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\title{NMR Line Shape in Anisotropic Superconductors \\
in Inclined Magnetic Fields.}

\author{{\it \ Efremova S.A., Proshin Yu.N., Tsarevskii S.L.} \\
Kazan State University, Kremlevskaya, 18, Kazan 420008, Russia \\
{\em E-mail: Sergey.Tsarevskii@ksu.ru}
}

\date{{\small Received December 5, 1997\\ Revised December 23, 1997\\ Accepted
December 23, 1997}}


\begin{document}

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\markright{{\sc Magnetic Resonance in Solids.  Electronic Journal  }
{\bf 1}, {\sc 4 (1997)}\hfill}
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\maketitle

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\vbox{\parbox{2.51in}{\LARGE \bf {\it \ \ \ Volume} 1  \\  {\it No.} 4, 1997}
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\begin{center}
{\bf Abstract}
\end{center}

The nuclear magnetic resonance line shapes within a primitive cell of the
vortex lattice of the type II anisotropic superconductors in inclined
magnetic fields are constructed taking into account a change of the local
magnetic field inhomogeneity and skin - effect near the superconductor
surface. The results of the numerical calculations show that the change of
the direction of an external magnetic field modifies the parameters of the
nuclear magnetic resonance line shapes. The derivative of the power of the
absorption energy with respect to the magnetic field is calculated.
It allows to obtain more detailed information about parameters
of the superconductor such as an anisotropy parameter.


\begin{description}
  \item[PACS]: 74.60.-w; 74.60.Ge
  \item[Keywords]: high-$T_c$ superconductivity, nuclear magnetic resonance,
vortex lattice
\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

\section{Introduction.}

     The nuclear magnetic resonance ($NMR$) is well known as a
useful tool for the investigation of high-$T_c$ superconductors
properties. To interpretate the $NMR$ line shape the following
three important circumstances are necessary to take into account:
the first, homogeneous width of the line, the second, inhomogenenuity of
a local magnetic field $h({\bf r})$ in a superconductor, the third,
peculiarities of a penetration of a microwave magnetic
field in a superconductor. Because the variable electromagnetic
field penetrates into a superconductor on depth by an order of the
$\lambda $ ( $\lambda $ - the penetration depth of a magnetic
field in a superconductor) \cite{r1} only, it is necessary to
take into account the inhomogenenuity of the magnetic field
$h({\bf r})$ in a narrow area near the
superconductor surface. However, the inhomogenenuity of the magnetic
field in a vortex lattice near the surface of the type II
superconductor considerably differs from the inhomogenenuity
$h({\bf r})$ in a bulk superconductor \cite{r2}. In work
\cite{r3} in the case of the perpendicular orientation of the external
homogeneous magnetic field, the lineshape of the $NMR$ line is constructed
using a real change of the inhomogenenuity of the vortex lattice near
the superconductor surface and was shown that these changes strongly effect
on parameters of the $NMR$ line. Besides, the surface effects
drastically change the conclusions about the of the vortex
lattice and parameters of the superconductor that usually are
taken from the analysis of the $NMR$ lineshape \cite{r4}.


As High temperature superconductors are strongly anisotropic it
is interesting to consider the changes of the $NMR$ linehshapes
parameters in the inclined external magnetic fields to the
surface of the superconductor. In present work the $NMR$
lineshape is considered in view of the surface effects that
depend on angle $\theta $ between normal to a surface of the
superconductor and direction of the external magnetic field. It
is shown that the lineshape essentially is modified for various
$\theta $.


\section{Calculation of a distribution function.}

     We consider the anisotropic type II superconductor occupying
the half-space $z<0$ in external magnetic field which is directed
at angle $\theta $ ( $\theta $ - the angle between vector $H$ and
axis $z$) assuming the axis $z$ is parallel to axis $c$ of the
superconductor. If $H_{c1}<H<H_{c2}$ ( $H_{c1},H_{c2}$ - the low and
upper critical fields respectievely ) the local magnetic field
$h({\bf r})$ penetrate into the superconductor as quantum
Abricosov vortices and becomes sharply inhomogeneous. It has also
the short range order in the general case of the anisotropic
vortex lattice and the lattice period is defined by value $H$. It is
possible to expand the field $h({\bf r},\theta )$ into Fourier
series on reverse vectors ${\bf G}$ of the vortex lattice. In
work \cite{r5}, the analytical expression for the
Fourier-component of the local magnetic field as functions $z$
and $\theta -h_{{\bf G}}(z,\theta )$ is obtained on the basis of
London equations using the appropriate boundary conditions. The
map of magnetic field $h({\bf r},\theta )$ in sections $z$= const
for the certain values of $\theta $ can be restored by
summarizing the Fourier- series using the values
$h_{{\bf G}}(z,\theta )$.


     The next step of the calculations is the determination of
the distribution function $\rho (h,z,\theta )$ of the
local magnetic field $h=|{\bf h}|$ in the elementary cell of the
vortex lattice for the thin (in comparison to $\lambda$) layer
that stands on the distance $z$ from the surface of the
superconductor. The function $\rho (h,z,\theta )$ is found by the
calculation of the relative number of points in the plane $(x,y)$ in
the elementary cell of the vortex lattice for which the value of
the local magnetic field is ranged between $h$ and $h+dh$
($dh=(H-hmin)/100$, where $H=2$ - value of the external magnetic
field; the value of the minimal field in the superconductor
$h_{min}=1.8767$ was defined on depth $z=-5.0$ ( regime
''bulk''). Two wings of the function will correspond the maximum
of the magnetic field which is placed in the centre of the
vortex, and the minimum in the valley of the relief of the
distribution of the field. The peaks of the function correspond
saddle points. For the analysis $\rho (h,z,\theta )$ the maps of
the distribution of the magnetic field for the different
distances from the surface $z$ and angle $\theta $ are used. The
elementary cell of the vortex lattice is partitioned into
$512\times 512$ points at which $h({\bf r},\theta )$ were
calculated. For example, the functions $\rho (h,z,\theta )$ are
presented in Fig.1 for some values of $z$ and $\theta $. Here and
further dimensionless units are used :the distance is defined in
terms of $\lambda $ and the magnetic field - in terms
of $\Phi_0/\lambda ^2$, where $\Phi _0$ - the quantum of the magnetic
flux. The functions $\rho (h,z,\theta )$ are calculated for the
superconductor with the parameter of the anisotropy
$\Gamma =25$ ( $\Gamma =m_3/m_1$, $m_1=m_2,m_3$ -
main meanings of
''mass-tensor'' ). The parameter $\Gamma =25$ reflects the
anisotropy of the high-temperature superconductor $Y-Ba-Cu-O$
with $T_c=90K$. The distribution function of the local
magnetic field $\rho (h,z,\theta )$ was calculated with step from
$\Delta z=0.05$ from the surface of the superconductor to
$z=-0.7$. Then the step was increased because on depth of the
order $\lambda /2$ the function $\rho (h,z,\theta )$ approaches
to the distribution function density of the magnetic field in the
unlimited superconductor (curve ''bulk'') and practically not
varies on the interval from $z\leq -0.5$ to $ z=-5.0 $. As it is
shown in Fig.1 the distribution function of the local
magnetic field essentially depends on the distance from the
surface of the superconductor but also is changed with angle
$\theta $. In the case of
the inclined external magnetic field there is additional
peak in the $\rho(h,z,\theta )$ . It is connected with the
fact that in the relief of the distribution of the magnetic field
the saddle points have different heights.

\begin{figure}[tb]
\centerline{\psfig{figure=ris1d.eps,width=4in,clip=}
}
\caption{distribution function of the local magnetic field
$\rho (h,z,\theta )$ in the elementary cell of the
vortex lattice in the superconductor (in any units). The dashed
line corresponds $\theta =0$. The straight line is designed for
the case $\theta =\pi /6$. For the mode ''bulk'' there
corresponds $z=-5.0$. Along abscissa axis the meanings are drown
of the field in terms of $100(h-hmin)/(H-hmin)$.}
\label{Fig1}
\end{figure}


\section{Analysis of $NMR$ lineshape.}

Analysing the $NMR$ lineshape we take into account that the
electromagnetic microwave field penetrating into the
superconductor is changed both on size and on phase. In type II
superconductors due to the screening by superconducting currents
an alternating field penetrates into the superconductor on depth
~$\symbol{126}\lambda $ decreasing at amplitude. However, it phase
is changed on the greater depthes \cite{r1}. As a result \cite{r6},
for the $NMR$ in the type II superconductor case, if homogeneous
broadening $\Delta $ is much less a spread of the local fields the
absorbing microwave capacity is proportional to the imaginary
part of the microwave susceptibility $\chi "$ and lineshape is
defined by the distribution of the local magnetic
field. Let us calculate the power of the alternating magnetic
field which is absorbed by resonant nuclear spins if they are located in
a narrow layer $z,z+dz$. It is clear that it is proportional to
$\symbol{126}exp(2z/\delta )\rho (h,z)dz$. The exponential
multiplier takes into account that the amplitude of the alternating
magnetic field is decreased exponentially with the
removing from the superconductor surface. The $\delta $ is
equal to depth of the penetration of the alternating magnetic
field with frequency $\varpi $ in the superconductor. Let us
consider that the homogeneous broadening is described by
Lorentzian with width $\Delta $. The power of the alternating
field which is absorbed by all resonant spins as a function of the
external homogeneous magnetic field $H$ for the isotropic
$g-$factor can be written \cite{r3}:


\begin{equation}
P(H,\theta )=C\int dh\cdot \frac \Delta {\Delta ^2+(H-h)^2}\int
dz\cdot \rho (h,z,\theta )\exp \left( \frac{2z}\delta \right)
\label{e1}
\end{equation}


Here the constant $C$ is defined by the normalization condition
$\int P(H,\theta)dH=1$. In Fig.2 the calculated NMR line is
presented using the (\ref{e1}) ( the absorption energy
$P(H,q)$) for the anisotropic superconductor with parameters
$\Gamma =25,\Delta =1,\delta =1$ in the case when $\theta =\pi
/6$.) For comparison the NMR lineshape of the superconductor with
the same parameters but without taking into account changing of
the inhomogeneous field with the removing from the surface
of the superconductor, i.e. when $\rho (h,z,\theta
)=\rho(h,-\infty ,\theta )$ (dashed line). As one can see the taking
into account of the surface effects changes essentially the lineshape
in the case of inclined fields as well as in the case when the
magnetic field $H$ is parallel $z$-axis. In Fig.3 the NMR lines
are given for the different angles $\theta $ from $0$ to $\pi /2$
with step $\pi /12$. There is a small displacement of the
top-field peak and it is more pronounced for the low-field peak.
After appearance of the deviation angle $\theta $ the lineshape
sharply changed its own characteristics. The additional peak is
visible obviously and it is decreased with the increasing
$\theta $ and almost disappears at $\theta =\pi /3$. The lines of
the absorption coincide practically at $\theta =\pi /3$ and
$\theta =5\pi /12$ (i.e. when $\theta \rightarrow \pi /2$).


\begin{figure}[tb]
\centerline{\psfig{figure=ris2d.eps,width=4in,clip=}}
\caption{Dependence of the absorption energy $P(H,\theta )$ for
$\theta=\pi /6.$ The dashed line corresponds to the bulk mode
without account of the surface effects. The straight line is
depicted with taking into account the surface effects $(\Delta
=1,\delta =1)$. The designationg of the abscissa axis is the same
as in Fig.1.} \label{Fig2}
\end{figure}


\begin{figure}[tb]
\centerline{\psfig{figure=ris3d.eps,width=4in,clip=}}
\caption{The dependence of the absorption for various meanings of
angle $\theta $ in view of the surface effects ($\Delta =1$,
$\delta =1$, mode skin ): $a-\theta =0;b-\theta =\pi /12$;
$c-\theta =\pi /6,d-\theta =\pi /4$ , $e-\theta =\pi /3$ and
$\theta =5\pi /12$. The designations on x-axis meet in Fig.1.}
\label{Fig3} \end{figure}


\begin{figure}[tb]
\centerline{\psfig{figure=ris4d.eps,width=4in,clip=}}
\caption{A curve $dP/dH$ for a case $\theta =\pi /6$. The
horizontal line means $dP/dH=0$, the designations on x-axis
correspond Fig.1. The dotted line - mode bulk. The continuous
line is designed in view of the surface effects ($\delta =1$,
mode skin ).$\Delta:a-0.75,b-1$, $c-1.25,d-1.75$. $a)$ ''bulk''
-$A/B=1.99$, ''skin'' -$ A/B=4.08 $, $b)$ ''bulk'' - $A/B=2.15$,
''skin'' - $A/B=4.54$, $c)$ ''bulk''- $A/B=2.16$, ''skin'' -
$A/B=4.92$, $d)$ ''bulk'' - $A/B=2.26$, ''skin'' - $ A/B=5.53$.}
\label{Fig4}
\end{figure}


\begin{figure}[tb]
\centerline{\psfig{figure=ris5d.eps,width=4in,clip=}}
\caption{$dP/dH$ for various $\theta $ in view of the surface
effects ($\delta $= 1, $\Delta $= 1, mode ''skin'' ):
$a-\theta=0,b-\theta =\pi /12,c-\theta =\pi /6,d-\theta =\pi
/4,e-\theta =\pi /3$ and $\theta =5\pi /12$. The horizontal line
means $dP/dH=0$, the designations on it correspond Fig.1.}
\label{Fig5}
\end{figure}


The changes of the absorption features of the microwave energy
are well appreciable on the lineshape of the derivative of
absorption energy with respect to magnetic field $dP/dH$. It is
shown in Fig.4 for the different angles and various meanings of
$\Delta $ with and without account of the changes of the local
field near surface of the superconductor. $\theta=\pi /6$.
Under increasing the parameter $\Delta $ that describes homogeneous
broadening two narrow low-field peaks are merged into one broaden
and additional top-field wide peak is more seen on diagrams and
it corresponds to the absorption energy of the magnetic field
near surface. The main importance of all curves is that the
parameter of the line assymmetry $A/B$ ($A/B$- relation of the
key low-field peak to the key top-field peak in the derivative of
the absorption energy with respect to the magnetic field) grows
approximately 2 times if to take into account the change of the
inhomogeneity of the magnetic field near surface of the
superconductor. For comparison in Fig.5 the curves $dP/dH$ are
performed for the superconductor with parameters $\Gamma
=25,\Delta=1,\delta =1$ for various angles $\theta $. It is clear
seen here that with increasing of the angle the parameter $A/B$ is
sharply augmented. These significant changes of the NMR lineshape
can essentially modify conclusions about the type of the vortex
lattice. The results show that the inhomogeneity of the distribution
of the magnetic field is appreciably changed for the different orientation
 of external field $H$ with respect to the boundary of the surface of the
anisotropic superconductor. So, the parameters of the line can be
vary in perceptible limits. Thus the $NMR$ method is very
sensitive to the features of the magnetic field distribution of the
type II superconductor. The $NMR$ probing in the inclined
magnetic fields will allow to obtain more detailed information
about the parameters of the superconductor (such as a parameter
of anisotropy $\Gamma $).




\newpage
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\bibitem{r4}  MacLauglin D. {\it Sol. St. Phys.} {\bf 31}, 1 (1976)

\bibitem{r5}  Kogan V.G., Simonov A.Yu., and Ledvij M.
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\bibitem{r6}  Kochelaev B.I., Khusainov M.G.
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\end{thebibliography}

\end{document}
