The high-field phase diagram of the cuprates derived from the Nernst effect
Yayu Wang$^1$
N. P. Ong$^1$
Z. A. Xu$^{1,\dagger}$
T. Kakeshita$^2$
S. Uchida$^2$
R. Liang$^3$ and W. N. Hardy$^3$
Measurements of the Nernst signal in the vortex-liquid state of the cuprates to high fields (33 T) reveal that %%@ vorticity extends to very high fields even close to the zero-field critical temperature $T_{c0}$. In %%@ overdoped $La_{2-x}Sr_xCuO_4$ (LSCO) we show that the upper critical field $H_{c2}(T)$ curve %%@ does not end at $T_{c0}$, but at a much higher temperature. These results imply that $T_{c0}$ %%@ corresponds to a loss in phase rigidity rather than a vanishing of the pairing amplitude. An intermediate field %%@ $H^*(T)\ll H_{c2}(T)$ is shown to be the field scale for the flux-flow resistivity.
Hifieldphase_web.TEX
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\title{The high-field phase diagram of the cuprates derived from the Nernst effect} 
\author{Yayu Wang$^1$, N. P. Ong$^1$, Z. A. Xu$^{1,\dagger}$, T. Kakeshita$^2$, S. Uchida$^2$, %%@
D. A. Bonn$^3$, R. Liang$^3$ and W. N. Hardy$^3$}      
\address{
$^1$Department of Physics, Princeton University, Princeton, New Jersey 08544. 
}
\address{$^2$School of Frontier Sciences, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, %%@
Japan.}
\address{$^3$Department of Physics and Astronomy, University of British Columbia, Vancouver, BC V6T %%@
1Z1, Canada.}
\date{\today}      

\begin{abstract}
Measurements of the Nernst signal in the vortex-liquid state of the cuprates to high fields (33 T) reveal that %%@
vorticity extends to very high fields even close to the zero-field critical temperature $T_{c0}$.  In %%@
overdoped $\rm La_{2-x}Sr_xCuO_4$ (LSCO) we show that the upper critical field $H_{c2}(T)$ curve %%@
does not end at $T_{c0}$, but at a much higher temperature.   These results imply that $T_{c0}$ %%@
corresponds to a loss in phase rigidity rather than a vanishing of the pairing amplitude.  An intermediate field %%@
$H^*(T)\ll H_{c2}(T)$ is shown to be the field scale for the flux-flow resistivity.
\end{abstract}
\pacs{74.40.+k,72.15.Jf,74.72.-h,74.25.Fy}
\maketitle                   % Produces the title.

In the cuprate superconductors, strong fluctuations of the order parameter~\cite{Fisher,Blatter} and the %%@
extreme field scales make the task of constructing the experimental phase diagram a formidable challenge.  %%@
While the important vortex-solid melting line $H_m(T)$ is now well-known~\cite{Zeldov,Schilling,Liang}, %%@
there is great uncertainty in the region above $H_m$.  A key difficulty is the {\em loss of long-range phase %%@
rigidity} in the transition to the vortex-liquid state.  In the vortex-liquid state, the flux-flow resistivity $\rho$ %%@
rises very rapidly to the (extrapolated) normal-state value~\cite{Chien} even though substantial condensate %%@
strength and pairing amplitude remain.  This makes $\rho$ an unreliable `diagnostic' of the field-suppression %%@
of the pairing amplitude.  By contrast, the Nernst and Ettinghausen effects can probe the existence of %%@
vorticity in the superfluid with undiminished sensitivity even in intense fields.  From Nernst measurements in  %%@
$\rm La_{2-x}Sr_xCuO_4$ (LSCO) and $\rm YBa_2Cu_3O_y$ (YBCO) in fields up to 33 T, we have %%@
uncovered an anomalous property of $H_{c2}(T)$ near their zero-field critical temperature $T_{c0}$.  %%@
This anomaly is related to the existence of vortex-like excitations high above $T_{c0}$~\cite{Xu,Wang}, %%@
and is central to the key question of whether the Meissner transition in zero field is caused by the collapse of %%@
long-range phase coherence or the vanishing of the pairing amplitude (see also Corson {\it et %%@
al.}\cite{Corson}).  In YBCO, previous Ettinghausen~\cite{Palstra} and Nernst ~\cite{Ri,Clayhold} %%@
measurements were performed in lower fields.  Measurements on LSCO have been extended recently to %%@
high fields~\cite{Capan}.

In the Nernst experiment, vortices moving with velocity ${\bf v}$ down a thermal gradient $-\nabla %%@
T\parallel {\bf\hat{x}}$ generate a Josephson voltage which is observed as a transverse $E$-field $E_y = %%@
Bv_x$, with $\bf B$ the mean flux density.  The vortex-Nernst effect is well-explored in low-$T_c$ %%@
superconductors.  Figure \ref{L20}b shows the Ettinghausen-Nernst effect in %%@
Pb\underline{In}~\cite{Vidal}.  The depinning of the vortex lattice by a supercurrent ($\bf J\parallel %%@
\hat{x}$) leads to a large transverse heat current $J^h_y\parallel {\bf \hat{y}}$ that rises to a maximum and %%@
then decreases to zero at $H_{c2}$ (the weak fluctuation tail above $H_{c2}$ is not relevant here).  By %%@
Onsager reciprocity, the Nernst signal $e_y\equiv E_y/|\nabla T|$ shares the same profile as $J^h_y$ in Fig. %%@
\ref{L20}b~\cite{Onsager}.  To compare with theory, $e_y$ is divided by $\rho(H)$ to obtain the %%@
`transport line-entropy' $s_{\phi} = \phi_0 e_y/\rho$ ($Ts_{\phi}$ is the heat energy carried by a %%@
unit-length vortex line).  In Pb\underline{In}, the linear decrease in $s_{\phi}$ near $H_{c2}$ matches that %%@
of the magnetization $M(T,H) = -[H_{c2}(T)-H]/1.16[2\kappa^2-1]$ ($\kappa$ is the Ginzburg-Landau %%@
parameter), as given by the Caroli-Maki expression \cite{Maki,Vidal}
\begin{equation}
s_{\phi}(T,H) = \frac{\phi_0}{T} |M(T,H)| L_D(T) \label{Maki}
\end{equation}
(here $L_D(T)$ decreases gradually with $T$ from 1 at $T_{c0}$).

We start with the overdoped regime, where the field profiles of $e_y$ most resemble those in %%@
Pb\underline{In}. In Fig. \ref{L20}a, we display curves of $e_y$ vs. field $H$ in Sample 1 [$\rm %%@
La_{2-x}Sr_xCuO_4$ (LSCO) with $x$ = 0.20 and $T_{c0}$ = 28 K] (for experimental details see Ref. %%@
\cite{Wang}).  At each temperature $T$, the signal rises steeply above $H_m$, attaining a prominent %%@
maximum before decreasing.  The total data set defines experimentally the region in $H$ and $T$ where %%@
vorticity is strongly present.  As in Pb\underline{In}, the monotonic decrease at high fields reflects the field %%@
suppression of the condensate strength.  At high fields, all the curves below 14 K are observed to follow a %%@
common curve towards zero (broken line).  The trend implies that the low-$T$ traces are all wedged %%@
between the field axis and the broken line.  Hence all the low-$T$ curves vanish at the intercept of the %%@
common curve with the field axis (45-50 T), which corresponds to $H_{c2}(0)$ .

\begin{figure}				% Fig 1
\includegraphics[width=9cm]{LNvsH-LSvsH.eps}
\caption{\label{L20} (al) The field dependence of $e_y\equiv E_y/|\nabla T|$ at indicated $T$ in Sample 1 %%@
(LSCO with $x$ = 0.20, $T_{c0}$ = 28 K).  Below 25 K, $e_y$ rises to a sharp peak before decreasing %%@
montonically.  Broken line indicates asymptotic behavior at large-$H$ and low $T$.  (b) Plots of the %%@
transverse heat flux $J^h_y$ and $s_{\phi}$ vs. $H$ in an Ettinghausen experiment in Pb\underline{In} at %%@
1.5 K (by reciprocity, $J^h_y = e_y/JT$, with $J$ the applied supercurrent).  $H_d$ is the depinning field %%@
(modified from Vidal \protect\cite{Vidal}a).  (c) The field dependence of the transport line-entropy %%@
$s_{\phi}$ obtained from $e_y$ and $\rho$ in Sample 1 (LSCO, $x$ = 0.20).  The high-field linear %%@
extrapolations (broken lines) provide estimates of $H_{c2}(T)$.  Fits to Eq. \ref{Maki} give $\kappa$ = %%@
670 and 500 at 26 and 18 K, respectively.  (d) The temperature dependence of $\rho$ and $e_y$ in %%@
Sample 1 in a fixed field (33 T).  The broken line is the extrapolation of $\rho_N$.  Note that $e_y$ %%@
(measured at 33 T) goes to zero at $\sim$26 K instead of $\sim$9 K (the `knee' in $\rho$).
}
\end{figure}


Going to higher $T$, we immediately encounter an anomaly.  Conventionally, the $H_{c2}$ line goes %%@
linearly to zero at $T_{c0}$.  Hence $e_y$ ought to be finite in a field interval that $\rightarrow$ 0 as %%@
$T\rightarrow T_{c0}^-$.  In sharp contrast, we find that, close to $T_{c0}$, the magnitude of $e_y$ %%@
remains large and nearly unchanged up to intense fields.  There is no evidence for a field scale (flagged by %%@
$e_y\rightarrow 0$) that decreases to zero regardless of how close we are to $T_{c0}$.  To make this %%@
point quantitative, we convert $e_y$ to $s_{\phi}$ using $\rho(T,H)$ measured in the same sample.  The %%@
line entropy rises steeply to a sharp maximum before decreasing monotonically (Fig. \ref{L20}c).  The linear %%@
decrease at high fields, strikingly similar to that in Pb\underline{In}, allows $H_{c2}(T)$ to be %%@
determined~\cite{fit}.  These results show that $H_{c2}(T)$ decreases with rising $T$, {\em but remains at %%@
extraordinarily high values} as $T\rightarrow T_{c0}$ (see Fig. \ref{Lcont}).  [We neglect a small %%@
correction from the negative Nernst coefficient $\nu_N\simeq$ -40 nV/KT of the holes in overdoped LSCO %%@
(Fig. 3 of Ref. \cite{Xu}).  Accounting for the background increases $s_{\phi}$ and $H_{c2}(T)$ by $\sim %%@
10\%$. ]

\begin{figure}				% Fig 2
\includegraphics[width=8cm]{Lcontcolor.eps}
\caption{\label{Lcont} The measured values of $e_y(T,H)$ in Sample 1 (LSCO, $x$ = 0.20) represented %%@
as a contour plot in the $T$-$H$ plane.  Values of $e_y$ (in $\mu$V/K) at successive contour lines are %%@
shown on the left column (white arrows).  The lower and upper white lines are the melting field $H_m$ and %%@
`ridge' field $H^*$, respectively.  $H_{c2}(T)$ values estimated from Fig. \ref{L20}c are shown as solid %%@
circles.  $T_{c0}$ is indicated in the lower-right corner.  
}
\end{figure}

A powerful way to summarize the anomalous trends described is to represent $e_y(T,H)$ as a contour plot %%@
in the $T$-$H$ plane (Fig. \ref{Lcont}).  In the vortex-solid phase below the melting line $H_m$ (lower %%@
white curve), $e_y=0$ because the vortices are pinned.  As the vortex solid melts across $H_m$, $e_y$ %%@
increases steeply up to the `ridge' which represents a field scale that we call $H^*(T)$ (upper white curve).  %%@
To the right of the ridge, $e_y$ falls monotonically.  We focus on the two interesting regimes: the low-$T$ %%@
limit and the region near $T_{c0}$.  

As $T\rightarrow 0$, the two fields $H_m(T)$ and $H^*(T)$ increase rapidly towards values approaching %%@
$H_{c2}(0)$~\cite{melting}.   In this large-$H$, low-$T$ region, the contours are nearly horizontal, as %%@
required by the asymptotic high-field behavior (broken curve in Fig. \ref{L20}a).  This implies that the %%@
$H_{c2}$ line is nearly horizontal below $\sim$12 K, as anticipated above.  At temperatures $T\le %%@
T_{c0}$, the weak attenuation of $e_y$ to high fields corresponds to the nearly vertical orientation of the %%@
contours.  Clearly, there is no evidence for a putative $H_{c2}$ line that terminates at $T_{c0}$.  The %%@
values of $H_{c2}(T)$ derived from $s_{\phi}$ are plotted in Fig. \ref{Lcont} as solid %%@
circles.  

The anomalous features of the Nernst signal become more pronounced when we go to the underdoped %%@
regime.  A major change from the overdoped case is apparent in the field profiles of $e_y$.  We show in %%@
Fig. \ref{Y650} results in underdoped YBCO (Sample 2, with $y = 6.50$ and $T_{c0}$ = 50 K).  As %%@
$H$ increases above $H_m$, $e_y$ rises rapidly, but attains a very broad maximum that extends %%@
undiminished to 30 T.  In comparison with the curves in Fig. \ref{L20}a, we see that $e_y$ in Sample 2 %%@
approaches zero only at fields considerably higher than 30 T.  If we now convert $e_y$ to $s_{\phi}$ using %%@
the measured $\rho$, the broad profile translates to a line entropy that extrapolates to zero at fields much %%@
higher than our maximum field.  At 40 K, we may estimate the lower bound $H_{c2}(T) \ge$60 T, which is %%@
very high for a temperature so close to $T_{c0}$ = 50 K.  These features are common to all the %%@
underdoped cuprates we have investigated to date (for results in LSCO and Bi 2201, see Ref. %%@
\cite{Wang}).


\begin{figure}				% Fig 3
\includegraphics[width=7cm]{NH-Y650.eps}
\caption{\label{Y650} Variation of the Nernst signal $e_y\equiv E_y/|\nabla T|$ with field $H$ at fixed $T$ %%@
in Sample 2 (YBCO with $y$ = 6.50, $T_{c0}$ = 50 K).  Above 40 K, the measurements were %%@
performed up to 14 T only.  The insert shows $e_y$ vs. $H$ in Sample 3 (YBCO, $y$ = 6.99) close to its %%@
$T_{c0}$ = 93 K. Both are detwinned crystals grown in barium zirconate crucibles.
}
\end{figure}
The contour plot of $e_y$ for Sample 2 is displayed in the main panel Fig. \ref{Ycont}.  In comparison with %%@
Fig. \ref{Lcont}, the melting line is now considerably lower relative to $H^*$.  Further, the contour lines in %%@
the region around $T_{c0}$ are more spread out.  The Nernst signal retains considerable strength at 70 K, %%@
indicating that vorticity strongly persists up to 20 K above $T_{c0}$.  The contour plots show rather clearly %%@
the {\em continuity} between the high-temperature fluctuation phase observed by Xu {\em et %%@
al.}~\cite{Xu,Wang} and the vortex-liquid state below $T_{c0}$.  The 2 regimes smoothly merge together, %%@
and no phase boundary or `crossover' line separating them is apparent in the $T$-$H$ plane.  As mentioned %%@
above, the upper critical field is estimated to be above 60 T even at 40 K.  The much higher field scale in %%@
underdoped YBCO is also apparent if we compare the contour plots in Figs. \ref{Lcont} and \ref{Ycont}.  %%@
In overdoped LSCO, the contours close at fields above 13 T (the field scale of the `island') whereas, in %%@
underdoped YBCO, they do not close until $H$ exceeds 30 T.

\begin{figure}			% Fig 4
\includegraphics[width=8cm]{Contour650-700color.eps}
\caption{\label{Ycont} [Main Panel] Contour plot of $e_y$ in the $T$-$H$ plane in Sample 2 (YBCO, %%@
$y$ = 6.50).   The upper (lower) white curve represents the `ridge' field $H^*$ (melting field $H_m$).    %%@
Insert shows the corresponding contour plot in Sample 3 ($y$ = 6.99).  Note that the region between %%@
$H_m$ and $H^*$ is compressed to a thin sliver.  
}
\end{figure}
For comparison, we also display the contours in near-optimal YBCO (Sample 3 with $y$ = 6.99 and %%@
$T_{c0}$ = 93 K).  As shown by the vertical orientation of the contour lines from 85 to 95 K, as well as %%@
the field profiles in the insert in Fig. \ref{Y650}, $e_y$ is virtually field independent up to 14 T.  Even for %%@
near-optimal YBCO, there is no evidence for an $H_{c2}$ line that decreases to zero at $T_{c0}$ (this %%@
calls into question the interpretation of `$H_{c2}$' values inferred from bulk magnetization %%@
results~\cite{Welp}).

The observation of vortex-like excitations high above $T_{c0}$ by Xu {\em et al.} ~\cite{Xu,Wang} raised %%@
the question whether the cuprate superconducting transition at $T_{c0}$ actually corresponds to the loss of %%@
long-range phase rigidity, as opposed to the vanishing of the Gorkov pairing amplitude ${\cal F}({\bf %%@
x,x}')$~\cite{Emery}.   The present results provide strong support for the former.  A prominent anomaly at %%@
temperatures near $T_{c0}$ is the remarkably weak field dependence of $e_y$ above the field scale %%@
$H^*(T)$, which implies that the condensate strength extends undiminished to very high fields.  In %%@
underdoped cuprates, this anomalous behavior persists to our lowest temperatures (Fig. \ref{Y650}).  %%@
However, in overdoped samples, the profiles sharpen up below 25 K to reveal pronounced field %%@
suppression at high fields (Fig. \ref{L20}a).  Values of $H_{c2}(T)$ inferred from the line-entropy profiles %%@
show that the reason for the anomaly is that the $H_{c2}$ vs. $T$ line does not terminate at $T_{c0}$, but %%@
at a much higher temperature.   

This seems to us to be compelling evidence that, at $T_{c0}$, the steep increase of $\rho$ and the collapse %%@
of the Meissner effect, correspond to a pre-emptive loss of phase rigidity (as has been advocated by Emery %%@
and Kivelson \cite{Emery} and others) rather than the vanishing of ${\cal F}({\bf x,x}')$.  This scenario is %%@
qualitatively distinct from that in bulk low-$T_c$ superconductors.  Although $\rho$ saturates rapidly to %%@
$\rho_N$ above $T_{c0}$ (in zero field), there remains sufficient condensate density to support vorticity.  %%@
In Sample 1, the measured $H_{c2}$ curve implies that we need to apply fields above 30 T to suppress all %%@
vorticity near $T_{c0}$.  The field scale is even higher in underdoped YBCO.  

A hallmark of the vortex-liquid state in cuprates is that $\rho$ rapidly increases and `saturates' to the %%@
extrapolated normal-state value ($\rho_N$) even when $H\ll H_{c2}(T)$. To illustrate this, we plot the $T$ %%@
dependence of $\rho$ and $e_y$ in Sample 1 with $H$ fixed at 33 T in Fig. \ref{L20}d (insert).  Clearly, %%@
the large Nernst signal indicates that the region from 5 to 25 K is in the vortex-liquid state (see Fig. %%@
\ref{Lcont}).  However, between 15 and 30 K, $\rho$ virtually lies on the extrapolated $\rho_N$ curve.  %%@
Hence reliance on $\rho$ alone would lead to a serious underestimate of `$H_{c2}$'.  

In all cuprates studied, the ridge field $H^*(T)$ line is ubiquitous.  Over a broad range of $T$, the ratio %%@
$\rho/\rho_N$ has the value 0.3-0.4 on the curve $H^*$ vs. $T$.   $H^*(T)$ serves as the boundary %%@
separating the low-field regime in which $\rho$ rapidly increases with $H$, from the high-field regime in %%@
which $\rho$ slowly asymptotes to the extrapolated $\rho_N$.   Hence $H^*$ represents an intrinsic field %%@
scale that controls dissipation in the vortex liquid state (attempts to find $H_{c2}$ using $\rho$ alone %%@
invariably turn up a curve akin to $H^*$ rather than the higher $H_{c2}$).  If we identify a length scale %%@
$\xi^*(T)$ by the ratio $\xi^*(T)/\xi_0(T) \sim \sqrt{H_{c2}(T)/H^*(T)}$, we find that at low $T$, %%@
$\xi^*/\xi_0\approx 1$.  As $T$ increases, however, the ratio rapidly increases (to 1.8 and 3.7 at 15 and %%@
25 K, respectively).  Since $H^*$ determines the onset of strong flux-flow dissipation, $\xi^*$ represents a %%@
radius larger than the vortex core radius $\xi_0$ ($\simeq$2.6 nm).  Packing vortices closer than the outer %%@
radius $\xi^*$ leads to a crossover to the high-dissipation state in which $\rho$ asymptotes to $\rho_N$.  %%@
However, as superfluidity and its associated 2$\pi$ phase winding survive right up to the inner radius $\xi_0$ %%@
\cite{Wen}, the Nernst signal is observable to the higher field scale $H_{c2}$ ($s_{\phi}$ is strongly %%@
attenuated in magnitude because of the reduced supercurrent for $\xi^*<r<\xi_0$)~\cite{fit}. A length scale %%@
extending outside the vortex core has been observed in tunneling experiments \cite{Pan}.

We thank S. Hannahs for valuable assistance at the National High Magnetic Field Lab. (a facility supported %%@
by NSF and the state of Florida).  We acknowledge useful discussions with P.W. Anderson, A. J. Millis and %%@
Z. Y. Weng.  This research is supported by a NSF grant (NSF-DMR 98-09483), the Office of Naval %%@
Research (N00014-01-0281), the New Energy and Industrial Tech. Develop. Org. (NEDO), Japan, the %%@
Natural Sciences and Engineering Research Council, and the Canadian Institute for Advanced %%@
Research.

\noindent
$^\dagger$ {\em Present address of ZAX: Department of Physics, Zhejiang University, Hangzhou, %%@
China}
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cond-mat0205299.extracted_bib
\bibitem{Fisher} D. S. Fisher, M. P. A. Fisher, and D. A. Huse, Phys. Rev. B. {\bf 43}, 130
    (1991). 

\bibitem{Blatter} G. Blatter, M. V. Feigel'man, V. B. Genshkenbein, A. I. Larkin, and V. M. Vinokur, Rev. %%@
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