Collapse of the vortex-lattice inductance and shear modulus at the melting transition in untwinned $%%@ YBa_2Cu_3O_7$.
Peter Matl$^{1,*}$
N. P. Ong$^1$
R. Gagnon$^2$
and L. Taillefer$^{2,\dagger}$
The complex resistivity $\hat{\rho}(\omega)$ of the vortex lattice in an untwinned crystal of 93-K $%%@ YBa_2Cu_3O_7$ has been measured at frequencies $\omega/2\pi$ from 100 kHz to 20 MHz in a 2-Tesla %%@ field $H\parallel c$, using a 4-probe RF transmission technique that enables continuous measurements %%@ versus $\omega$ and temperature $T$. As $T$ is increased, the inductance ${\cal L}_s(\omega) ={%%@ Im}\; \hat{\rho}(\omega)/ \omega$ increases steeply to a cusp at the melting temperature $T_m$, and then %%@ undergoes a steep collapse consistent with vanishing of the shear modulus $c_{66}$. We discuss in detail %%@ the separation of the vortex-lattice inductance from the `volume' inductance, and other skin-depth effects. %%@ To analyze the spectra, we consider a weakly disordered lattice with a low pin density. Close fits are %%@ obtained to $\rho_1(\omega)$ over 2 decades in $\omega$. Values of the pinning parameter $\kappa$ and %%@ shear modulus $c_{66}$ obtained show that $c_{66}$ collapses by over 4 decades at $T_m$, whereas %%@ $\kappa$ remains finite.
Vortexshear02web.tex
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\title{Collapse of the vortex-lattice inductance and shear modulus at the melting transition in untwinned $\rm %%@
YBa_2Cu_3O_7$.} 
\author{Peter Matl$^{1,*}$, N. P. Ong$^1$, R. Gagnon$^2$, and L. Taillefer$^{2,\dagger}$}      
\address{$^1$Department of Physics, Princeton University, Princeton, New Jersey 08544}
 
\address{$^2$Department of Physics, McGill University, 3600 University
Street, Montr\'eal, Quebec, Canada H3A 2T8}

\date{\today}      % Deleting this command produces today's date.

%\newcommand{\ip}[2]{(#1, #2)}
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\begin{abstract}
The complex resistivity $\hat{\rho}(\omega)$ of the vortex lattice in an untwinned crystal of 93-K $\rm %%@
YBa_2Cu_3O_7$ has been measured at frequencies $\omega/2\pi$ from 100 kHz to 20 MHz in a 2-Tesla %%@
field $\bf H\parallel c$, using a 4-probe RF transmission technique that enables continuous measurements %%@
versus $\omega$ and temperature $T$.  As $T$ is increased, the inductance ${\cal L}_s(\omega) ={\rm %%@
Im}\; \hat{\rho}(\omega)/ \omega$ increases steeply to a cusp at the melting temperature $T_m$, and then %%@
undergoes a steep collapse consistent with vanishing of the shear modulus $c_{66}$.  We discuss in detail %%@
the separation of the vortex-lattice inductance from the `volume' inductance, and other skin-depth effects.  %%@
To analyze the spectra, we consider a weakly disordered lattice  with a low pin density.  Close fits are %%@
obtained to $\rho_1(\omega)$ over 2 decades in $\omega$.  Values of the pinning parameter $\kappa$ and %%@
shear modulus $c_{66}$ obtained show that $c_{66}$ collapses by over 4 decades at $T_m$, whereas %%@
$\kappa$ remains finite. 
\end{abstract}
\pacs{74.60.Ge,74.72.Bk,72.15.Gd,64.70.Dv}

\maketitle                   % Produces the title.


%
\section{Introduction}\label{intro}
The investigation of the response of vortices in type II superconductors to an alternating driving current has %%@
had a long and fruitful history, beginning with the experiment of Gittleman and Rosenblum (GR) on PbIn and %%@
Nb alloys \cite{Gitt}.  In cuprate superconductors, these investigations have been performed on $\rm %%@
YBa_2Cu_3O_7$ (YBCO) over a broad range of frequencies extending from 100 kHz to a few THz.  At %%@
frequencies $\omega$ below a few 100 MHz, the vortex response is obtained by directly measuring the %%@
sample's complex resistivity $\hat{\rho}(\omega)$.  At microwave frequencies (10 to 100 GHz), cavity %%@
perturbation techniques have been the primary approach, although bolometric absorption techniques have %%@
been more useful at the high microwave end.  

The occurence of the vortex solid-to-liquid melting transition in the cuprates is one of the most interesting %%@
phenomena in the investigation of `vortex matter' in superconductors~\cite{Blatter}.  The melting transition at %%@
$T_m$ has been investigated in some detail by high-resolution measurements of the magnetization %%@
\cite{Zeldov,Welp}, heat capacity \cite{Schilling} and $dc$ resistivity \cite{Kwok,Safar}.  
In any solid-to-liquid transition, the key quantity of interest is the shear modulus which vanishes in the liquid.  %%@
The resistivity profile ($\rho$ vs. $T$) in a crystal of $\rm Bi_2Sr_2CaCu_2O_8$ in which alternating %%@
strong and weak pinning channels are created by ion irradiation was interpreted as a loss of shear strength at %%@
$T_m$~\cite{Kes}.  The effect of the vanishing shear modulus at $T_m$ on $\rho$ when the field $\bf H$ %%@
is tilted relative to the twin boundaries in twinned YBCO crystals was also demonstrated %%@
~\cite{Kwokmod}.  

The standard approach to investigating how the shear modulus changes in a system undergoing a %%@
solid-to-liquid transition is by observing its {\em dynamical response} to an oscillating driving force.  The %%@
frequency dependence of $\hat{\rho}(\omega)$ in the vortex state in YBCO has been investigated by %%@
several groups at frequencies in the 1 to 100 MHz range.  Initial measurements using thin-film samples found %%@
that $\hat{\rho}(\omega) = \rho_1(\omega)+j\rho_2(\omega)$ displays strong, featureless dispersion at %%@
these frequencies \cite{Koch,Wu1,Anlage}.  However, the response in high-purity single crystals is %%@
qualitatively different.  Wu, Ong, Gagnon and Taillefer (WOGT) \cite{Wu2} investigated the $ac$ response %%@
in untwinned crystals of YBCO in the MHz range, and found that the inductance vs. $H$ exhibits a %%@
step-wise change at the melting field.  As a liquid cannot produce an inductive response, the step-wise %%@
change in the inductance is direct evidence for a collapse of the shear modulus $c_{66}$ at %%@
$T_m$.  

The 2-probe experiments of WOGT were performed at fixed $T$ versus $H$.  This approach cannot be %%@
extended to measurements in which $T$ is continuously varied in fixed $H$ because variations in the %%@
background signal (from $T$-dependent stray cable reactances) swamp the sample signal.  We report %%@
results obtained by the use of an RF 4-probe technique that circumvents this obstacle.  The 4-probe %%@
technique allows high-resolution impedance measurements versus each of the three quantities $\omega$, %%@
$H$ and $T$.  We find that the collapse of the inductance is even more abrupt when $T$ is scanned at fixed %%@
$H$.  While we largely confirm the {\em fixed}-$T$ results of WOGT, the swept-$T$ experiments provide %%@
a more direct picture of the behavior of the vortex resistivity and inductance as $T$ is increased above the %%@
melting line.  With the broader perspective, we correct a previous inference regarding a `remanent' shear %%@
modulus in the vortex liquid state.  A detailed discussion of how the vortex-lattice inductance is separated %%@
from the `volume' inductance is provided.  Together with the measurements of WOGT, the present results %%@
provide a complete experimental picture of the linear response of the pinned lattice in a high-purity cuprate %%@
crystal to an ac driving current in the MHz frequency range.  The two distinguishing features are the strong %%@
dispersion of $\hat{\rho}(\omega)$ observed even at low $\omega$ (just below the melting temperature %%@
$T_m$) and the striking abrupt collapse of the inductance at $T_m$.  

To understand the spectra, we have adopted a model in which the vortex lattice is weakly disordered by %%@
pinning to a low density of pins \cite{Schmid,Larkin}.  A mean-field solution proposed by Ong and Wu %%@
(OW)~\cite{Ong} is used to fit the measured resistivity spectra.  By measuring the damping viscosity %%@
$\eta(T)$ independently, we reduce the number of adjustable parameters to just 2 numbers at each $T$. %%@
We show that the OW solution can achieve close fits to $\rho_1(\omega)$ extending over 2 decades in %%@
$\omega$.  The fits enable us to find both $\kappa$ and $c_{66}$ (to within a multiplicative constant) at %%@
each $T$.  The latter exhibits a remarkable 4-decade collapse to zero within 1 K of $T_m$.   

In Sec. \ref{exper} a summary of the experimental approach is given. Section \ref{skin} discusses %%@
skin-depth effects and the `volume' inductance.  In Sec. \ref{fixedfreq}, we report measurements versus %%@
$T$ and $H$ at constant $\omega$.  The central results -- the spectra of $\rho_1$ and $\rho_2/\omega$ -- %%@
are reported in Sec. \ref{spectra}.  Section \ref{fits} explains the fits of the spectra to the weakly disordered %%@
lattice model, and Sec. \ref{modulus} discusses the temperature dependence of the shear modulus near %%@
$T_m$.  Appendix A describes the 4-probe technique, while Appendix B summarizes the impedance %%@
calculation for a sample with elliptical cross-section.  Appendix C summarizes the linear response of the %%@
weakly disordered lattice to an $ac$ current.

\section{\label{exper}Experimental details}
As in WOGT \cite{Wu2}, a weak, alternating current ${\bf J} {\rm e}^{j\omega t}$ is applied within the %%@
$\rm CuO_2$ plane $({\bf J} \parallel {\bf a})$ in the presence of a field $\bf H\parallel \hat{c}$.  The %%@
average velocity response of the vortices is detected as a complex resistivity $\hat{\rho}(\omega) = %%@
\hat{v}(\omega)B/J_0$.  

Because contact resistances on YBCO crystals are typically a fraction of an Ohm, they add a %%@
$T$-dependent background that dominates the sample signal.  In an experiment in which the field $H$ is %%@
swept at constant $T$ (as in WOGT), this large background may be subtracted, by reference to the %%@
zero-field data at each $T$.  However, when $T$ is swept at constant $H$, the contact background leads %%@
to substantial errors.  The problem is exacerbated by the $T$ dependence of the dielectric filling in the long %%@
coax cables.  We minimized this by using special cables with low-density Polytetrafluoroethylene (teflon) %%@
dielectric.  The higher propagation velocity (0.85$c$ versus 0.66$c$) is also an advantage.  

The 4-probe design was developed to remove the background altogether.  The sample (impedance %%@
$Z_s=R_s +j\omega L_s$) shunts the inner and outer conductors of both the incident and transmission %%@
coaxial cable (see Appendix A).  If $Z_s\ll Z_0$ (the line impedance), the incident signal is strongly %%@
reflected, and the transmitted signal is highly sensitive to small changes in $Z_s$.  By phase-detecting the %%@
latter we may determine $Z_s$.  By inserting broad-band transformers in both the incident and transmission %%@
cables, we can achieve true 4-probe measurement at frequencies up to 50 MHz (Appendix A describes the %%@
calibration procedure).

The measurements were made on a detwinned crystal of YBCO (of dimension $a$ = 1.11 mm, $b$ = 0.41 %%@
mm, and $c$ = 55 $\mu$m), in which the $dc$ resistivity displays a sharp transition (width of 0.2 K) at %%@
$T_c$ = 93.3 K.  The RF resistivity was measured with the RF current density $\bf J\parallel a$, and $\bf %%@
H\parallel c$.  Although all measurements were performed in a screened room, radio broadcast signals are %%@
picked up as weak resonance lines in the spectra.  Parasitic line resonances, detected as weak resonances, %%@
are the hardest to eliminate.  However, they can be minimized by empirically adjusting the grounding strap %%@
configuration. 

With the 4-probe method, we may vary independently each of the variables $H$ (0-8 Tesla), $\omega$ %%@
(100 kHz to 20 MHz), and $T$.  To confirm that we are in the linear-response regime, we checked that %%@
curves of $\hat{\rho}(\omega)$ measured with incident power at -15.5 dBm and at 4.5 dBm are closely %%@
similar (except for a higher noise content in the -15.5 dBm curve for $\rho_2$).  We report all %%@
measurements in terms of the sample's complex resistivity $\hat{\rho}(\omega) = Z_s(\omega)(bc/a') \equiv %%@
\rho_1(\omega) + j\rho_2(\omega)$, where $a'$ = 0.68 mm is the separation of the voltage contacts.  In %%@
place of $\rho_2(\omega)$, it is preferable to discuss the (specific) inductance defined by ${\cal L}_s \equiv %%@
\rho_2/\omega$ (note that ${\cal L}_s =L_s(bc/a')$).  Dividing $\rho_2$ by $\omega$ isolates the %%@
divergent behavior of ${\cal L}_s$ as $\omega\rightarrow 0$, thereby highlighting the dramatic collapse of %%@
${\cal L}_s$ at the lattice melting temperature.  

\section{\label{skin}Skin-depth and volume inductance}

In zero magnetic field, the impedance $Z_s(\omega)$ of a superconductor (in the form of a cylinder) may be %%@
calculated from the 2D Helmholtz equation $(\nabla^2+\hat{\kappa}^2){\bf A}=0$, where %%@
$\hat{\kappa}^2 = \lambda^{-2}+ 2j\delta^{-2}$ (here, $\lambda$ is the London penetration length and %%@
$\delta$ the skin-depth determined by the quasiparticle conductivity).  

First, we consider the normal-state impedance ($T>T_c$) by letting $\lambda\rightarrow \infty$, and %%@
replacing $\delta$ by the normal-state skin-depth $\delta_n= \sqrt{2\rho_n/\omega\mu_0}$ (where %%@
$\rho_n$ is the normal-state resistivity and $\mu_0$ the vacuum permeability).  When $\omega$ is large %%@
enough that $\delta_n\ll \sqrt{\cal A}$, $J$ is confined to the skin-depth, and the effective $R_s(\omega)$ %%@
increases (${\cal A}$ is the cross-section area).  For a circular cross-section, $Z_s(\omega)$ is given by %%@
\cite{Marion}
\begin{equation}
Z_s(\omega) = \frac{\omega\mu_0\ell\delta_n}{2\pi r_0\sqrt{2j}}
\frac{J_0(\hat{x})}{J_1(\hat{x})},
\label{Zcirc}
\end{equation}
where $\hat{x} \equiv -jr_0\sqrt{2j}/\delta_n$, $J_m(\hat{x})$ is the Bessel function of order $m$, and %%@
$r_0$ ($\ell$) is the radius (length) of the sample. The case of elliptical cross-section is treated in Appendix %%@
B. 

Equation \ref{Zcirc} shows that, at large $\omega$, $R_s(\omega)\rightarrow \rho\ell/(2\pi r_0\delta_n)$, as %%@
expected. More important for our inductance discussion, the RF field energy at high $\omega$ is confined to %%@
the skin-depth.  Therefore, the inductance $L_s(\omega)$ decreases monotonically to zero as %%@
$\omega\rightarrow \infty$.  As this inductance expresses the RF energy stored in the volume of a {\em %%@
straight} cylinder, we refer to it as the `volume' inductance $L_{vol}(\omega)$ to distinguish it from the %%@
vortex-lattice inductance which is the main focus of this work (and from stray, geometric inductances that %%@
arise from coil-coil flux linkage).  From Eq. \ref{Zcirc}, we find that 
\begin{equation}
\lim_{\omega\rightarrow 0} L_{vol} = \frac{1}{2}\frac{\mu_0 \ell}{4\pi}.
\label{Ln}
\end{equation}
For a high-purity Cu wire ($r_0$ = 0.4 mm, $\ell$= 3 mm) at 300 K, $L_{vol}(\omega)$ equals 120 pH at %%@
low $\omega$, but falls to a tenth of this value at 20 MHz.  

Below $T_c$, the penetration of the RF fields (in zero $H$) is set by the London length, i.e. %%@
$\hat{\kappa}\rightarrow \lambda^{-1}$.  As the in-plane $\lambda$ in YBCO is very short ($\sim 0.2 %%@
\mu$m), we may regard the stored RF field energy to be nominally zero.  Hence the change in $L_s$ (in %%@
zero field) between $T_c$ and, say 60 K, provides a direct measurement of $L_{vol}$.  Applying Eq. %%@
\ref{Ln} to a cylinder with circular cross-section area equal to that in our crystal, we find that %%@
$\lim_{\omega\rightarrow 0} L_{vol}$ = 34 pH (corresponding to ${\cal L}_{vol} = 1.2\times 10^{-13}$ %%@
Hcm).  This tiny inductance may be measured with about 2$\%$ resolution by our 4-probe %%@
method.  

In the resistive state ($H\gg H_{c1}$), the RF penetration is determined by $\rho_1(\omega)$ (i.e. %%@
$\hat{\kappa}^2 = 2j\delta_f^{-2}$, where the flux-flow skin-depth $\delta_f= %%@
\sqrt{2\rho_1/\omega\mu_0}$).  As $\rho_1$ may be as large as 8 $\mu\Omega$cm for $H$= 2 T, the %%@
volume inductance $L_{vol}$ nearly recovers its full value given by Eq. \ref{Zcirc}.  However, in our %%@
measurements, the inductance increase is actually far in excess of $L_{vol}$.  The excess inductance arises %%@
purely from the vortex lattice.  

Returning to the resistive component $R_s(\omega)$, we have converted it to $\rho_1$ by the assumption %%@
that $\bf J$ is uniform over the cross-section (i.e. $\rho_1(\omega)= R_s(\omega)bc/a'$).  However, if %%@
$\delta$ is comparable to the half-thickness $c/2$, $R_s(\omega)$ is significantly enhanced by the %%@
skin-depth effect, and $\rho_1$ will be overestimated.  We now address this concern.  As the aspect ratio %%@
$b/c\simeq 8.3$, we cannot rely on Eq. \ref{Zcirc} for quantitative guidance.  In Appendix B, we have %%@
calculated $Z_s(\omega)$ for a normal-state sample with an elliptical cross-section using Mathieu functions.  %%@
The variation of $R_s(\omega)$ and ${\cal L}_{vol}(\omega)$ is shown in Fig. \ref{elliptical}) for a sample %%@
with this high ellipticity.  We assumed the normal-state resistivity $\rho_n = 4 %%@
\mu\Omega$cm. 
%Figure 1 (elliptical) 
\begin{figure}
\includegraphics[width=4cm]{elliptical.eps}
%\centerline{\psfig{file=Matl1/elliptical.eps,height=2.7in,width=2.8in}}
\caption{\label{elliptical} Calculated impedance $Z_s(\omega)= R_s(\omega)+ j\omega L_s(\omega)$ of a %%@
conducting cylinder of elliptical cross-section and resistivity $\rho = 4{\rm \mu\Omega cm}$.  The %%@
inductance $L_s$ is entirely volume inductance $L_{vol}$. The parameters $\xi_0\simeq 0.108$ and $h= %%@
250 \mu$m are selected to match the sample cross-section.}
\end{figure}

As the results depend only on $\delta_n$, values for $R_s(\omega)$ and $L_{vol}(\omega)$ at other values %%@
of $\rho_n$ are obtained by re-scaling the $x$-axis.  For example, at 85.9 K, $\rho_1$ measured in a field %%@
of 2 T with $\omega/2\pi = 3$ MHz equals $\sim 1 \mu\Omega$cm.  Re-scaling the frequency axis in Fig. %%@
\ref{elliptical} by a factor of 4, we see that at $\omega/2\pi$ = 12 MHz, $R_s(\omega)$ and %%@
$L_{vol}(\omega)$ deviate by only 5$\%$ and 3$\%$, respectively, from their low-frequency (i.e. %%@
uniform-distribution) values.  At the lowest $T$ ($<$ 84 K), however, the deviations may be up to 20$\%$ %%@
for $\omega/2\pi>$10 MHz, but these low $T$-high-$\omega$ data will not play any significant role in our %%@
analysis.  Throughout the low-$\omega$ region near the melting transition, the correction is less than 0.5 %%@
$\%$, so that the uniform-$J$ assumption is valid.  

\section{\label{fixedfreq} Fixed-frequency measurements}
All the data reported here are taken either in zero field or in a fixed field of 2 T. In the resistive state, the %%@
inductance of YBCO varies strongly with $\omega$ and $T$.  We first display the variation of ${\cal %%@
L}_s(\omega,T,H)$ versus $T$ measured at selected $\omega$ in a 2-Tesla field as well as in zero field %%@
(main panel of Fig. \ref{LvsT}).  The inset shows the corresponding curves for $\rho_1$.  

In the main panel, we have plotted the specific inductance ${\cal L}_s = \rho_2/\omega$ with an arbitrary %%@
origin, because the total observed inductance $L_{obs}$ is the sum of the sample contributions ($L_{vol}$ %%@
and the vortex lattice term $L_v$) and an arbitrary background $L_{bg}$ that arises from small %%@
uncertainties in setting `null' in the phase setting of the lock-in amplifiers), viz.
\begin{equation}
L_{obs} = L_v + L_{vol} + L_{bg}
\label{Lobs}
\end{equation}
We take the portion of the curve at 1 MHz-0T below 90 K as our reference, and identify sample inductance %%@
signals relative to this reference curve. 

%Figure 2 (LvsT)
\begin{figure}
\includegraphics[width=7cm]{LvsT.eps}
%\centerline{\psfig{file=Matl1/LvsT.eps,width=3.4in}}
\caption{\label{LvsT} (Main panel) Temperature dependence of the total sample inductance ${\cal %%@
L}_s(\omega,T,H)= \rho_2(\omega,T,H)/\omega$ measured in a 2-Tesla field ($\bf H\parallel c$) at the %%@
frequencies indicated.  Here, `15' means 15.4 MHz and $H$ (in Tesla) is in parantheses. As $T$ %%@
approaches $T_m$ from below, ${\cal L}_s$ (measured at 1-4 MHz) rises to a sharp cusp before %%@
collapsing to the value ${\cal L}_{vol}$ observed in the normal state.  For comparison, the zero-field traces %%@
taken at 1 and 15.4 MHz are also shown.  The inductance scale has an arbitrary origin (because of the term %%@
$L_{bg}$ in Eq. \ref{Lobs}).  The portion  of the curve 1 (0T) below 90 K serves as our reference line.  %%@
Inductance signals measured relative to this reference are identified as contributions from the sample.  The %%@
step-change at $T_c$ in zero $H$ reflects the Meissner expulsion of the stored RF field energy (reduction %%@
of volume inductance ${\cal L}_{vol}$).  The inset shows $\rho_1$ measured at the same %%@
$\omega$.}
\end{figure}

%Figure 3 (rhoeta)
\begin{figure}
\includegraphics[width=7cm]{rho1hifreq.eps}
\caption{\label{rhoeta} The high frequency resistivity $\rho_1(\omega)$ measured at $\omega$ = 15.4 %%@
MHz versus applied field $\bf H\parallel \hat{c}$ at temperatures close to $T_c$ in untwinned YBCO ($\bf %%@
J\parallel \hat{c}$).  The minimum is caused by the `peak effect' which appears close to the melting field %%@
$H_m$.  At each $T$, the initial slope of $\rho_1$ vs. $H$ is used to determine the viscosity $\eta_a(T)$.   %%@
The $T$ dependence of $\eta_a$ is plotted in the insert ($\eta_b$ obtained from a second sample is also %%@
plotted).  Near $T_c$, $\eta_a\sim(1-T/T_c)^{1.5}$, while $\eta_b\sim (1-T/T_c)$. 
}       
\end{figure}
We first discuss the 1 MHz curve in zero field. As $T$ increases from 60 K, ${\cal L}_s(\omega_1,T,0)$ is %%@
nearly $T$-independent until, at $T_c$, it undergoes a step-increase to the normal-state value %%@
($\omega_1/2\pi$ = 1 MHz).  Clearly, the step corresponds to the collapse of the Meissner effect at $T_c$, %%@
as discussed above.  The difference between ${\cal L}_s(\omega_1,T,0)$ measured at 100 and 60 K gives %%@
${\cal L}_{vol} = 1.0\;\times 10^{-13}$ Hcm, rather close to the value 1.2 $\times 10^{-13}$ Hcm %%@
estimated below Eq. \ref{Ln}.  Hence we identify the step change in the curve ${\cal L}_s(\omega_1,T,0)$ %%@
with the maximum (specific) volume inductance ${\cal L}_{vol}$ of our sample. The curve ${\cal %%@
L}_s(\omega_{15},T,0)$ measured at $\omega_{15}/2\pi$ = 15.4 MHz shows a closely similar step, as %%@
expected from Meissner flux-expulsion. 

%Figure 4 (rho1)
\begin{figure}
\includegraphics[width=6cm]{rho1_comb.eps}
%\centerline{\psfig{file=Matl1/rho1.eps,width=3.4in}}
\caption{\label{rho1} (a) The frequency dependence of $\rho_1(\omega,T,H_0)$ at temperatures 80.6 to %%@
88.9 K in a 2-Tesla field ${\bf H}_0\parallel{\bf c}$.  The curves for $\rho_1$ become increasingly %%@
$\omega$ dependent as $T_m (\sim$ 88.5 K) is approached from below.  In the vortex liquid state above %%@
$T_m$, $\rho_1$ rapidly approaches a nominally $\omega$-independent value (curve at 88.9 K).  (b) A %%@
subset of the data in upper panel plotted against $\log \omega$ to show the low-$\omega$ region.  The solid %%@
lines are fits discussed in Sec. \ref{fits}.
}       
\end{figure}

When a field $H_0$= 2 T is applied, the sample inductance is dominated by that of the vortex lattice.  We %%@
follow the curve at 1 MHz and 2 T, starting at 60 K.  As $T$ increases, ${\cal L}_s(\omega_1,T,H_0)$ %%@
increases rapidly, rising to a cusp just below the melting temperature of the lattice $T_m\simeq$ 88.5 K.  %%@
Curves at higher frequencies show a similar but reduced peak.  Above $T_m$, ${\cal L}_s$ collapses %%@
rapidly to a value very close to the normal state value.  Because of the finite value of the RF vortex resistivity %%@
$\rho_1$ (inset), expulsion of the RF fields from the interior is now determined by the flux-flow skin-depth %%@
$\delta_f= \sqrt{2\rho_1/\mu_0\omega}$ ($\delta_f\gg \lambda$).  At these temperatures, $\delta_f > c/2$, %%@
so the RF fields are uniform in the vortex liquid state; there is no change in ${\cal L}_s$ in crossing $T_c$ = %%@
93.5 K.  

The sharp increase in ${\cal L}_s$ as $T$ approaches $T_m$ from below is one of our main results.  The %%@
collapse of the inductance above $T_m$ is the constant-$H$-scan version of the collapse observed by %%@
WOGT in constant-$T$ scans versus $H$.  

[As mentioned in Sec. \ref{intro}, the present swept-$T$ scans provide a much clearer picture of how the %%@
inductance evolves into the normal-state value.  In WOGT \cite{Wu1}, the finite value of ${\cal L}_s$ %%@
above the melting field $H_m$ was interpreted as evidence for a small, residual shear rigidity in the vortex %%@
liquid state (their Fig. 1b).  This inference is incorrect.  As discussed here, the residual inductance above %%@
$H_m$ is simply the volume inductance ${\cal L}_{vol}$.  Within our resolution, the shear rigidity in the %%@
vortex liquid state is zero.  This removes the need for the arbitrary `background' liquid term $\rho_2^b$ %%@
introduced by WOTG \cite{Wu2}.]

At our highest frequencies, the lattice response is predominantly resistive.  As discussed below, the friction %%@
term in this limit dominates the lattice forces and pinning forces, so that $\rho_1$ is proportional to the %%@
reciprocal of the viscosity $\eta(T)$.  Figure \ref{rhoeta} (main panel) displays the field dependence of %%@
$\rho_1(\omega)$ measured at the frequency $\omega/2\pi$ = 15.4 MHz at temperatures close to $T_c$ %%@
(the current is along the $a$-axis).  At each $T$, we have fitted the initial slope to the (free) flux-flow %%@
expression $\rho_f = B\phi_0/\eta$ in order to find $\eta(T)$ (plotted in the insert)  [$\phi_0 = h/2e$ is the %%@
flux quantum].   We measured $\eta(T)$ in 2 untwinned crystals.  In the geometry ${\bf J\parallel a}$, the %%@
uncertainty in determining $\eta_a$ is large because the broad `peak effect' restricts the range of $H$ over %%@
which $\rho_1$ vs. $H$ is truly linear.  More reliable results are obtained (for $\eta_b$) in the geometry %%@
${\bf J\parallel b}$ where the peak effect is much narrower.  The two sets of viscosity $\eta_a$ and %%@
$\eta_b$ are shown in Fig. \ref{rhoeta} (insert).  The viscosity data are used in the fits discussed in Sec. %%@
\ref{fits}. 


\section{\label{spectra} Frequency dependence of the complex resistivity}
Figure \ref{rho1} displays the frequency dependence of $\rho_1(\omega,T,H_0)$ at temperatures from %%@
80.6 to 88.9 K ($H$ is fixed at 2 T).  At temperatures $T<T_m\sim$ 88.5 K, the value of $\rho_1$ in the %%@
limit $\omega\rightarrow 0$ is zero.  However, above $T_m$, the limiting value becomes finite.  At the %%@
highest $\omega$, $\rho_1$ approaches the Bardeen-Stephen value for free flux-flow $\rho_f$ (inset of Fig. %%@
\ref{LvsT}).


Below $T_m$, $\rho_1$ is strongly dispersive.  Close to $T_m$, $\rho_1$ displays a crossover from a %%@
sharply increasing region at low $\omega$ to a gradual region at high $\omega$.  The `knee' feature %%@
separating the two regions rapidly moves to high frequencies as $T$ falls below $T_m$ by a few K, %%@
eventually moving out of our frequency window. 

%Figure 5 (Llogf)
\begin{figure}
\includegraphics[width=9cm]{LLOGF.eps}
%\centerline{\psfig{file=Matl1/LLOGF.eps,width=3.4in}}
\caption{\label{Llogf} The frequency dependence of the sample inductance ${\cal L}_s(\omega,T,H_0)$ %%@
measured in a 2-T field ${\bf H}_0\parallel{\bf c}$.  ${\cal L}_s$ (referenced to the zero-field inductance %%@
curve) is the sum of ${\cal L}_v$ and ${\cal L}_{vol}$.  The upper and lower panels are measurements %%@
taken below and above 87.95 K, respectively.  As $T\rightarrow T_m$ from below (a), the vortex term %%@
${\cal L}_v$ increases sharply, especially at low $\omega$.  Just below $T_m$ (b), it collapses to zero, %%@
leaving the nominally flat spectrum that we identify as the volume inductance ${\cal %%@
L}_{vol}$.}
\end{figure}


Figure \ref{Llogf} displays ${\cal L}_s(\omega,T,H_0)$ versus $\omega$ for temperatures below and %%@
above $T_m$ (with $H_0$ = 2 T).  As explained following Eq. \ref{Lobs}, we identify the sample %%@
inductance using the zero-field curve (below 90 K) as reference, viz. 
$${\cal L}_s(\omega,T,H) = [L_{obs}(\omega,T,H)-L_{obs}(\omega,T,0)](bc/a').$$  As discussed in %%@
Sec. \ref{fixedfreq}, ${\cal L}_s$ rises to a sharp cusp as $T\rightarrow T_m^-$, before undergoing a %%@
steep collapse above $T_m$.  The most pronounced dispersion occurs at low $\omega$ where ${\cal %%@
L}_s$ shows divergent behavior in the limit $\omega\rightarrow 0$.  

Although the spectra of the inductance are not used in our fitting process in Sec. \ref{fits}, we discuss %%@
various contributions to the observed inductance signal (which includes the volume inductance) for the sake %%@
of completeness.  We may estimate ${\cal L}_{vol}(\omega,T,H)$ by appealing to the observed $\rho_1$ %%@
and the calculated curves in Fig. \ref{elliptical}.  Note that ${\cal L}_{vol}$ can never exceed 0.12 pHcm.  %%@
At high frequencies, $L_v$ is negligible, so that the curve at 15.4 MHz (2T) in Fig. \ref{LvsT} may be taken %%@
as that for ${\cal L}_{vol}$ vs. $T$ at large $\omega$ (we ignore the slight dip near $T_m$).  Using the %%@
values of $\rho_1$ measured at 15.4 MHz (Fig. \ref{rho1}), we find this interpretation is consistent with the %%@
calculations in Fig. \ref{elliptical}.

We may also use the observed values of $\rho_1$ to estimate ${\cal L}_{vol}$ at low frequencies.  At the %%@
temperatures of interest (85-90 K), $\rho_1$ is sufficiently large, even at 200 KHz, to satisfy the %%@
uniform-$J$ condition, i.e. ${\cal L}_{vol}$ is close to its maximum value 0.12 pHcm.  With increasing %%@
$\omega$, ${\cal L}_{vol}$ decreases slowly to its high-$\omega$ value as seen in Fig. \ref{LvsT}.  For %%@
example, at 88.2 K (in Fig. \ref{Llogf}b), our estimate of the volume inductance spectrum is a curve that %%@
starts at 0.12 pHcm at low $\omega$ and decreases slowly to $\sim$0.06 pHcm at high $\omega$. The %%@
divergent vortex term is easy to distinguish from this small, featureless background.   

This concludes the purely experimental part of the report.  The spectra in Figs. \ref{rho1} and \ref{Llogf}, %%@
complemented by the measurements of WOTG~\cite{Wu2}, constitute a rather complete quantitative %%@
description of the linear-response of the vortex lattice in untwinned YBCO at RF frequencies.  In the %%@
following sections, we interpret the measurements within a specific model, in order to extract the temperature %%@
dependence of the vortex-lattice shear modulus.

\section{\label{fits} Fits to the spectra}
\subsection{Discussion of the model}
In the model used by GR \cite{Gitt}, the vortex equation of motion is
\begin{equation}
\eta \dot{\bf u} + \kappa{\bf u} = {\bf J\times \hat{z}} \phi_0,
\label{GR}
\end{equation}
where $\bf u$ is the vortex displacement from equilibrium, and $\eta$ and $\kappa$ are the damping %%@
viscosity and the Labusch parameter, respectively, and $\bf B\parallel \hat{z}$.  Measurements of %%@
$\hat\rho$ in thin-film samples \cite{Koch,Wu1} and in untwinned crystals \cite{Wu2} of YBCO are in %%@
strong disagreement with the Lorentzian response $\hat{\rho} = (B\phi_0/\eta)[1-j\omega_p/\omega]^{-1}$ %%@
predicted by Eq. \ref{GR} (here $\omega_p\equiv \kappa/\eta$).  In particular, the dramatic dispersion %%@
observed when the lattice melting line is approached, in either constant-$H$ scans (WOGT) or in the %%@
constant-$T$ scans reported here, cannot be reproduced by Eq. \ref{GR}, even if one replaces $\kappa$ %%@
by an `effective' parameter that depends on $\omega$ and $B$.

In the context of the vortex glass~\cite{Fisher}, Dorsey \cite{Dorsey} has derived scaling relations for the %%@
$\omega$-dependent conductivity $\hat{\sigma}(\omega)$ at the vortex glass-to-liquid transition, viz. %%@
$\hat{\sigma}(\omega)\sim (-i\omega)^{\alpha}$.  In {\em thin-film} samples of YBCO, the measured %%@
spectra do in fact display power laws in $\omega$ over 2-3 decades (1-500 MHz)~\cite{Wu1}.  The %%@
situation in untwinned crystals is very different, however.  The spectra reported here (and in %%@
WOGT~\cite{Wu2}) do not follow power laws at all.  Further, the melting transition is abrupt and weakly %%@
first-order rather than the continuous transition of the vortex glass.  Under the high-purity conditions found in %%@
untwinned YBCO crystals, the observed spectra  {\em are more consistent with that of a weakly disordered %%@
vortex lattice with random pins} than a vortex glass (as we now show).

In their theory of the depinned vortex lattice, Schmid and Hauger \cite{Schmid} and Larkin and Ovchinikov %%@
~\cite{Larkin} represented the rigidity of the moving periodic structure by the lattice force matrix ${\bf %%@
D}^s_{l,m}$.  As discussed below, the non-Debye spectrum of $\hat{\rho}(\omega)$ and its rapid change %%@
vs. $T$ in the vicinity of $T_m$ reflect a rapidly changing length-scale in the problem.  To describe this, it is %%@
necessary to retain the $\bf q$ (wavevector) dependence of the Fourier transform $D_s({\bf q})$ of ${\bf %%@
D}^s_{l,m}$. (Clem and Coffey \cite{Clem} have shown that approximating $D_s({\bf q})$ by a constant %%@
$D$ merely reproduces a Debye-like spectrum.)  With inclusion of the pinning forces, the equation of %%@
motion is~\cite{Schmid}
\begin{equation}
\eta{\dot{\bf u}}_{\bf l} + 
\sum_{{\bf l}'} \bf D_{{\bf l,l}'}\cdot \bf u_{{\bf l}'}
+ \kappa\sum_{{\bf i}} {\bf u_i} \delta_{{\bf l,i}} 
= {\bf J\times \hat{z}} \phi_0,
\label{eq:eta}
\end{equation}
where the sites with pins are indexed by $\bf i$ (the pinning force is short-ranged).  To match the strong %%@
dispersion observed, OW assumed that the pin distribution is sparse~\cite{Ong} ($R_0\gg a_B$, where %%@
$R_0$ is the average pin separation, and $a_B = \sqrt{\phi_0/B}$ the lattice spacing).  Appendix C %%@
summarizes the solution of OW.

In response to the driving current ${\bf J}e^{j\omega t}$, the magnitude and phase of the vortex velocity %%@
(averaged over the sample) produces the ac voltage that determines $\hat{\rho}$.  At low $\omega$, the %%@
inter-vortex forces dominate the friction force $\eta \dot{\bf u}$, so the velocity response is predominantly %%@
inductive ($\omega{\cal L}_v \gg\rho_1$).  In the opposite limit of large $\omega$, $\eta \dot{\bf u}$ is %%@
dominant. As ${\cal L}_v\rightarrow 0$, $\rho_1$ approaches the free-flow value $B\phi_0/\eta$.  Thus %%@
${\cal L}_v$ is a measure of the average restraining force on each vortex.  It is largest if the restraining %%@
forces are dominant, but vanishes when viscous damping dominates.  

The main feature of the solution is that the lattice propagator $G({\bf R},\omega)$, which transmits %%@
information on the vortex displacement at site $\bf R$ to its surroundings, has an {\em effective range} %%@
$R_G = a_B\sqrt{c_{66}/\eta\omega}$ that varies as $1/\sqrt{\omega}$ (Eq. \ref{p}).  At low %%@
frequencies, $R_G\gg a_B$, which implies that the motion of any one vortex is strongly correlated with that %%@
of a huge number of neighbors, a subset of which are pinned (the 2D correlation volume is $\sim R_G^2$).  %%@
Hence the velocity response is large and inductive.  With increasing $\omega$, the average vortex is %%@
restrained by fewer and fewer pinned vortices as $R_G$ shrinks.  The in-phase (dissipative) component of %%@
the average velocity increases, while the inductance drops.  When $\omega$ exceeds the characteristic %%@
frequency 
\begin{equation}
\omega_{66}= \frac{4\pi c_{66}}{\eta}\frac{a_B^2}{R_0^2},
\label{w66}
\end{equation}
$R_G$ falls below the average pin spacing $R_0$.  The majority of vortices now respond as if they are %%@
free, and $\rho_1$ rapidly approaches the Bardeen-Stephen free flux-flow value $\rho_f = B\phi_0/\eta$ %%@
while ${\cal L}_v$ decreases to zero.  

The crossover is observed as the knee in  $\rho_1$ at $T$ just below $T_m$ (see Fig. \ref{rho1}a).  As %%@
$T$ approaches $T_m$ from below, the rapid softening of $c_{66}$ causes this crossover to occur at %%@
progressively lower frequencies.  Finally, in the liquid state above $T_m$, the vanishing of $c_{66}$ implies %%@
that virtually all the vortices are decoupled from the pins: $\rho_1$ equals $\rho_f$ at all $\omega$ in the RF %%@
range, while ${\cal L}_v$ is zero.

\subsection{The fitting procedure}  
As discussed in Appendix C, the OW solution has 4 adjustable parameters at each $T$, viz. $\kappa(T)$, %%@
$c_{66}(T)$, $\eta(T)$ and $R_0$.  Empirically, the fits are not sensitive to the particular choice of $R_0$ %%@
(as long as $R_0\gg a_B$).  Following WOGT, we set $R_0 = 7\;a_B$ for $B$ = 1 T.

As discussed in Sec. \ref{fixedfreq}, the viscosity $\eta(T)$ may be obtained from the initial slope of %%@
$\rho_1$ vs. $H$ measured at 15.4 MHz (Fig. \ref{rhoeta}).  Adopting the viscosity data in the insert of %%@
Fig. \ref{rhoeta}, we then have only {\em two} adjustable parameters at each $T$, viz. $\kappa(T)$ and %%@
$c_{66}(T)$ [equivalently, $\omega_p(T)$ and $\omega_{66}(T)$].  As we have a {\em continuous} %%@
spectrum measured over 2 decades in frequency to find 2 numbers, the fitting problem is evidently strongly %%@
over-constrained.  

The fitting procedure consists in selecting seed values for $\omega_p$ and $\omega_{66}$ so that the %%@
quantity $\hat{\cal{S}}(\omega)$  (Eq. \ref{S}) can be computed by summing the propagator %%@
$G_{2D}({\bf R_i},\omega)$ (Eq. \ref{G2Da}) over an area much larger than $R_G^2$ (the pins are %%@
arrayed in a triagular superlattice of spacing $R_0$, and the sum is typically extended to 12 impurity `shells' %%@
from the origin).  With $\hat{\cal{S}}(\omega)$, we may compute the complex resistivity %%@
$\hat{\rho}(\omega)$, viz. 
\begin{equation}
{\hat\rho}(\omega)  =  \left( \frac{B\phi_0}{\eta}\right)
\left[ 1 - \left( \frac{\omega_p a^2 } {{\rm j}\omega R_0^2}  \right) 
\frac{1}{1+\hat{\cal{S}}(\omega)}\right].
					\label{rho}
\end{equation}

The calculated curve is then compared with the measured spectra of $\rho_1(\omega)$, and the fit %%@
parameters are changed in an iterative manner until the fit is optimized (the inductance data are not used in %%@
the optimization).  For spectra near $T_m$, the high-frequency curvature is most sensitive to $\omega_p$ %%@
whereas the curvature at low $\omega$ is largely determined by the choice of $\omega_{66}$, as expected %%@
on physical grounds (the value of $\eta$ mainly fixes the overall scale of $\rho_1$).  As shown in Fig. %%@
\ref{rho1}b, close fits to the data are achieved over the 2-decade range of $\omega$.  At low $\omega$, %%@
the calculated $\rho_1$ is capable of matching the curvature of the measurements to a surprising degree %%@
(with only 2 adjustable parameters).  The close agreement achieved in Fig. \ref{rho1}b offers encouraging %%@
support for the validity of the model.

\subsection{Calculated inductance}
With the parameters $\omega_p$ and $\omega_{66}$ optimized at each $T$, we may calculate the %%@
vortex-lattice inductance ${\cal L}_v$.  As discussed above, in the limit $\omega\rightarrow 0$, the %%@
extended range of the propagator $G_{2D}(R,\omega)$ implies that each vortex is restrained by a great %%@
many pins.  Hence the velocity response is purely inductive.  The calculated ${\cal L}_v$ displays the strong %%@
divergence observed in the RF measurements.  In Fig. \ref{FitL}, we compare at 4 selected $T$ the %%@
calculated ${\cal L}_v$ (solid lines) with the observed ${\cal L}_v$ (open symbols).  

The sample inductance is the sum of the vortex-lattice term ${\cal L}_v$ and the volume inductance ${\cal %%@
L}_{vol}$, viz.
\begin{equation}
{\cal L}_s(\omega,T,H) = {\cal L}_v(\omega,T,H) + {\cal L}_{vol}(\omega,T,H).
\label{Ls}
\end{equation}
The volume term cannot be measured directly at low $\omega$ because it is much weaker than the vortex %%@
term.  However, at frequencies above $\sim$ 3-10 MHz (depending on $T$), it accounts for nearly all of %%@
the sample inductance (see Sec. \ref{fixedfreq}).  Because of this uncertainty, we used the spectra of %%@
$\rho_1$ to optimize the fit parameters.
%Figure 6 (FitL)
\begin{figure}
\includegraphics[width=7cm]{rho2vsf.eps}
\caption{\label{FitL} Comparison of calculated $\rho_2(\omega)/\omega$ (solid lines) with the measured %%@
vortex inductance ${\cal L}_v(\omega,T,H_0)$ (open symbols) over the frequency range 100 kHz to 20 %%@
MHz in a field $H_0$ = 2 T.  $\rho_2(\omega)/\omega$ is calculated from Eq. \ref{rho} using the values of %%@
$\omega_p$ and $\omega_{66}$ derived from the fits to $\rho_1$ in Fig. \ref{rho1}b.  The estimated %%@
volume inductance has been subtracted from the total sample inductance (see text).  Above 87 K, ${\cal %%@
L}_v$ strongly diverges as $\omega\rightarrow 0$, while at 83.9 K, the divergence is not %%@
observed.}

\end{figure}

The contribution of the volume inductance has to be estimated to isolate the vortex-lattice term (Eq. %%@
\ref{Ls}).  In Sec. \ref{fixedfreq}, this was carried out using the observed $\rho_1$ together with %%@
calculations for a sample with elliptical section.  The estimated ${\cal L}_{vol}$ has been subtracted to %%@
isolate the vortex term ${\cal L}_v$ in the plots displayed in Fig. \ref{FitL}.  Although the uncertainties in %%@
the estimated ${\cal L}_{vol}$ are largest at low $\omega$, they have the least impact on the comparison %%@
because ${\cal L}_s$ diverges steeply in this limit.  The comparison (inset) shows good agreement between %%@
calculation and measurement above 87 K (no further refinements of the fit parameters were made in %%@
comparing with ${\cal L}_v$).  As $T$ approaches $T_m\simeq$ 88.5 K, the low-$\omega$ divergence in %%@
the observed ${\cal L}_v$ progresses to lower frequencies (see the curves at 87.9, 88.2 and 88.4 K in %%@
inset).  The calculated curves also match this progression.  At low $T$, however (83.9 K curve), the %%@
agreement is not as good because the uniform-$J$ assumption used to extract $\hat{\rho}$ is increasingly %%@
suspect.

\section{\label{modulus} Fit results}
\subsection{Collapse of the shear modulus}
The values of $\kappa(T)$ and $c_{66}(T)$, obtained from $\omega_p(T)$ and $\omega_{66}(T)$, %%@
respectively, are shown in panels (a) and (b) of Fig. \ref{figc66}.  In each case, we show two sets of values %%@
depending on whether $\eta_a$ or $\eta_b$ has been used in the fits.  We interpret the decrease of %%@
$\kappa$ as $T\rightarrow T_c$ (Panel (a)) as reflecting the decrease in condensation energy.  We note %%@
that $\kappa$ remains finite at the melting temperature $T_m$.  Hence our fit strongly argues against models %%@
in which the transition at $T_m$ is interpreted in terms of a vanishing $\kappa$.  

\begin{figure}   %Figure 7 (figc66)
\includegraphics[width=9cm]{c66norm.eps}
%\centerline{\psfig{file=Matl2/c66norm.eps,width=3.4in}}
\caption{\label{figc66} (a) Values of the Labusch parameter $\kappa$ obtained from the fits, using either %%@
$\eta_a$ (open circles) or $\eta_b$ (solid squares). The sharp peak (right scale) is the behavior of the %%@
sample inductance ${\cal L}_s = \rho_2/\omega$ measured at 1 MHz in a 2-T field [the zero of ${\cal %%@
L}_s$ is arbitrary (Sec. \ref{fixedfreq})]. (b) The normalized shear modulus $c_{66}$ (log scale) versus %%@
$T$ obtained from the fits, using either $\eta_a$ (open circles) or $\eta_b$ (solid squares) in Fig. %%@
\ref{rhoeta}.  Near $T_m$, $c_{66}$ decreases by 4 orders of magnitude within an interval of 0.5 K.  The %%@
solid line represent values of $c_{66}$ obtained from fits to $\hat{\rho}(\omega)$ with $\omega$ fixed at 1 %%@
MHz, and $H$=2 T, using $\eta_b$.  The dashed line is obtained with $\omega$ fixed at 2 MHz, but using %%@
$\eta_a$.
}
\end{figure}
The critical parameter is the shear modulus $c_{66}$ which undergoes a steep collapse by more than 3 %%@
decades within an interval of 1 K.  The collapse occurs over the background (gradual) decrease of %%@
$c_{66}$ associated with the power-law decrease of the condensate density as $T\rightarrow T_c$.  %%@
Because of the rapid collapse in $c_{66}$, the steep increase in ${\cal L}_s$ at temperatures below %%@
$T_m$ is interrupted.  Within a 1-K interval, ${\cal L}_s$ drops to its normal-state value (background %%@
curve in Fig. \ref{figc66}b).

Each of the discrete data points for $c_{66}$ is derived from a full spectrum for $\rho_1$.  To follow the %%@
collapse in more detail, we have also adopted a different procedure, using the 1 MHz curve for ${\cal %%@
L}_s$ and $\rho_1$ measured {\em continuously} vs. $T$, together with interpolated values of $\eta_a$ %%@
and $\eta_b$.  This allows us to determine $c_{66}$ as a continuous curve very close to $T_m$ (the fitting %%@
procedure becomes unstable at lower $T$).  We show these continuous curves as solid and broken lines in %%@
Fig. \ref{figc66}b.  At low $T$, the continuous-fit values are slightly larger than the spectra-based fits which %%@
are more accurate, but close to $T_m$, they match very well.  The continuous curve allows us to monitor %%@
the decrease of $c_{66}$ over 4 decades.

From Fig. \ref{figc66}, we conclude that the transition at $T_m$ is associated with a rapid collapse of the %%@
shear modulus.  However, the Labusch parameter $\kappa$ remains finite.  This provides firm evidence that %%@
the transition involves the {\em collapse of the shear modulus}, rather than the vanishing of the Labusch %%@
parameter. 

\subsection{Summary}
In summary, we have performed measurements of $\hat{\rho}(\omega)$ of the vortex lattice in an untwinned %%@
YBCO crystal to investigate the collapse of the lattice inductance at the melting temperature, using a %%@
high-resolution 4-probe RF technique.  At all $T$ investigated, the resistivity $\rho_1(\omega)$ is strongly %%@
dispersive.  Over the frequency range 100 kHz - 20 MHz, $\rho_1$ increases from zero to the free-flow %%@
value $B\phi_0/\eta$, with a cross-over frequency scale ($\omega_{66}$) that rapidly decreases towards %%@
zero as $T$ approaches the melting temperature $T_m$ from below.  In the solid phase, the inductance %%@
$\rho_2/\omega$ displays a steep divergence as $\omega\rightarrow 0$.  However, at $T_m$, this %%@
low-frequency divergence collapses to give an $\omega$-independent inductance in the vortex-liquid state %%@
above $T_m$.  The observed spectra are qualitatively different from those in thin-film YBCO, and %%@
incompatible with the predictions of vortex glass theory.  To extract the vortex-lattice shear modulus from %%@
these spectra, we have used the mean-field solution of the Schmid-Hauger-Larkin-Ovchinikov model to fit %%@
the complex resistivity spectra.  From the 2-parameter fit at each temperature, the normalized shear modulus %%@
$c_{66}(T)$ displays a remarkable 4-decade collapse towards zero at $T_m$.  Hence the observed %%@
collapse of the inductance is driven by the vanishing of the shear modulus rather than the vanishing of the %%@
pinning strength $\kappa$.

\section*{\label{expt} Appendix A: Experimental details}
We describe first the 2-probe method used by WOGT \cite{Wu2}.  The sample $Z_2$ bridges the inner %%@
and outer conductors of both the incident and transmission coax cables of line impedance $Z_0$ (Fig. %%@
\ref{expt}a).  The load resistors $Z_1$ and $Z_1'$ are 100-$\Omega$ thin-film resistors.  This %%@
configuration maximizes the sensitivity of the transmitted wave to small changes in $Z_2$ when $|Z_2|\ll %%@
Z_0$.  The incident wave is transmitted with transmission coefficient $\Gamma_T$, and reflected with %%@
reflection coefficient $\Gamma_R$, where
\begin{eqnarray}
\Gamma_T & = &\frac{2Z_0Z_2}{Z_2(2Z_0+Z_1+Z_1')+(Z_0+Z_1)(Z_0+Z_1')},\\
\Gamma_R & = & \Gamma_T \frac{Z_0+Z_1'}{Z_0+Z_1}+\frac{Z_1-Z_0}{Z_1+Z_0}.
\label{gamma2}
\end{eqnarray}
Phase-sensitive detection of the transmitted signal allows $Z_2$ to be determined (Fig. \ref{expt}a).  The %%@
incident power from a synthesizer (Hewlett Packard 3336C) is typically fixed at +4.5 dBm, corresponding %%@
to a current less than 5 mA at the sample. The transmitted signal is phase-detected by a high-frequency %%@
lock-in Extender/Enhancer (Palo Alto Research PAR100), which down-converts signals in the range 1-25 %%@
MHz to 1-50 kHz, suitable for a standard lock-in amplifier (Stanford Research SR830). 

%Figure 8 (expt)
\begin{figure}
\includegraphics[width=7cm]{MatlExpt.eps}
%\centerline{\psfig{file=Matl1/MatlExpt.eps,width=3.5in}}
\caption{\label{expt} (a) Schematic diagram of the measurement circuit.  The sample shunts the inner and %%@
outer conductors, so that the transmitted signal is highly sensitive to slight changes in the sample impedance %%@
$Z_2$. Transformers T1 and T2 are inserted to decouple grounds in the 4-probe technique.  The lower %%@
figure (b) shows the labelling of the contact impedances $Z_{si},Z_{si}'$, which may greatly exceed $Z_2$ %%@
in amplitude.}
\end{figure}

To modify the technique for 4-probe measurements (Fig. \ref{expt}a), we inserted an RF transformer T1 of %%@
bandwidth 10 kHz - 150 MHz (Mini-Circuits T1-6-KK81) at the output of the synthesizer.  The transmitted %%@
signal passes through a second transformer T2 before entering the PAR100.  In Fig. \ref{expt}b, the contact %%@
impedances are represented by the impedances $Z_{si}$ and $Z_{si}'$. The transmission and reflection %%@
coefficients are now
\begin{eqnarray}
\Gamma_T & = &\frac{2\xi_2}{(1+\xi_T)(1+\xi_T')-\xi_2^2},\label{gamma4t}\\
\Gamma_R & = &\frac{(\xi_T-1)(1+\xi_T')-\xi_2^2}{(1+\xi_T)(1+\xi_T')-\xi_2^2},
\label{gamma4r}
\end{eqnarray}
where $\xi_T = (Z_1+Z_{s1}+Z_2+Z_{s3})/Z_0, \;\;\xi_T' = (Z_1' + Z_{s1}'+Z_2+Z_{s3}')/Z_0 $, and %%@
$\xi_2=Z_2/Z_0$.

While the 4-probe method provides much higher resolution, a drawback is that the inserted transformers %%@
introduce strong reflections.  The largest reflection comes from the upstream transformer T1.  A wave %%@
reflected from the sample is reflected again at T1 and adds an $\omega$-dependent contribution to the %%@
original transmitted wave.  

The effect of the multiply reflected wave on the total transmitted signal at the detector is %%@
expressed as 
\begin{equation}
\Gamma_T^{obs} \equiv \frac{V_{out}}{V_{in}} = {\rm e}^{jk\ell_{tot}}\Gamma_T\Gamma_T^1
\left[1 + {\rm e}^{j2k\ell_1} \Gamma_R\Gamma_R^1 + \cdots \right],
\label{mult} 
\end{equation}
where $\Gamma_T^1$ and $\Gamma_R^1$ are, respectively, the transmission and reflection coefficients of %%@
T1.  The path-lengths from T1 to the detector, and from T1 to the sample, are called $\ell_{tot}$ and %%@
$\ell_1$, respectively.  Higher-order reflection contributions (notably from T2) are indicated by $\cdots$.  In %%@
addition, we are now also sensitive to slight $\omega$-dependent deviations from unity of the PAR100's %%@
transfer function $\hat{g}(\omega)$. 

To compensate for these two background contributions, we used a reference combination %%@
$Z_r,Z_{1,r},Z'_{1,r}$ tailored to have a reflection coefficient $\Gamma_R^r$ nearly equal to that of the %%@
sample (hereafter, $\Gamma_R^s$).  As $ |Z_s| \ll 1$, the sample's $\Gamma_R^s$ is very close to 0.33, %%@
with $Z_1$ and $Z_1'$ = 100 $\Omega$. The best choice for the reference combination is $Z_r$ = 10 %%@
$\Omega$, and $Z_{1,r}=Z'_{1,r}$ = 90.9 $\Omega$, which has an $\omega$-independent %%@
$\Gamma_R^r$ of 0.3344, and a transmission coefficient $\Gamma_T^r$= 0.044.  

With the sample in place, the signal presented at the input of the SR 830 lock-in is given by $V_s = %%@
\hat{g}(\omega)V_{out}(Z_s)$, where $V_{out}$ is given by Eq. \ref{mult}.  If the sample (and load %%@
resistors) are replaced by the reference combination, we have $V_r = \hat{g}(\omega)V_{out}(Z_r)$.  %%@
Dividing these two equations removes $\hat{g}(\omega)$.  This leaves the multiple reflection factor (quantity %%@
in $[\cdots]$ in Eq. \ref{mult}).  However, as $\Gamma_R^s$ and $\Gamma_R^r$ are nearly identical by %%@
design, the leading term in $[\cdots]$ is the same (the other terms are down by a factor $\Gamma_T^r\ll %%@
1$).  Hence we have 
\begin{equation}
\frac{V_s}{V_r} = \frac{\Gamma_T^s}{\Gamma_T^r}[1 + {\cal O}(\Gamma_T^s)],
\label{vs}
\end{equation}
where $\Gamma_T^s$ and $\Gamma_T^r$ are given by Eq. \ref{gamma4t} with $Z_2=Z_s$ and $Z_r$, %%@
respectively.  Our procedure is to measure and store the curve of $V_r$ vs. $\omega$ at each temperature %%@
of interest.  Then $V_s$ is measured versus $\omega$.  Since $\Gamma_T^r$ is known, $\Gamma_T^s$ %%@
(and $Z_s$) may be calculated from $V_s$ and $V_r$ using Eq. \ref{vs}.  We checked the reliability of the %%@
procedure using a second reference consisting of just a single 50-$\Omega$ resistor connecting the inner %%@
conductors of the two coax cables (this also has $\Gamma_R^r$ = 0.333, but a much larger %%@
$\Gamma_T^r$ = 0.667). 

\section*{\label{impedance} Appendix B: Impedance of a conductor with elliptical %%@
cross-section}
The impedance $Z_s(\omega)$ of a conductor with elliptical cross-section is obtained by solving the 2D %%@
Helmholtz equation $(\nabla^2+\hat{\kappa}^2){\bf A}=0$ in elliptical coordinates $(\xi,\eta)$, where %%@
$\hat{\kappa}^2 = 2j\delta_n^{-2}$.  With ${\bf A}(\xi,\eta) = \psi(\xi)\phi(\eta)\hat{\bf z}$, the Helmholtz %%@
equation separates into the two Mathieu equations, viz. \cite{McL}
\begin{eqnarray}
\frac{d^2\psi}{d\xi^2} - [a(\hat{q}) -2\hat{q}\cosh 2\xi]\psi = 0,\\
\frac{d^2\phi}{d\eta^2} + [a(\hat{q}) -2\hat{q}\cos 2\eta]\phi = 0,
\label{Mathieu}
\end{eqnarray}
where $\hat{q}\equiv -\hat{\kappa}^2h^2/4$ ($h$ is the foci spacing).  With the boundary condition %%@
$A(\xi_0,\eta) = A_S$ (i.e. a constant on the surface $S$), the vector potential may be expanded in terms %%@
of the Matthieu functions as
\begin{equation}
A(\xi,\eta) = \sum_{n=0}^{\infty} c_{2n}Ce_{2n}(\xi,\hat{q})ce_{2n}(\eta,\hat{q}).
\end{equation}
$Ce_{2n}$ is expressed by the expansion 
$Ce_{2n}(\xi,\hat{q})= \sum_{r=0}^\infty A^{(2n)}_{2r} \cosh(2r\xi)$
[the same expansion holds for $ce_{2n}$, but with $\cos(2r\eta)$ in place of $\cosh(2r\xi)$].  

The Mathieu functions are not tabulated because the separation parameter $a(\hat{q})$ and the coefficients %%@
$A^{(2n)}_{2r}$ are functions of $\hat{q}$.  For a given $\hat{q}$, $a(\hat{q})$ satisfies the continued %%@
fraction  
$a = -(\hat{q}^2/2)/[1-(a/4)-(\hat{q}^2/64)/[1-\cdots)]]$ \cite{McL}.  We truncated the continued %%@
fraction at the 12$^{th}$ order.  The 12 roots \{$a_n(\hat{q})$\} generate the full matrix %%@
$A^{(2n)}_{2r}$ with $(n,r)\;=1,\cdots 12$.  

Finally, with the ac voltage $V(\omega) = -j\omega A_S$ and the ac current $I = %%@
-\hat{\kappa}^2/\mu_0\int_S d\xi d\eta A(\xi,\eta)$, the impedance may be expressed as 
\begin{equation}
Z_s(\omega) = \frac{j\omega\ell\mu_0}{(2\pi)^2} \left[ \sum_n \frac{(A^{(2n)}_0)^2}{{\cal L}_{2n}} %%@
\frac{Ce_{2n}'(\xi_0,\hat{q})} {Ce_{2n}(\xi_0,\hat{q})} \right]^{-1},
\end{equation}
where $Ce_{2n}'= \partial Ce_{2n}/\partial \xi$, and ${\cal L}_{2n} = \int_0^{2\pi} %%@
ce^2_{2n}(\eta,\hat{q})d\eta$.  In Fig. \ref{elliptical}, we display the $\omega$ dependence of %%@
$R_s(\omega)$ and $\cal{L}(\omega)$ calculated with parameters $\xi_0\simeq 0.108$ and $h= 250 %%@
\mu$m, matching the sample cross-section, and $\rho_1 = 4{\rm \mu\Omega cm}$.


\section*{\label{Ong} Appendix C: The solution of Ong and Wu}
We summarize the equations in Ref. \cite{Ong}.  The displacement ${\bf u_l}$ of a vortex at site $\bf l$ is %%@
described by the equation
\begin{equation}
\eta \dot{\bf u_l} + 
\sum_{s,\bf m} {\bf D}^s_{\bf{l,m}} \cdot {\bf u_m}
+ \kappa\sum_{pins\;\bf i} {\bf u_i} \delta_{{\bf l,i}} 
= {\bf J \times \hat{z}}\phi_0.
		\label{eta}
\end{equation}
Only a subset of vortices (at sites indexed by $\bf i$) are affected by the pins.  The remaining vortices are %%@
`free', apart from being restrained by the lattice forces represented by ${\bf D}^s_{\bf{l,m}}$ ($s= T, L$ %%@
indexes the polarization mode).  The displacement at an arbitrary site $\bf l$ is related to the displacement of %%@
the pinned vortices by
\begin{eqnarray}
{\bf u_l}(\omega) & = & \left( \frac{J\phi_0}
{\rm j\omega\eta}\right){\bf\hat y} - \frac{\kappa}{N} \sum_{pins\;\bf i} 
{\bf u_i}(\omega)\cdot  \nonumber\\
&  &  \times\sum_{s,{\bf q}}
\frac { {\bf\hat e}_s({\bf q}) {\bf \hat e}_s({\bf q}) 
{\rm e}^{ j\bf q \cdot({\bf R_l}- {\bf R_i} ) } }
{ [D_s(\bf q) + \rm j\omega\eta] },
					\label{ul}
\end{eqnarray}
where ${\bf {\rm\hat{e}}}_s({\bf q})$ is the unit vector for the mode $s$.  

At the mean-field level, the displacement of the vortex at a pin-site ${\bf u}_{imp}$ is
\begin{equation}
{\bf u}_{imp}(\omega)  = {\bf\hat y} \left( {J\phi_0\over{\rm j\omega\eta}} 
\right) \frac{1}{1+ \hat{\cal{S}}(\omega)},
\label{uimp}
\end{equation}
where all the interaction effects are in the term
\begin{equation}
\hat{\cal{S}}(\omega) \equiv \frac{\omega_p a^2 }{{\rm j}\omega R_0^2} + 
\kappa[ G(0,\omega) + \sum_{\bf i}' G({\bf R_i},\omega)],	
			\label{S}
\end{equation}
and $\omega_p \equiv \kappa/\eta$ is the pinning frequency.  The lattice propagator  $G({\bf R},\omega)$ %%@
is defined as \cite{Schmid,Ong}
\begin{equation}
G({\bf R},\omega) = \frac{1}{N} \sum_{s{\bf q}} 
\frac { ({\bf{\hat y} \cdot {\hat e}}_s({\bf q}) )^2
{\rm e}^{ {\rm j}{\bf q\cdot R} } } 
{ [D_s({\bf q}) + {\rm j}\omega\eta]}.
					\label{eq:G}
\end{equation}
In Eq. \ref{S}, the first term represents the uniform $\bf q$=0 mode of the transmitted elastic forces.  The %%@
remaining terms are elastic forces transmitted to the pinned vortex at the origin $\bf 0$ arising from its own %%@
displacement ($G(0,\omega)$), or the displacement of other pinned vortices $\sum_{\bf i}' G({\bf %%@
R_i},\omega)$ (the prime indicates that the $\bf i=0$ term is left out).

Neglecting the longitudinal modes, the transverse force matrix is expressed as \cite{Schmid}
\begin{equation}
D_T({\bf q}) = [c_{66}  q^2 + c_{44} q_{z}^2]a_B^2.  \label{DT}
\end{equation}
where $q$ is the in-plane component of ${\bf q}$, viz. ${\bf q} = (q,q_z)$, and $c_{66}$ and $c_{44}$ %%@
are the shear and tilt moduli, respectively.  

In the 2D limit, the propagator simplifies to the form (after angular-averaging within the $ab$ plane and %%@
neglecting the tilt modulus $c_{44}$) 
\begin{equation}
\kappa G_{2D}({\bf R}_{\perp},\omega) = g \int_{0}^{Q}dq 
\frac{qJ_0(qR) } { [q^2 + {\rm j} p^2] }.
				\label{G2D}
\end{equation}
\noindent
where $Q=\sqrt{4\pi/a_B^2}$, $J_0(x)$ is the zeroth order Bessel function, $R=|{\bf R}_{\perp}|$, with %%@
${\bf R = (R}_{\perp}, R_z)$, and $g$ the coupling constant $\kappa/4\pi c_{66}$. The frequency %%@
dependence appears only in the characteristic wavevector 
\begin{equation}
p = \sqrt{ {\eta\omega\over{c_{66}a_B^2} } }   \label{p} 
\end{equation}
which serves as a cut-off of the logarithmic divergence in the 2D case.  Setting $R=0$ in Eq. \ref{G2D}, we %%@
have 
\begin{equation}
\kappa G_{2D}(0,\omega) = g \left[ \ln \sqrt{\left( 1+ 
{Q^4\over{p^4}} \right)}  - {\rm j} \arctan{({Q^2\over{p^2}})} \right].    
			\label{G2D0}
\end{equation}
As the propagator $G_{2D}(R,\omega)$ is used repeatedly in the sum in Eq. \ref{S}, it is convenient to %%@
adopt the approximation (accurate to a few percent)  
\begin{eqnarray}
\kappa G_{2D}^a(R,\omega) &=& g  \left\{ \ln \sqrt{\left[\frac
{(\gamma R)^4 + p^{-4}} {(\gamma R)^4 + Q^{-4} } \right]} \right. \nonumber\\
&& \left. -{\rm j} \arctan({Q^2\over{p^2}}) \frac{1}
{ [1+(\beta pR)^2]^2 } \right\},
 		\label{G2Da}
\end{eqnarray}
\noindent
with $\beta = 0.52$ and $\gamma = 0.85$.  Equations \ref{G2D0} and \ref{G2Da} were given incorrectly %%@
in OW \cite{Ong} ($cpf$ their Eqs. 18 and 19).  




%%%%%%%%%%%%
%\newpage
\section*{Acknowledgments}
We thank Hui Wu for generous assistance and helpful advice.  N.P.O. acknowledges support from the U.S. %%@
Office of Naval Research (Contract N00014-01-0281) and the New Energy and Industrial Tech. Develop. %%@
Org. (NEDO), Japan.  R.G. and L.T. are funded by NSERC of Canada and FCAR of Quebec. L. T. %%@
acknowledges support from the Canadian Inst. for Advanced Research and the A.P. Sloan %%@
Foundation.  

% now the references. delete or change fake bibitem. 
\bigskip\noindent
$^*${\em Present address of P.M.}: Koch Industries, 20 East Greenway Plaza, Houston, TX %%@
77046.\newline
$^\dagger${\em Present address of L. T.: Department of Physics,
University of Toronto, Toronto, Ontario, CANADA  M5S 1A7.}   
\begin{references}
\bibitem{Gitt} J.I. Gittleman and B. Rosenblum, Phys. Rev. Lett. {\bf 16}, 734
(1966).
\bibitem{Blatter} For a review, see G. Blatter, M. V. Feigel'man, V. B. Genshkenbein, A. I. Larkin, and V. %%@
M. Vinokur, Rev. Mod. Phys. {\bf 66}, 1125 (1994).
\bibitem{Zeldov} E. Zeldov, D. Majer, M. Konczykowski, V. B. Geshkenbein, V. M. Vinokur, and H. %%@
Shtrikman, Nature {\bf 375}, 373 (1995).
\bibitem{Welp} U. Welp, J. A. Fendrich, W. K. Kwok, G. W. Crabtree, and B. W. Veal
     Phys. Rev. Lett. {\bf 76}, 4809 (1996). 
\bibitem{Schilling} A. Schilling, R. A. Fisher, N. E. Phillips, U. Welp, D. Dasgupta, W. K. Kwok,  and G. %%@
W. Crabtree, Nature {\bf 382}, 791 (1996).

\bibitem{Kwok} W. K. Kwok, S. Fleshler, U. Welp, V. M. Vinokur, J. Downey, G. W. Crabtree, and M. %%@
M. Miller,   Phys. Rev. Lett. {\bf 69}, 3370 (1992); W. K. Kwok {\em et al.}, Phys. Rev. Lett. {\bf 72}, %%@
1088 (1994).
\bibitem{Safar} H. Safar, P. L. Gammel, D. A. Huse, D. J. Bishop, J. P. Rice, and D. M. Ginsberg,      %%@
Phys. Rev. Lett. {\bf 69}, 824 (1992).

\bibitem{Kes} H. Pastoriza and P. H. Kes, Phys. Rev. Lett. {\bf 75}, 3525 (1995).
\bibitem{Kwokmod} W. K. Kwok, J. A. Fendrich, V. M. Vinokur, A. E. Koshelev, and G. W. Crabtree,    %%@
Phys. Rev. Lett. {\bf 76}, 4596 (1996).
\bibitem{Wu2} Hui Wu, N. P. Ong, R. Gagnon and L. Taillefer, Phys. Rev. Lett. {\bf 78}, 334  %%@
(1997).
\bibitem{Koch} H. K. Olsson, R.H. Koch, W. Eidelloth, and R. P. Robertazzi,  Phys. Rev. Lett. {\bf 66}, %%@
2661 (1991).
\bibitem{Wu1} Hui Wu, N. P. Ong and Y. Q. Li, Phys. Rev. Lett. {\bf 71}, 2642 (1993). 
\bibitem{Anlage} Dong H Wu, J. C. Booth and Steven M. Anlage, Phys. Rev. Lett. {\bf 75}, 525 %%@
(1995).


\bibitem{Fisher} D. S. Fisher, M. P. A. Fisher, and D. A. Huse, Phys. Rev. B {\bf 43}, 130 %%@
(1991).
\bibitem{Dorsey} Alan T. Dorsey, Phys. Rev. B {\bf 43}, 7575 (1991).
\bibitem{Schmid} Albert Schmid and Wolfgang Hauger, Jnl. Low Temp. Phys. 
{\bf 11}, 667 (1973). 
\bibitem{Larkin} A. I. Larkin and Yu. N. Ovchinikov, Zh. Eksp. Teor. Fiz. {\bf 65}, 1704 (1973) [Sov. %%@
Phys. JETP {\bf 38}, 854 (1974)].
\bibitem{Ong} N.P. Ong and Hui Wu, Phys. Rev. B {\bf 56} 458 (1997).

\bibitem{Clem} J. R. Clem and Mark W. Coffey, Phys. Rev. B {\bf 46}, 14662 (1992).

\bibitem{Marion} See for e.g., {\it Classical Electromagnetic Radiation}, 3$^{rd}$ ed., M. A. Heald and J. %%@
B. Marion (Saunders College Pub. N.Y., 1995), p. 190.

\bibitem{McL} N. W. McLachlan, {\em Theory and applications of 
Mathieu functions} (Oxford, 1947), Chs. III and XVII.


%%%%%%
\end{references}


\end{document}               


%%%%%%%%%%%%%%%%%%

cond-mat0205466.extracted_bib
\bibitem{Gitt} J.I. Gittleman and B. Rosenblum, Phys. Rev. Lett. {\bf 16}, 734
(1966).

\bibitem{Blatter} For a review, see G. Blatter, M. V. Feigel'man, V. B. Genshkenbein, A. I. Larkin, and V. %%@
M. Vinokur, Rev. Mod. Phys. {\bf 66}, 1125 (1994).

\bibitem{Zeldov} E. Zeldov, D. Majer, M. Konczykowski, V. B. Geshkenbein, V. M. Vinokur, and H. %%@
Shtrikman, Nature {\bf 375}, 373 (1995).

\bibitem{Welp} U. Welp, J. A. Fendrich, W. K. Kwok, G. W. Crabtree, and B. W. Veal
     Phys. Rev. Lett. {\bf 76}, 4809 (1996). 

\bibitem{Schilling} A. Schilling, R. A. Fisher, N. E. Phillips, U. Welp, D. Dasgupta, W. K. Kwok,  and G. %%@
W. Crabtree, Nature {\bf 382}, 791 (1996).


\bibitem{Kwok} W. K. Kwok, S. Fleshler, U. Welp, V. M. Vinokur, J. Downey, G. W. Crabtree, and M. %%@
M. Miller,   Phys. Rev. Lett. {\bf 69}, 3370 (1992); W. K. Kwok {\em et al.}, Phys. Rev. Lett. {\bf 72}, %%@
1088 (1994).

\bibitem{Safar} H. Safar, P. L. Gammel, D. A. Huse, D. J. Bishop, J. P. Rice, and D. M. Ginsberg,      %%@
Phys. Rev. Lett. {\bf 69}, 824 (1992).


\bibitem{Kes} H. Pastoriza and P. H. Kes, Phys. Rev. Lett. {\bf 75}, 3525 (1995).

\bibitem{Kwokmod} W. K. Kwok, J. A. Fendrich, V. M. Vinokur, A. E. Koshelev, and G. W. Crabtree,    %%@
Phys. Rev. Lett. {\bf 76}, 4596 (1996).

\bibitem{Wu2} Hui Wu, N. P. Ong, R. Gagnon and L. Taillefer, Phys. Rev. Lett. {\bf 78}, 334  %%@
(1997).

\bibitem{Koch} H. K. Olsson, R.H. Koch, W. Eidelloth, and R. P. Robertazzi,  Phys. Rev. Lett. {\bf 66}, %%@
2661 (1991).

\bibitem{Wu1} Hui Wu, N. P. Ong and Y. Q. Li, Phys. Rev. Lett. {\bf 71}, 2642 (1993). 

\bibitem{Anlage} Dong H Wu, J. C. Booth and Steven M. Anlage, Phys. Rev. Lett. {\bf 75}, 525 %%@
(1995).



\bibitem{Fisher} D. S. Fisher, M. P. A. Fisher, and D. A. Huse, Phys. Rev. B {\bf 43}, 130 %%@
(1991).

\bibitem{Dorsey} Alan T. Dorsey, Phys. Rev. B {\bf 43}, 7575 (1991).

\bibitem{Schmid} Albert Schmid and Wolfgang Hauger, Jnl. Low Temp. Phys. 
{\bf 11}, 667 (1973). 

\bibitem{Larkin} A. I. Larkin and Yu. N. Ovchinikov, Zh. Eksp. Teor. Fiz. {\bf 65}, 1704 (1973) [Sov. %%@
Phys. JETP {\bf 38}, 854 (1974)].

\bibitem{Ong} N.P. Ong and Hui Wu, Phys. Rev. B {\bf 56} 458 (1997).


\bibitem{Clem} J. R. Clem and Mark W. Coffey, Phys. Rev. B {\bf 46}, 14662 (1992).


\bibitem{Marion} See for e.g., {\it Classical Electromagnetic Radiation}, 3$^{rd}$ ed., M. A. Heald and J. %%@
B. Marion (Saunders College Pub. N.Y., 1995), p. 190.


\bibitem{McL} N. W. McLachlan, {\em Theory and applications of 
Mathieu functions} (Oxford, 1947), Chs. III and XVII.


%%%%%%
