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\title{Temperature dependencies of g-factors in GdBa$_2$Cu$_3$O$_{6+x}$ crystal.}

\author{{\it \ R. M. Eremina } \\
Kazan Physical-Technical Institute, 420029 Sibirskii trakt 10/7, Russia \\
{\em }
}
\date{{\small Received November 22, 1997\\ Accepted
December 11, 1997}}

\begin{document}

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\markright{{\sc Magnetic Resonance in Solids.  Electronic Journal  }
{\bf 1}, {\sc 3 (1997)}\hfill}
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\maketitle

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\vspace*{1in}

\begin{center}
{\bf Abstract}
\end{center}

The temperature dependencies of the g - factors in GdBa$_2$Cu$_3$O$_{6+x}$
crystal are successfully explained using the suggestion that the big
macroscopic magnetization of the samples due to the high concentration of Gd$%
^{3+}$ ions (100 \%) is responsible for the ESR signal. The difference of
the temperature behavior of g-factors on the 35 GHz and 9.3 GHz is due to a
different fraction of the penetration depth of the field in the
superconductor.

\begin{description}
  \item[PACS]: 74.25.Jb; 74.72.Bk
  \item[Keywords]: high-T$_{c}$ superconductors, EPR
\end{description}

\bigskip
Selected for publication by Programme Committee of youth 
scientific school "Actual problems of a magnetic resonance 
and its applications: Magnetic resonance 
in high $T_c$ materials", Kazan, November 20-22, 1997


\newpage


In present communication the trying was undertaken to explain the unusual
behavior of the ESR spectra in high-T$_c$ superconductor GdBa$_2$Cu$_3$O$%
_{6+x}$ that were observed by Baranov and Badalyan \cite{baran}. In spite of
a big attention attracted by those results \cite{barans}, their origin is
still puzzled for high-T$_c$ community. In papers \cite{baran}-\cite{barani}
was noted that the g-factors of the ESR line depend strongly on the
temperature and their dependencies are different at X- and Q-bands.

\begin{figure}[htbp]
\centerline{\psfig{figure=fmrs1.eps,clip=}}
\caption{The temperature dependencies of the g-factors in GdBa$_2$Cu$_3$O$_{6+x}$.
Circles Q-band; triangles X-band; open circles and triangles correspond
magnetic field is parallel to $c$ - axes; filled circles and triangles,
correspond magnetic field is perpendicular to $c$ - axes. The calculated g -
factors is presented by straight line.}
\label{FigZB}
\end{figure}


In Fig. \ref{FigZB} is shown the g-factor temperature dependencies for GdBa$_2$Cu$_3$O$
_{6+x}$ crystal \cite{barans}. Circles represent Q-band; triangles represent
X-band; open circles and triangles correspond magnetic field is parallel to $
c$ - axes; filled circles and triangles, correspond magnetic field is
perpendicular to $c$ - axes. The samples have been heated up to 500-800$^0$C
and then rapidly quenched to 300 or 77K by dropping into water or liquid
nitrogen. The magnetic resonance signal appears below 40K and its intensity
increases more rapidly that it would be expected according to the Curie law.
The signal intensity corresponds the concentration of the spins more than 10$
^{20}$. It is natural to connect this fact with appearance in the sample the
new metastable copper-oxygen clusters or magnetic polarons.

Here we suggest the present ESR signal is directly related to macroscopic
magnetization. And the sample shape influences on the resonance field $B_0$
\cite{korabl,kittel,kondal}. In experiments \cite{baran}-
\cite{barani} the samples are the thin plates. In this case, the conditions
for the resonance fields at the operating frequency $\omega _0/2\pi $ are
\cite{kittel}, $\omega _0=\gamma (B_0+4\pi M)$ and $\left( \frac{\omega _0}
\gamma \right) ^2=B_0(B_0-4\pi M)$, where $\gamma $ is the gyromagnetic
ratio and the $M$ are the magnetizations at the respective resonance fields $
B_0$ for the applied magnetic field parallel and perpendicular to the $c$
axis respectively.

The magnetization of the single crystal GdBa$_2$Cu$_3$O$_{6+x}$ contains two
terms: the first is the magnetization of the Gd$^{3+}$ ions and the second
is the magnetization of the copper-oxygen clusters \cite{hs,Aristov,optica,erem}.
The copper-oxygen cluster with S=2 has been used for the fitting of the low
temperature ESR signal intensity in YBa$_2$Cu$_3$O$_{6.25}$ and it
deviation from Curie law \cite{rerem}. The temperature
dependence of the magnetization has been calculated as follows:
\begin{equation}
M=-\frac{Ng\beta H}V\frac{\sum_{i=-S}^{S} m_i \exp (-\frac{\varepsilon_i}{kT})}
{\sum_{i=-S}^{S} \exp (-\frac{\varepsilon_i}{kT})}
\label{fir}
\end{equation}
where N - is a number of spins, V - is a sample volume, $\varepsilon _i$ are
the energy level, m$_i$ - are the projections of the magnetic moment on the
chosen direction.

The spin Hamiltonian for the Gd$^{3+}$ ion (S=7/2) can be written:
\begin{equation}
H=b_2^0O_2^0+b_4^0O_4^0+b_4^4O_4^4+g\beta B_0S  \label{spin}
\end{equation}
The crystal field parameters are $b_2^0=-18000MHz,b_4^0=-180MHz,b_4^4=600MHz$
\cite{shaltel}. The effective g-values of $Gd^{3+}$ is 2.17 and was measured
in the samples with temperature below 100 K \cite{kessler}.

Among all possible copper-oxygen clusters the most stable has $S=2$. It
energy spectrum is described by spin-Hamiltonian \cite{RME}:
\begin{equation}
H=DS_z^2+g\beta B_0S  \label{sph}
\end{equation}
where $D=0.08meV,g_{\| }=2.25,g_{\bot }=2.05$ \cite{rerem}.

I interpreted the experimental data using formulas (\ref{spin}), (\ref{sph})
and taking into account that single crystals GdBa$_2$Cu$_3$O$_{6+x}$ are the
II-type superconductors and the magnetic field penetrates into the sample by
vortexes \cite{Abrikosov}. The fraction of the normal phase in the sample is
proportional $\xi ^2n\lambda $, where $\xi $ - a coherence length, $n$ - a
vortexes density and $\lambda $ is a penetration depth.

Let me consider now the ratio of the normal phase fractions in the case of
the parallel and perpendicular sample orientations in the Q-band
experiments. The values $\xi _{\| }=4A$, $\xi _{\bot }=12A$ are taken
from \cite{Sokolov}, and $\lambda _{\| }=3\mu m$, $\lambda _{\bot
}=0.6\mu m$ are from \cite{Athan}. For the temperature $T=1.6K$ the magnetic
resonance fields in Q-band for the parallel and perpendicular orientations
equal $B_{\| }(Q) \sim 18000\cdot Oe$, $B_{\bot }(Q)\sim
9000\cdot Oe$ respectively. Therefore, the ratio of the normal phase
fractions is near 1 that means the ratio of the spin concentration $N/V$ in
the expression for the magnetization is constant for the parallel
orientation the same as for the perpendicular. The situation is changed
drastically in the case of the analysis of the experimental results in the
X-band. The vortexes density decreases because $B_{\| }(X)\sim
3000\cdot Oe$, $B_{\bot }(X)\sim 3000\cdot Oe$. In comparison to Q -
band the normal phase fraction for the parallel orientation decreases six
times at the same time as for the perpendicular orientation the fraction
decreases three times only. Consequently, the same fall is for the spin
concentration $N/V$ that gives the contribution to the macroscopic
magnetization $M$.

The calculated temperature dependencies of g - factors is presented in Fig.,
straight line. The effective gyromagnetic ratios of the magnetic polaron is
accepted to be equal $\left( \frac{\gamma h}\beta \right) _{\| }=2.17$, $
\left( \frac{\gamma h}\beta \right) _{\bot }=2.1$. The Gd$^{3+}$ ions
concentration is 56$\cdot 10^{20}$, and copper-oxygen clusters - 31$\cdot
10^{20}$ (Q-band). As one can see, the calculated results well agree with
experimental data \cite{baran}-\cite{barani}.

In conclusion, the results of the present work can be formulated as follows:

i. The temperature dependencies of the g - factors observed in \cite{baran}-
\cite{barani} can be successfully explained using the suggestion that the
big macroscopic magnetization of the samples due to the high concentration
of Gd$^{3+}$ ions (100 %) is responsible for the ESR signal.

ii. The difference of the temperature behavior of g-factors on the 35 GHz
and 9.3 GHz is due to a different fraction of the penetration depth of the
field in the superconductor.

I am grateful to Dr. I. N. Kurkin to bring my attention on the role of the
demagnetization factors and paper \cite{korabl}. This work was supported by
ISSEP ''Soros graduate students '' program (Grant No. a97-2593).

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