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\begin{document}
\title{Intrinsic anomalous behaviour of an 'exotic' superconductor $URu_{2}Si_{2}$ }
\author{ Seema Bahinipati\\
        \\
        Department of Physics \\
        University of Cincinnati \\
        Cincinnati, Ohio 45221\\[1em]}
\date {February 25, 2002 }
\maketitle

\medskip
\begin{abstract}

$URu_{2}Si_{2}$ is an 'atypical' heavy fermion superconductor,which has 
unusual superconducting and normal state physical properties.These 
properties include a moderately high specific heat coefficient,large value 
of electrical resistivity and large magnetic susceptibility.Tranport,thermal
and magnetic data for $URu_{2}Si_{2}$ indicate an energy gap of 
$\sim11 meV $ over a portion of the Fermi surface.It also exhibits anisotropic 
Josephson effects in point contacts with Nb.Most recent works suggest 
that below $T_{N}\sim17 K$,the $^{29}$Si NMR line in $URu_{2}Si_{2}$ exhibits 
a previously unobserved field-independent nearly isotropic contribution to 
the line width,which increases to $\sim12G$ as $T \rightarrow 0$.
\end{abstract}
\newpage
\section{Introduction}
Superconductors,materials that have no resistance to the flow of 
electricity,below a certain temperature,are one of the great frontiers of 
scientific discovery.Since 1911,when superconductivity was first observed 
by H.K.Onnes in Hg at 4 K,several classes of new superconductors have been discovered.These include high-T${c}$ superconductors(with$T_{c}\sim138 K$) and several 'atypical superconductors'like ceramic superconductors,organic superconductors,Borocarbides,heavy fermions and so on.{\bf'Heavy Fermions'} are intermetallic compounds in which the effective mass of the conduction electrons is several hundred times greater than that of a free electron.One of the constituents of such compounds is usually a massive f-atom element with partially filled f-orbital(eg. U),and the rest are non-f atom elements.Superconductivity in such compounds was first discovered in 1979(Steglich et.al,1979).They exhibit low-temperature superconductivity,usually in the range of 
Type {\bf I} superconductors,which severly limits their usefulness.However,they possess certain anomalous properties- both normal state and superconducting 
,due to which they are often called "exotic superconductors".We will study these properties in one such heavy-fermion superconductor,$URu_{2}Si_{2}$.
\section{Crystallography of $URu_{2}Si_{2}$}
$URu_{2}Si_{2}$ is one of the heavy fermion superconductors,in which the 
effective mass of the conduction electrons is roughly 1000 times the mass 
of an electron.It was reported to be superconducting at $\sim1.5 K$ in 
the year 1985 by Schlabitz and his co-workers.\\
Crystal system:Tetragonal\\
Space group:P4/mmm\\
Lattice parameters:(in \AA)\\
a=b=4.129\\
c=9.575\\
\begin{figure}[htb]
\epsfysize=2.0in
\centerline {\epsffile{figurer1.ps}}
%\centerline{\psfig{figure=figurer1.ps,height=1.5in}}
\caption[]{\em{Crystallographic structure of $URu_{2}Si_{2}$\cite{B}}}
%\label{models}
\end{figure}
\section{ Resistivity and Magnetic susceptibility }
Electrical resistivity $\rho$ vs temperature T data between $\sim 80 K$ 
and 300 K is shown in Fig.2(a).The shape of  curve is similar to that of many 
hybridized f-electron rare-earth and actinide compounds,particularly the 
rapid decrease in with decreasing temperature. has a large magnitude of 
2.46 m$\Omega$-cm at 300 K.There is an abrubt drop in below 1.7 K,due to the 
onset of superconductivity.The small ,but sharp peak at 17.2 K resembles the 
type of anomaly that would be expected for a charge- or spin- density 
wave(CDW or SDW) transition.The anomaly in at 17.2 K is insensitive to 
applied magnetic fields,shifting downwards in temperature by only $\sim0.1 K$
in a field of 6T(see inset of Fig.2(a)).There is an approximate linear 
dependance of with T between $T_{c}$ and $\sim8 K$.The magnetic susceptibility $\chi$ vs T of $URu_{2}Si_{2}$ between 2 and 300 K is shown in Fig.2(b).There is a rounded maximum near 55 K in the $\chi(T)$ data,which correlates with a similar maximum of 70 K in the $\rho$(T) data.A small but distinct change in slope in at 18 K (see inset of Fig.2(b)) accompanies the phase transition.The increase in $\rho$ and decrease in $\chi$ as T is decreased through 18 K could result from a change in the Fermi-surface topology associated with the formation of 
a CDW or SDW.
\begin{figure}[htb]
\epsfysize=2.0in
\centerline {\epsffile{figurer2.ps}}
%\centerline{\psfig{figure=figurer2.ps,height=2in}}
\caption[]{\em{(a)Electrical resistivity $\rho$ vs temperature T for 
$URu_{2}Si_{2}$ between 80 mK and 300 K.Inset:$\rho$ vs T between 80 mK and 
20 K in magnetic fields of 0 and 6T.(b)Magnetic susceptibility $\chi$ vs 
temperature for $URu_{2}Si_{2}$ between 2 and 300 K.Inset:$\chi$ vs T in the vicinity of the phase transition at $\sim 17.5 K$\cite{B}}}
%\label{models}
\end{figure}
\section{Specific heat}
$URu_{2}Si_{2}$ has a large value of specific heat coefficient 
,$\gamma$ =75mJ/mole-$K^{2}$.Fig.3 shows the plot of C/T vs $T^{2}$ between 
0.6 and 500 $K^{2}$.A linear extrapolation to 0 K of the C/T vs $T^{2}$ data 
in the normal state above 1.5 K yields a value of $\sim 65.5 mJ/mole-K^{2}$.
This represents the 0 K value of the specific-heat coefficient $\gamma$ in 
the absence of superconductivity.This value is compatible to the value given 
by Schlabitz and co-workers(75mJ/mole-$K^{2}$).\\*
\begin{figure}[htb]
\epsfysize=2.0in
\centerline {\epsffile{figurer3.ps}}
%\centerline{\psfig{figure=figurer3.ps,height=2in}}
\caption[]{\em{specific heat of C divided by temperature T vs $T^{2}$ for 
$URu_{2}Si_{2}$ between 0.6 and 500 $K^{2}$.Inset:Estimated specific heat 
$\delta$C vs T associated with the apparent CDW or SDW transition at 
$\sim 17.5$ K.The solid line represents the function A exp(-$\Delta$/T) where 
A =9890 J/mole-K and $\Delta$ =129 K\cite{B}}}
%\label{models}
\end{figure}
\section{Bulk Superconductivity below 1.5 K}
Fig.4(a) and 4(b),respectively,are $H_{c2}$ vs T and C/T vs T data that 
pertain to the superconducting state.The $H_{c2}$ vs T is very steep 
with an initial slope of (-d$H_{c2}$/dT) at $T_{c}$ =9.2T/K.Using the 
relation(-d$H_{c2}$/dT) at $T_{c}$ =4.44$\gamma\rho$ for Type{ \bf II} 
superconductors,with $\gamma$ =65.5 mJ/mole-$K^{2}$ and $\rho(2K) =1.04\times
10^{-4}\Omega$-cm,we calculate (-d$H_{c2}$/dT) at $T_{c}$ = 6.2T/K,in agreemen%t withthe measured value.This suggests that the large value of  $\rho$ might b%e intrinsic to the specimen.The C/T vs T data show that the superconductivity %of $URu_{2}Si_{2}$ is a bulk phenomenon.
\begin{figure}[htb]
\epsfysize=2.0in
\centerline {\epsffile{figurer4.ps}}
%\centerline{\psfig{figure=figurer4.ps,height=2in}}
\caption[]{\em{(a)Upper critical magnetic field $H_{c2}$ vs temperature 
for  $URu_{2}Si_{2}$.(b)Specific heat C divided by temperature T vs T 
for  $URu_{2}Si_{2}$.\cite{B}}}
%\label{models}
\end{figure}
\newpage
\section{Partially gapped Fermi surface}
The phase transition in $URu_{2}Si_{2}$ at 17.5 K appears to involve the development of a CDW or SDW out of the heavy-electron system,which opens an energy gap of $\sim 11$ meV over a portion of the Fermi surface.The portion of the Fermi surface that is not gapped by the CDW or SDW is removed by the superconductivity that occurs below 1.5 K,resulting in an energy gap $\sim 0.01$ meV over the remainder of the Fermi surface.
\section{Anisotropic Josephson effects in point contacts between 
$URu_{2}Si_{2}$ and Nb}
$URu_{2}Si_{2}$ exhibits extremely anisotropic Josephson current 
unlike a conventional superconductor.This is a 
characteristic and intrinsic feature of $URu_{2}Si_{2}$. Point contacts were fabricated by pressing etched Nb needles onto the surface of single crystals of $URu_{2}Si_{2}$.Fig.5(a) and 5.(b) show point-contact characteristic obtained for a contact aligned along the a-b direction and along the c-direction respectively.Fig.5(a) shows a pronounced structure at about 0.1mA,which appears to be below the superconducting transition temperature of  Nb of $T_{c}$ $\sim 9.2 K$.Below the superconducting transition temperature of  $URu_{2}Si_{2}$ the contact resistance drops again at low bias and becomes zero within experimental 
resolution.However,the contacts obtained in c-direction show a comletely 
different behavior,as shown in Fig.5(b).The structure of dV/dI occurs at 
about 0.1meV,as in a-b contacts.But,there is no indication of 
additional structure at and below the superconducting transition temperature
of $URu_{2}Si_{2}$.Thus,experimental study of point contacts between Nb and  $URu_{2}Si_{2}$ show that a Josephson current is observed below the transition 
temperature of $URu_{2}Si_{2}$ in contacts aligned along the a-b direction of 
the tetragonal structure,whereas it is absent in contacts alinged along the 
c direction.
\begin{figure}[htb]
\epsfysize=2.0in
\centerline {\epsffile{figurer5.ps}}
%\centerline{\psfig{figure=figurer5.ps,height=2in}}
\caption[]{\em{(a)Differential resistance dV/dI of a point contact 
between $URu_{2}Si_{2}$ and Nb vs bias current I applied parallel to the 
a-b direction at various fixed temperatures given in the figure 
.Inset:the same data at T$\sim 0.4 K$ shown in a plot of V vs I.
(b)Differential resistance dV/dI of a point contact between $URu_{2}Si_{2}$ 
and Nb vs bias current I applied along c-direction at various fixed	
temperatures given in the figure.\cite{C}}}
%\label{models}
\end{figure}
\newpage
\section{Unconventional order parameter}
The extreme anisotropy of Josephson current in $URu_{2}Si_{2}$ contacts with Nb is very unusual in a metallic point contact and provides strong evidence for an unconventional order parameter(OP) in $URu_{2}Si_{2}$ with a symmetry such that Josephson current along the c direction is zero.It has an even-parity $B_{1g}$ (d-wave) symmetry,shown in Fig.6,which shows two line nodes and maximum gap values along the a and b axes,unlike s-wave symmetry in usual superconductors.
\begin{figure}[htb]
\epsfysize=2.0in
\centerline {\epsffile{figure6.ps}}
%\centerline{\psfig{figure=figure6.ps,height=2in}}
\caption[]{\em{A possible order parameter for $URu_{2}Si_{2}$($B_{1g 
}$symmetry)."+" and "-" signs refer to the phase of the OP\cite{C}}}
%\label{models}
\end{figure}
\section{'Hidden Order' in $URu_{2}Si_{2}$}
NMR study at ambient pressure and low magnetic field strengths(below 6T) show 
an unambiguous,field-independent nearly isotropic component $\lambda$ of the
linewidth which increases below $T_{N}$ to about 12G at 4.2 K.Results suggest 
that $\lambda$ is not related to the static magnetization of the sample, 
rather it is due to the coupling of $^{29}$Si nulcei and the "hidden order"in the system.The sample used was a fine powder(particle size $\sim 50\mu$m)embedded in Stycast 1266 epoxy and oriented in a field of 9.4T.Alignment of order 
$90\%-95\%$ was estimated for the c-axis orientation by comparing the magnetic 
susceptibility for both transverse (H $\perp$ c) and (H $\parallel$ c)
external fields with that of a single crystal under similar conditions.
Spectra for (H $\perp$ c) and (H $\parallel$ c)(Fig.7.(a) and 7(b)
respectively) consisted of a single narrow line.Each line is fitted to a 
Lorentzian function of half width at half maximum(HWHM) (H,T),which we find 
can be written as $\Gamma^{2}$(H,T) = $\Gamma^{2}_{m}$(H,T) + $\lambda^{2}$(T).
Here $\Gamma_{m}$ is the contribution due to the sample magnetization(a term proportionalto H), and $\lambda$ is the new contribution to the linewidth that vanishes above $T_{N}$.Fig.7(a) shows that the line shape doesnot change with the field orientation $\theta$.Separating $\Gamma$ into two components is compelled by its field dependence,which is shown in Fig.7(c).The slope($\sim\ 6.4G/T)$ (dashed-line fit) for H $\parallel$ c is more than 10 times greater than that for H $\perp$ c due to  the large magnetic anisotropy of the system.\\
\begin{figure}[htb]
\epsfysize=2.0in
\centerline {\epsffile{figurer7.ps}}
%\centerline{\psfig{figure=figurer7.ps,height=2in}}
\caption[]{\em{$^{29}$Si NMR spectra in $URu_{2}Si_{2}$ at T = 14.5 K for (a)
H $\perp$ c, (b) H $\parallel$ c.Curves fit to Lorentzian functions of 
HWHM $\Gamma$(H,T) and (c)Field dependence of the $^{29}$Si linewidth $\Gamma$ in $URu_{2}Si_{2}$.Circles:H $\perp$ c,T =14.5 K(open)and T=40 K(filled).Filled triangles: H $\parallel$ c,T =20 K.Open triangles:H $\parallel$ c,T=14.5 K.Squares: T =4.2 K.Curves:one-(dashed) and two-(solid)component fits to the linewidth \cite{C}}}
%\label{models}
\end{figure}
The above results can be discussed using two possible arguments,neither of 
which turns out to be satisfactory:\\
\noindent(i)disordered U-moment freezing,and\\
(ii)inhomogeneous antiferromagnetism.\\
Combined NMR,ND,$\mu$SR experiments rule
out these scenarios.On the contrary,the suggestion of orbital antiferromagnetism as the hidden order and source of the extra $^{29}$Si linewidth
by Chandra et al. is a more viable possibility.
\section{Conclusion}
It is self-evident by now that the heavy fermion superconductor$URu_{2}Si_{2}$
possesses intrinsic anomalous properties.Understanding the normal state 
and superconducting behaviour of this material has proven to be very 
challenging and hence,a substantial amount of work has been done in this
field over the years.Indeed,the physics of this material is very young and exciting.
\section{Acknowledgement}
I wish to acknowledge my gratitude to Prof. M. Jarrell for encouraging me to 
choose this topic.I would also like to thank all my friends for their support.\

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