5_9_2.tex

%translator Savrov, date 09.04.13

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\Work
{Study of domain structure of ferrimagnetic film of iron-yttrium garnet}
{Study of domain structure of ferrimagnetic film of iron-yttrium garnet}
{Various domain structures in ferrimagnetic film placed in magnetic field are observed via the Faraday effect.}

\vspace{8pt}
\textbf{Domain structure of ferromagnet.}
The magnetic moments of atoms of a ferromagnet below the Curie temperature $T_\textrm{C}$ are parallel due to exchange interaction; this is a quantum-mechanical effect which combines the Coulomb interaction and the Pauli exclusion principle. 

According to the Pauli principle there can be not more than two electrons with opposite spins in the same quantum state. This is true for electron states in molecules and solids. Suppose that two neighboring atoms have one unpaired electron each. If the spins of these electrons are antiparallel their wavefunctions tend to overlap and form a single quantum state. If the spins are parallel the overlap is suppressed and the electrons tend to remain in their individual states. Obviously, the electrostatic energy of these spin configurations is different. For instance, this is the case for hydrogen molecule. The energy of solid state also depends on spin configuration (or magnetic moments since each spin is related to magnetic moment) of electrons and atoms. This energy is called the {\it exchange energy.}

In a ferromagnet the exchange energy is minimal when the magnetic moments of electrons are parallel, which results in spontaneous magnetization at $T<T_\textrm{C}$.

The exchange interaction is isotropic, so if only this interaction is present in a ferromagnet the direction of its magnetization is arbitrary. However any crystal is anisotropic. Because of the anisotropy a ferromagnet is always magnetized in some direction. Usually the direction coincides with a principal axis. For instance, the magnetization of a cubic ferromagnet can be directed along the cube edge, a side diagonal, or the main cube diagonal. In a uniaxial crystal the magnetization is directed either along the principal axis (the so called \emph{easy axis}) or in the plane perpendicular to the principal axis (the \emph{easy plane}). Thus a typical ferromagnet can be easily magnetized in one direction while there are directions in which magnetization is difficult. 

The origin of magnetic anisotropy is easy to understand. Magnetization emerges as the sum of magnetic moments of atoms (ions) of a crystal. Electrons responsible for magnetization have both spin and orbital angular momentum and the latter depends on the structure of crystal lattice. Thus the spin magnetic moments are directed along certain crystallographic axes due to spin-orbital interaction.

Macroscopically the exchange interaction is specified by exchange energy$W_\textrm{e}$, and the anisotropy of ferromagnetic properties is specified by the energy of anisotropy $W_\textrm{a}$. Obviously, both energies depend on magnetization. Let us expand $W_\textrm{a}$ and $W_\textrm{e}$ in powers of $\textbf{M}$ and its component $M_z$. For simplicity consider a uniaxial ferromagnet, then
$$
W_\textrm{e}={\alpha { M}^2\over2}+...  \eqMark{9_1_28}
$$
and
$$
W_\textrm{a}={\beta { M_z}^2\over2}+...,  \eqMark{9_1_29}
$$
where $\alpha$ and  $\beta$~are the expansion coefficients called the exchange constant and the constant of anisotropy, respectively. Notice that Eqs.~(\refEquation{9_1_28}) and~(\refEquation{9_1_29}) contain only even powers of magnetization because the energy must be positive. Besides, the exchange interaction is isotropic, so it depends only on the magnetization magnitude, and magnetization is axial vector.

Any real sample of ferromagnet is of finite size. Such a sample becomes a permanent magnet at $T<T_\textrm{C}$ due to exchange interaction. Therefore the sample is surrounded by magnetic field which energy $W_\textrm{m}$ is called the \emph{energy of magneto-dipole interaction}. (A permanent magnet can be considered as a <<magnetic dipole>> composed of two magnetic charges. Of course, magnetic charges do not exist but it is often convenient to introduce them in order to draw an analogy between electrostatics and magnetostatics.) The term $W_\textrm{m}$ is also called the \emph{energy of demagnetizing field} because the magnetic induction outside the sample opposes the magnetization vector. If the sample magnetization is saturated (all magnetic moments are aligned), $W_\textrm{m}$ is proportional to the ferromagnet volume. In general $W_\textrm{m}$ is proportional to the magnetic induction outside the sample.

The competition between three interactions determines the magnetic structure of a ferromagnet sample. In particular, the competition leads to formation of domains of spontaneous magnetization. In the absence of external field a ferromagnet sample is divided into the domains each of which is magnetized to saturation. The domain magnetization is random and mostly directed along an easy axis. There is a transition layer between domains (\emph{domain wall}) in which the magnetization vector gradually rotates (see~\refFigure{9_2_1}).
%
\fFigure{Magnetization inside domain wall.}9_2_1
{4.3cm}{3.cm}{pic/domen1.eps}
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The domain structure of a sample can be quite complicated. It depends on the crystal anisotropy, on its size, and shape. A particular domain structure corresponds to the minimum of the sum of exchange energy, energy of anisotropy, energy of demagnetization field, and the energy of domain walls. 

Suppose that magnetization is nonuniform over a sample. Then it is possible to create a configuration for which the energy $W_\textrm{m}$ vanishes. In such a configuration the magnetization significantly deviates from the directions corresponding to the minimal energy of anisotropy, so $W_\textrm{a}$ is proportional to the sample volume. One can see that a completely nonuniform magnetization is not energetically favorable either. 

Simultaneous minimization of the energy of exchange, anisotropy, and demagnetizing field is not possible. In 1935 L.~Landau and E.~Lifshitz showed that the minimum corresponds to the domain structure described above: a sample is divided into domains of uniform magnetization separated
by domain walls which are of macroscopic size, i.e. the wall thickness significantly exceeds the interatomic distance. 
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\fFigure{Domain structure of ferromagnetic plate: {\it a}~magnetic flux is not closed, {\it b}~magnetic flux is closed via prismatic surface closing domains.}9_2_2
{5.08cm}{5.84cm}{pic/l09_2_02.eps}
%
It is not difficult to observe the domains. In 1932 F.~Bitter, L.~Chamos, and P.~Tissen observed magnetic domains in microscope. In their experiment a fine powder of ferromagnetic particles was sprayed on the surface of a ferromagnet. The particles concentrated mostly near the domain walls where the magnetization is nonuniform and there is a magnetic field. The method is now known as the method of powder figures

Consider formation of a domain structure in an ideal (without defects) uniaxial ferromagnetic plate which surface is perpendicular to the anisotropy axis ($Z$-axis). Assume that the plate thickness (along $Z$) is $h$ while a perpendicular dimension is very large. According to Eq.~(2) for $\beta<0$ and in the absence of external magnetic field the magnetization is either along or opposite $Z$. Obviously, the state of an equal number of domains with ${M_z=+M_0}$ and ${M_z=-M_0}$ is energetically favorable providing the domains alternate as it is shown in~\refFigure{9_2_2}{\it a}. The energy of this state is the sum of the energy of demagnetizing field, which is mostly concentrated near the surface, and the energy of domain walls. This particular domain structure belongs to the open magnetic flux type (the magnetic field lines do not close inside the sample). However, in a uniaxial ferromagnet a structure of the closed magnetic flux type is often more energetically favorable (see~\refFigure{9_2_2}{\it b}). The difference is the presence of triangular domains closing the magnetic flux inside the crystal. As a result, the magnetizing field outside the ferromagnet disappears. But the energy of magnetic anisotropy increases because the magnetization of the triangular domains is perpendicular to the direction of the minimal energy of magnetic anisotropy. Which particular structure is realized depends on the energy of anisotropy (the value of $\beta$ in Eq.~(2)).

\vspace{1ex}
\textbf{Magnetic domains in ferrimagnetic film.}
Similar to ferromagnet a spontaneous ordering of magnetic moments takes place in ferrimagnet.
The word <<ferri>> indicates that there are two or more magnetic sublattices which moments are opposite like in antiferromagnet. However, their magnitudes are different and the net magnetic moment is nonzero. Thus a ferrimagnet is similar to a ferromagnet in several aspects: it has a spontaneous magnetization and the Curie point. A typical ferrimagnet is a ferri-garnet which chemical composition is 3Re$_2$O$_3\cdot$5Fe$_2$O$_3$, where Re stands for a rare-earth element, e.g. Y or Ho.

In a uniaxial crystal film one can often observe meander domains, i.e. randomly curved stripes in the film plane. They form because the domain walls do not have a preferred direction in the film plane. The stripes bending is due to minor film inhomogeneities, thermal randomization effects, etc. (see~ Fig.~.3). The equilibrium stripe width $d\sim\sqrt h$, where $h$~is the film thickness (see the Appendix for the details).

The meander structure remains energetically favorable even in an external magnetic field  perpendicular to the film surface. The greater the field, the larger are the domains with {\bf
M} along the magnetic induction {\bf B} while the size of the domains with {\bf M} opposite to {\bf B}declines. At some {\bf B} the size of the latter shrinks to $l_0=\sigma/M_0^2$, where $\sigma$~is the energy of domain wall per unit area and $M_0$~is the domain magnetization. When this happens a stripe splits into separate magnetic bubbles (MB) of circular shape (see~\refFigure{9_2_3}). The magnetostatic interaction between MBs is repulsive (actually they are small magnets with their magnetic moments aligned; no wonder they repel each other) so they spread evenly over the film, usually MBs form a hexagonal lattice.

MBs possess peculiar properties which distinguish them from another domain structure. For instance, the internal magnetic field must be zero in the meander structure, while the internal field in the MB structure must be nonzero since the domain wall is curved.
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\begin{cFigures}
\Figure
{Magnetic bubbles in a film.}9_2_3
[t]{4.47cm}{2.58cm}{pic/L09_2_03.eps}\quad
\Figure
{Isolated magnetic bubble in a film. Arrow corresponds to magnetization.}9_2_4
[t]{6.2cm}{2.72cm}{pic/L09_2_04.eps}
\end{cFigures}
% 
Otherwise the structure is unstable. An MB resembles a gas bubble in liquid. Bubble stability requires the pressure inside the bubble to be greater than that of liquid otherwise it collapses. The stability of MB requires an internal magnetic field to exert an extra pressure on the curved domain wall. The name <<magnetic bubble>> is due to this analogy. The domain wall energy is like a surface tension in liquid, it shapes the domain into cylinder because another shape (e.g. elliptic) increases the energy.

Consider an isolated MB in some detail (see~\refFigure{9_2_4}). The equilibrium shape of MB is due to competition between the surface energy of domain wall, which tends to decrease the domain radius, and the magnetostatic energy tending to increase it since the demagnetizing field inside the domain is directed along the magnetization outside it. A magnetic pole on the MB butt has the opposite sign to that of the poles surrounding it thereby reducing the net demagnetizing field and the magnetostatic energy.
%
\hFigure{Fragmentation of meander structure into MBs when external magnetic field increases from 0 to 70\,Oe}9_2_5
{8.7cm}{4.7cm}{pic/L09_2_05.eps}
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%
\fFigure{Fragmentation of meander structure into MBs by compressing:
{\emph{a}}~is initial stripe domain; {\emph{b}--\emph{c}}~is undulant compression; {\emph{d}}~is breaking; {\emph{e}}~is formation of MBs}9_2_6
{3.6cm}{5.2cm}{pic/L09_2_06.eps}
%

To maintain MB in static equilibrium the net magnetic field described above must be balanced by an external magnetic field. In a small field MB structure becomes unstable and turns into  meander structure while in a large field the domain radius decreases and MBs vanish, this corresponds to a homogeneous state without domains. The diagram in~\refFigure{9_2_5} illustrates the behavior of domain structure in a thin ferrimagnetic film. Formation of MB structure is due to two reasons. Firstly, small film inhomogeneities (or magnetic impurities) of the size of MB favor their formation. Secondly, MB formation can be driven by an external factor, e.g. by dispersion of small ferromagnetic particles on the film surface or by placing the film in a pulsed magnetic field. 

Consider what happens when a pulsed magnetic film is applied. When the field increases the domain stripes shown in~\refFigure{9_2_5} narrow. However, narrowing is not uniform along the stripe (see~\refFigure{9_2_6}).

\vspace{1ex}

\textbf{The Faraday effect.} The domain structure of a ferrimagnetic film can be observed in a microscope by illuminating the film with a polarized light. The method is based on Faraday rotation.

%
\fFigure{Refraction index of light with different circular polarization versus frequency}9_2_7
{5.64cm}{2.43cm}{pic/L09_2_07.eps}
%
To describe the Faraday effect, i.e. rotation of the polarization vector of a light wave passing through a ferromagnet, we will use the classical theory of light dispersion. In this theory electrons of the external atomic shell (optical electrons) are considered as isotropic oscillators specified by a single eigenfrequency~$\omega_0$. The refraction index $n(\omega)$ is independent of polarization. In external magnetic field (in our case the magnetization field is already present in a material) the oscillator eigenfrequency splits. The oscillator eigenmodes in a magnetic field are the oscillations along the field with the same frequency~$\omega_0$ and a rotation in the perpendicular plane with a frequency $\omega_\pm=\sqrt{\omega_0^2+\Omega^2}\pm\Omega\Simeq\omega_0\pm\Omega$. The higher frequency $\omega_{+}=\omega_0+\Omega$ corresponds to electron rotation in the direction of the current generating the magnetic induction. Electron rotation of atomic oscillator with the Larmor frequency $\Omega = eB/(mc)$ is excited by a circularly polarized field propagating in the material.

The calculation similar to the calculation of refraction index in molecular optics gives
\vspace{-7pt}
$$
n_\pm(\omega)=n(\omega\pm\Omega)\Simeq
n(\omega)\pm\Omega {dn(\fomega)\over d\fomega} .\eqMark{9_1_30}
$$

\vspace{-0pt}
Since a linearly polarized wave can be represented as a linear superposition of two circularly polarized waves with oppositely rotating electric field we obtain for the rotation angle $\phi$ of the polarization plane (the wave propagates along $z$-axiz)
$$
\phi={\fomega\over 2c}(n_--n_+)z={\fomega\over c}\Omega{dn(\fomega)\over
d\fomega}z = \frac{e\fomega}{2mc} \frac{dn}{d\fomega} Bz  .\eqMark{9_1_31}
$$
One can see that $\phi$ is proportional to magnetic induction $B$ and the covered distance $z=l$.

The diagram in~\refFigure{9_2_7} shows the dependence of refraction index on frequency for the  waves with right and left circular polarization propagating along a magnetic field.

In the case of normal dispersion the refraction index grows with frequency 
($dn/d\omega>0$), while for anomalous dispersion it is the other way around.\looseness=-1

\vspace{1ex}

\textbf{Installation for observation of domain structure in ferrimagnetic film.}
The rotation angle of polarization plane of light passing through a magnetized film is  
$$
\alpha=\alpha_0h \frac{(\textbf{nM})}{M_s}. \eqMark{9_1_32}
$$
where $\alpha_0$~is the specific Faraday rotation of the material magnetized to saturation, {\bf n}~is a unit vector in the direction of light propagation, {\bf M}~is the magnetic moment per unit volume, ${M}_s$~is the magnetization of saturation, and $h$~is the film thickness.

An individual domain is magnetized to saturation. When light passes through a domain, the polarization plane rotates either by $+\alpha l$ or $-\alpha l$ depending on the direction of {\bf M}, whether it is parallel or antiparallel to that of light propagation; in other words, oppositely magnetized domains rotate the polarization plane in opposite directions.

For a typical epitaxial garnet structure $\alpha_0$ is about $(10^3\div 10^4)$\,deg/cm in the optical range, so $\alpha_{\rm f}=(1\div 10)$\,deg for a layer of $h=10$\,$\mu$m thick. The best contrast of the domain pattern is achieved when the angle between an analyzer and the crossed position of a polarizer is $\phi\Simeq2$\,deg. In so doing one sees a pattern of dark and light domains. According to Malus' law the intensity of light incident on the polarizer is reduced by factor $\cos^2\phi$, where $\phi$~is the angle between the light polarization plane and the analyzer axis.

Since the width of observed domains is several microns they can be observed in a microscope with a magnification of 100x. A microscope <<Áèîìåä Ñ-1>> is used in our installation. The installation is shown in~\refFigure{9_2_8}.

The current flowing through a magnetizing coil with the sample inside generates a constant magnetic field. When the field increases the domains magnetized along the field grow while the oppositely magnetized domains shrink. When the field reaches the saturation, the domain pattern completely vanishes because the sample now becomes the single domain.

\vspace{1ex}
\so{\textbf{Directions}}
\vspace{2pt}

\begin{Enumerate}{tab}
\Item.
In this experiment one studies behavior of the domain structure of a film of iron-yttrium garnet Y$_3$Fe$_5$O$_{12}$ in an external magnetic field. 
%
\fFigure{Installation for observation of magnetic domains: \emph{1}~is a light source; \emph{2}~is a concave mirror; \emph{3}~is a polarizer;  \emph{4}~is a magnetizing coil; \emph{5}~is a sample; \emph{6}~is an analyzer; and \emph{7}~is a microscope}9_2_8
{5.09cm}{10.6cm}{pic/L09_2_08.eps}
%
The angle between the polarizer and the analyzer is already adjusted and should not be altered. 

\Item.
Obtain the sharp image of the domain pattern at zero current by adjusting the microscope eyepiece. Gradually increase the current and observe how the pattern is evolving. Reverse the field direction and observe the pattern evolution.

\Item.
The size of the microscope field of vision is 1\,mm. Measure the domain width as a function of the magnetic induction. The calibration curve relating the induction and the current is provided. The domain size can be measured with the aid of a movable micrometer scale located in the field of vision. Plot the dependence
$$
{I_0-I\over I_0+I}\,,
$$
where $I$~is a current through the coil and $I_0$~is the current at which the dark stripes vanish. The current should be gradually increased or decreased during the measurement because of hysteresis. 

\Item.
To observe MBs increase the field until the sample becomes a single domain (this corresponds to a uniform field of vision), then slowly decrease the field and observe the MBs to emerge. Record the induction at which MBs become elliptical and turn into meander domains (MBs serve as their nuclei). This induction is called the induction of elliptical instability.

\Item.
Slowly increase the field and observe restoration of MBs and their collapse.

\Item.
Measure the MB size and its dependence on magnetic induction. Estimate the film thickness using the results.
\end{Enumerate}%

\begin{center}\so{\textsf{\small ËÈÒÅÐÀÒÓÐÀ}}\end{center}
{\small

1. \emph{Öèïåíþê\;Þ.\;Ì.} Êâàíòîâàÿ ìèêðî- è ìàêðîôèçèêà.\,---\,Ì.: Ôèçìàòêíèãà, 2006.

2. \emph{Êèíãñåï\;À.\;Ñ., Ëîêøèí\;Ã.\;Ð., Îëüõîâ\;Î.\;À.} Îñíîâû ôèçèêè. Ò.\,1.~--- Ì.: Ôèçìàòëèò, 2007.

3. \emph{Êðèí÷èê\;Ã.\;Ñ.} Ôèçèêà ìàãíèòíûõ ÿâëåíèé.\,---\,Ì.: Èçä. Ìîñê. óí-òà, 1985.

4. \emph{Ìèøèí\;Ä.\;Ä.} Ìàãíèòíûå ìàòåðèàëû.\,---\,Ì.: Âûñøàÿ øêîëà, 1981.

5. \emph{Î`Äåëë\;Ò.}  Ìàãíèòíûå äîìåíû âûñîêîé ïîäâèæíîñòè.\,---\,Ì.: Ìèð, 1978.
}

\vspace{2ex}
\textbf{\so{Appendix}}
\vspace{4pt}

{I. \it Îöåíêà ðàçìåðîâ ïîëîñîâîãî äîìåíà.}
The width of meander domain in a thin ferrimagnetic film can be estimated by considering the balance between the surface energy of domain wall and the energy of demagnetizing field outside the film. The magnetization vector {\bf M} in a thin film is perpendicular to its surface. To simplify the calculation assume that the domains are the parallel stripes of width {\it d} extending all the way along the film. The magnetization {\bf M} of the neighboring domains is opposite as it is shown in~\refFigure{9_2_9}.

The energy of domain wall in this configuration is 
$$
W_s=\frac{A}{d}h\sigma .
$$
Here $A$~is the film area, $h$~is the film width, and $\sigma$~is the wall energy per unit area.

To estimate the energy of the demagnetizing field created by the domains outside the film assume that the field lines are semicircles, as it is shown in~\refFigure{9_2_9}.
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\cFigure{Magnetic field of meander domains.}9_2_9
{7.64cm}{2.7cm}{pic/L09_2_09.eps}
%
The adopted approximation is good enough since it takes into account that the field lines are mostly concentrated near the surface. Thus, 
$$
W_{\rm m}=2\frac{A}{dL}\cdot\frac{\fpi d^2}{8} L\cdot\frac{(4\fpi M)^2}{8\fpi}=
\frac{\fpi^2}{2}AM^2d.
$$
Here $L$~is the domain length. The first factor equals the number of semicylinders formed by the field lines, the factor 2 takes into account that the film has two surfaces. The second factor is the volume of magnetic field semicylinder. The third one is the magnetic field energy density; here one sets $B=4\pi M$ because the normal component of magnetic induction must be continuous. It is easy to verify that the sum $W_s+W_{\rm m}$ is minimal at
$$
d_0={\sqrt{2h\sigma}\over\pi M}\,,
$$
i.e. the domain width is proportional to the square root of the film thickness.

\medskip
II. \emph{Dependence of domain size on external magnetic field.}
In an external magnetic field {\textbf{H}} the magnetization \textbf{M} tends to align with the field (recall that the energy of magnetic dipole is minimal in this position). Therefore the domains with \textbf{M} along the field grow and the domains of opposite orientation shrink. A macroscopic magnetization {{\textbf{I}}} is proportional to {\textbf{H}} and equal to the microscopic magnetization  \textbf{M} averaged over the film:
$$
\textbf{I}=\chi{\textbf{H}}=\textbf{M}{A_+-A_-\over A_++A_-}.
$$
Here $\chi$~is the magnetic susceptibility, $A_\pm$~is the area of the domains with \textbf{M} along and opposite to {\textbf{H}}, respectively. According to this equation the ratio of the areas is
$$
{A_-\over A_+}={\textbf{M}-\chi{\textbf{H}}\over\textbf{M}+\chi{\textbf{H}}}.
$$
Taking into account that {\textbf{H}} is proportional to the coil current {\it I} one can rewrite the last equality as
$$
{A_-\over A_+}={I_0-I\over I_0+I},
$$
where $I_0$~is the current at which the film becomes a single domain magnetized along {\textbf{H}}. One should verify this relation by measuring the ratio of the domain areas as a function of the current.