(Мs)
plots without any scaling along the ordinate axis into smooth Нс(Т) and F(Т) curves.

The results of fitting the Нс(Т) and F(Т) functions to the experimental data are shown in Figs.

2 and 4. The experimental points in Fig. 4 were obtained using the Ф(Т) curves (Fig. 3d and *e) *
corresponding to specimen’s cooling in a state far from that of magnetization saturation (we
assumed the point of crossing between the Ф(Т) curves and the x-axis to be Tс) and the temperature
dependence of the spontaneous FE (Fig. 3a). It is important to mention that the values of the
respective fitting parameters used for constructing the theoretical curves *1 and 2 differ by about *
10%, which in the order of magnitude corresponds to the uncertainty of the measurement results.

The same refers to the values of γ, obtained via fitting the curves calculated by (4) and (5) to the experimental data, which indicates an inherent consistency of the theoretical model used.

**Fig. 4. Temperature plots of the Ф/φ ratio obtained for different field strengths, Н: 3 (curve 1) and 2 Oe **
(curve 2). Solid lines – calculations via Eq. (5), filled circles and boxes – experiment.

From a comparison of the plots presented in Figs. 2 and 4 it is evident that formulas (4) and
(5) provide a quite good description of the results of two independent experiments: the visual
examination of variations in DS of the specimen in the course of its magnetization and the FE
temperature dependence. A quite conspicuous systematic deviation of the experimental points to the
left of the calculated F(Т) curves (Fig. 4) might be accounted for by the pinning effect resulting in
the slow response of the above-mentioned stepwise variation in the DB position with lowering the
temperature. Concerning the linear quasi-horizontal portions of the Ф(Т) plot, which are observed at
*Н > 4 Oe (Fig. 3b and c), it is evident that upon reaching the magnetization saturation field Н*с,
typical of a certain temperature interval, the DB displacement х= d/2 and hence the value of FE in
this temperature range level off as well.

To sum up, we can draw a conclusion that despite a number of provisos and assumptions made here to simplify the real situation, the model of the crystal DS variation with varying Н and Т allows us to correctly describe (at least qualitatively) all of the identified regular features of the FE behavior in Tb3Fe5O12 in the course of temperature transition via the magnetic compensation point.

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Note that the FE temperature dependences similar to the plots depicted in Fig. 3 were also observed elsewhere [11] in Dy3Fe5O12 in weak magnetic fields in the vicinity of Тс. The authors of [11], however, in order to interpret the results obtained made use of the DS theory from [5], i.e., they assumed that independent of the value of Н in a certain temperature interval, including Тс, the crystal transits into a single-domain state. It is clear that within the framework of the latter approach it is hard to interpret the increase in the height of the Ф(Т) curve and the linear quasi-horizontal portions appearing on it left and right of Тс with increasing Н (unfortunately, no discussion of these effects was made in [11]).

**Acknowledgments **

This work was supported in part by a grant ЁФ2-1, Uzbekistan

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Article accepted for publication 02.03.2017