The onset of the vortex-like Nernst signal above $T_c$ in $La_{2-x}Sr_xCuO_4$ and $%%@ Bi_2Sr_{2-y}La_yCuO_6$.
revtex
The diffusion of vortices down a thermal gradient produces a Josephson signal which is detected as the vortex Nernst %%@ effect. In a recent report, Xu {et al.}, Nature {406}, 486 (2000), an enhanced Nernst signal identified with %%@ vortex-like excitations was observed in a series of $La_{2-x}Sr_xCuO_4$ (LSCO) crystals at temperatures %%@ 50-100 K above $T_c$. To pin down the onset temperature $T_{\nu}$ of the vortex-like signal in the lightly doped %%@ regime ($0.03 \le x \le 0.07$), we have re-analyzed in detail the carrier contribution to the Nernst signal. By %%@ supplementing new Nernst measurements with thermopower and Hall-angle data, we isolate the off-diagonal Peltier %%@ conductivity $\alpha_{xy}$ and show that its profile provides an objective determination of $T_{\nu}$. With the new %%@ results, we revise the phase diagram for the fluctuation regime in LSCO to accomodate the lightly doped regime. In the %%@ cuprate $Bi_2Sr_{2-y}La_yCuO_6$, we find that the carrier contribution is virtually negligible for $y$ in the range %%@ 0.4-0.6. The evidence for an extended temperature interval with vortex-like excitations is even stronger in this system. %%@ Finally, we discuss how $T_{\nu}$ relates to the pseudogap temperature $T^*$ and the implications of strong %%@ fluctuations between the pseudogap state and the $d$-wave superconducting state.
OnsetNernstWeb.TEX
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\title{The onset of the vortex-like Nernst signal above $T_c$ in $\rm\bf La_{2-x}Sr_xCuO_4$ and $\rm\bf %%@
Bi_2Sr_{2-y}La_yCuO_6$.} 
\author{Yayu Wang$^1$, Z. A. Xu$^{1,\dagger}$, T. Kakeshita$^2$, S. Uchida$^2$, S. Ono$^3$, Yoichi %%@
Ando$^3$, and N. P. Ong$^1$}
\address{
$^1$Joseph Henry Laboratories of Physics, Princeton University, Princeton, New Jersey 08544, U.S.A..}
\address{
$^2$School of Frontier Sciences, University of Tokyo, Yayoi 2-11-16, Bunkyo-ku, Tokyo 113-8656, Japan. }
\address{
$^3$Central Research Institute of Electric Power
Industry (CRIEPI), Komae, Tokyo 201-8511, Japan}
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\begin{abstract}
The diffusion of vortices down a thermal gradient produces a Josephson signal which is detected as the vortex Nernst %%@
effect.  In a recent report, Xu {\it et al.}, Nature {\bf 406}, 486 (2000), an enhanced Nernst signal identified with %%@
vortex-like excitations was observed in a series of $\rm La_{2-x}Sr_xCuO_4$ (LSCO) crystals at temperatures %%@
50-100 K above $T_c$.  To pin down the onset temperature $T_{\nu}$ of the vortex-like signal in the lightly doped %%@
regime ($0.03 \le x \le 0.07$), we have re-analyzed in detail the carrier contribution to the Nernst signal.  By %%@
supplementing new Nernst measurements with thermopower and Hall-angle data, we isolate the off-diagonal Peltier %%@
conductivity $\alpha_{xy}$ and show that its profile provides an objective determination of $T_{\nu}$.  With the new %%@
results, we revise the phase diagram for the fluctuation regime in LSCO to accomodate the lightly doped regime.  In the %%@
cuprate $\rm Bi_2Sr_{2-y}La_yCuO_6$, we find that the carrier contribution is virtually negligible for $y$ in the range %%@
0.4-0.6. The evidence for an extended temperature interval with vortex-like excitations is even stronger in this system.  %%@
Finally, we discuss how $T_{\nu}$ relates to the pseudogap temperature $T^*$ and the implications of strong %%@
fluctuations between the pseudogap state and the $d$-wave superconducting state.  

\end{abstract}
\pacs{74.40.+k,72.15.Jf,74.72.-h,74.25.Fy}
}				
%End of Wideabs command

\section{Introduction}\label{intro}
Close to the the upper critical field line $H_{c2}(T)$ of a type-II superconductor, the vortices exist in the `vortex liquid' %%@
state as highly mobile excitations.  In this mobile state, the vortices readily flow in response to a weak applied %%@
temperature gradient towards the cooler end of the sample.  By the Josephson effect, the vortex motion generates an %%@
electric field $\bf E = B\times v$ that lies perpendicular to both the vortex velocity $\bf v$ and ${\bf B} = \mu_0{\bf %%@
H}$ (Fig. \ref{Expt}(a)).  In general, the appearance of a transverse $\bf E$ in the presence of a thermal gradient and %%@
magnetic field is known as the Nernst effect \cite{Stephen}.  In weak fields, the Nernst {\em signal} $E_y/|\nabla T|$ %%@
increases linearly with $B$, but at higher fields, the curve of $E_y$ vs. $T$ tends to develop negative curvature.  The %%@
Nernst coefficient $\nu$ is defined as the Nernst signal per unit $B$ in the weak-$B$ limit.  Because a %%@
field-antisymmetric $E_y$ may be measured to high resolution, the Nernst effect provides a highly sensitive probe for %%@
detecting vortices.

From the purview of conventional superconductivity, the search for vortices high above the critical temperature $T_c$ %%@
seems quite unrewarding.  Above $T_c$, the average value of the condensate density $n_s$ is zero.  Although %%@
fluctuation effects produce small evanescent droplets of superconductivity detectable by susceptibility and resistivity, the %%@
existence of a Nernst signal in the fluctuation regime above $T_c$ is not expected.  Indeed, vortex Nernst signals in the %%@
fluctuation regime do not seem to have been reported for any low-$T_c$ superconductor.  In the first Ettinghausen %%@
\cite{Stephen} experiment on a cuprate ($\rm YBa_2Cu_3O_7$), however, Palstra {\it et al.} \cite{Palstra} noted that %%@
the signal extended  $\sim$10 K above $T_c$.   
%Figure 1
\begin{figure}[htb]
\centerline{\psfig{file=Exptblue.eps,width=2.3in}}
\caption{(a) Geometry of the Nernst experiment in the vortex liquid state.  Vortices (disks with vectors) flow with %%@
velocity $\bf v$ down the gradient $-\nabla T\parallel {\bf \hat{x}}$.  Phase slippage induces a dc signal $E_y$ that is %%@
antisymmetric in $H$.  The lower panel (b) shows the currents $J_i = \sigma_{ij}E_j$ and $J'_i = \alpha_{ij}(-\partial_j %%@
T)$ produced by the $E$-field and thermal gradient, respectively, in the normal state.  The slight difference between their %%@
$y$-components engenders the carrier Nernst signal (angles $\theta$ and $\theta_{\alpha}$ are defined in the text).
}
\label{Expt}
\end{figure}\noindent
Later Nernst experiments on cuprates (restricted to optimally doped samples) found that the vortex signal extends above %%@
$T_c$ by roughly the same interval \cite{Hagen,Ri,Clayhold}.  Because the relative temperature interval $\sim 0.1$ is %%@
small, these observations did not provoke much theoretical comment.  More recently, Corson {\em et al.} \cite{Corson} %%@
investigated the conductivity in thin-film $\rm Bi_2Sr_2CaCu_2O_8$ at Terahertz frequencies and found that the kinetic %%@
inductance of the superfluid is observed as high as 25 K above $T_c$.  

In Nernst measurements on $\rm La_{2-x}Sr_xCuO_4$ (LSCO), Xu {\it et al.} \cite{Xu} observed that the Nernst %%@
signal remains anomalously enhanced 50-100 K above $T_c$, from which they inferred the existence of vortex-like %%@
excitations in the putative `normal-state'.  Starting at a relatively small and nearly $T$-independent value at high %%@
temperatures, $\nu$ begins to diverge at an onset temperature $T_{\nu}$, until it peaks near $T_c$ at a value 500 to %%@
1,000 times larger than the value at onset.  Because of the specific symmetry properties of the Nernst signal (${\bf E %%@
\parallel B\times }\nabla T$), Xu {\it et al.} argued that the excitations are vortex-like.   Moreover, they noted that the %%@
anomalous signal is very sensitive to Nd doping (which suppresses $T_c$ to $\sim$8 K), and {\em smoothly evolves to %%@
the familiar vortex signal} below $T_c$.  A closely similar extended fluctuation regime has also been observed in %%@
underdoped $\rm YBa_2Cu_3O_y$ \cite{Wang}.

The important issue whether the excitations are identical with magnetic vortices or are more exotic excitations (for e.g., %%@
holes bound to vorticity in a novel electronic state) is at present open.  Indeed, the notion that vortices of whatever origin %%@
can be detected 50-100 K {\em above} $T_c$ remains as surprising and challenging to us (and many in the community) %%@
as when they were first observed.  Hence, as in Ref. \cite{Xu}, we refer to them generically as {\em vortex-like} %%@
excitations in this paper.  We also refer to the region between $T_c$ and $T_{\nu}$ generically as the `fluctuation' %%@
regime.  (Discussion of this issue in relation to the pseudogap state is given in Sec. \ref{pseudo} .)

Our goal in this report is to sharpen the distinction of the vortex-like signal from that produced by normal-state charge %%@
carriers in order to address several issues raised by Ref. \cite{Xu} (we also provide extended discussions of many issues %%@
that were only touched on there).  The onset temperature of the vortex signal $T_{\nu}$ appears to increase with %%@
decreasing hole density $x$.  Is $T_{\nu}$ finite in the $x\rightarrow$ 0 limit?  If not, how does $T_{\nu}$ behave in %%@
this limit?  How significant is the normal-state carrier contribution to $\nu$ in the small-$x$ regime?  Can a more %%@
objective separation of the vortex signal be obtained through a better understanding of the normal-state Nernst effect?  %%@
Are these excitations seen in other cuprates? 

The small-$x$ regime requires a more careful analysis of the carrier Nernst signal because the normal-state thermopower %%@
is strongly enhanced.  We introduce a new method suitable for this limit and apply it to LSCO and the single-layer %%@
cuprate $\rm Bi_2Sr_{2-y}La_yCuO_6$ (Bi 2201).  We isolate the key quantity in the Nernst experiment -- the %%@
off-diagonal Peltier current -- and show that it provides an objective procedure for separating the vortex signal that is %%@
particularly suited for the small-$x$ regime.  The new results enable us to obtain a revised phase diagram for fluctuations %%@
in LSCO that is valid in both the small- and large-$x$ regimes.  In Bi 2201, we find that the vortex-like signal, relative to %%@
the normal carrier contribution, is even larger than in LSCO.  Hence, $T_{\nu}$ may be obtained directly from the %%@
original Nernst signal without the need for isolating the Peltier current.

\section{Isolating the Peltier current}\label{isolate}
To explain our procedure, we recall the various charge currents generated in a Nernst experiment (initially we consider %%@
the normal-state terms only).  We take the thermal gradient $-\nabla T\parallel {\bf\hat{x}}$ and the field $\bf H\parallel %%@
{\bf \hat{z}}$.  The Nernst signal is the $H$-antisymmetric electric field $\bf E\parallel {\bf\hat{y}}$ per unit gradient %%@
(Fig. \ref{Expt}).  

In zero field, the applied gradient drives a charge current density ${\bf J} = \alpha(-\nabla T)$ along the length of the %%@
sample (we define the Peltier conductivity tensor $\alpha_{ij}$ by $J_i = \alpha_{ij}(-\partial_j T)$ and write for brevity %%@
$\alpha_{ii} = \alpha$).  To satisfy the boundary condition $J_x=0$, there must exist an $\bf E$-field to drive a current %%@
$\sigma E_x$ in the opposite direction ($\sigma$ is the electrical conductivity).  Hence, the total current along ${\bf %%@
\hat{x}}$ is
\begin{equation}
J_x = \sigma E_x + \alpha(-\partial_x T).
\label{Jx}
\end{equation}
With $J_x$= 0, we have $E_x = -(\alpha/\sigma)(-\partial_x T)$, which is the signal detected in a thermopower %%@
experiment.  The thermopower coefficient is $S = \alpha/\sigma$.  

In finite field, the two currents in Eq. \ref{Jx} spawn Hall-type currents (antisymmetric in $H$) flowing along the %%@
$y$-axis (Fig. \ref{Expt}(b)).  One is the familiar Hall current $\sigma_{yx}E_x$ while the other is the {\em %%@
off-diagonal} Peltier current $\alpha_{yx}(-\partial_x T)$, where $\sigma_{yx}$ and $\alpha_{yx}$ denote the Hall %%@
and off-diagonal Peltier conductivity, respectively.  These two off-diagonal currents are opposite in direction and nearly %%@
equal in magnitude.  (The standard Boltzmann-theory expressions for these currents are given in the Appendix.)  Because %%@
of the experimental boundary condition $J_y$ = 0, any residual difference between the off-diagonal currents leads to a %%@
weak $E_y$ which is then detected as the Nernst signal.  Hence, we have 
\begin{eqnarray}
J_y &= & \alpha_{yx}(-\partial_x T)  + \sigma_{yx}E_x + \sigma E_y \nonumber\\
& = & \left[\alpha_{yx} - \sigma_{yx}\frac{\alpha}{\sigma}\right](-\partial_x T) + \sigma E_y = 0.
\label{Jy}
\end{eqnarray}
We have dropped a term $\alpha(-\partial_y T)$, which is important in conventional metals, but negligible in cuprates %%@
(see Appendix).  Using the Hall angle $\tan\theta = \sigma_{xy}/\sigma$ in Eq. \ref{Jy}, we obtain for the Nernst %%@
coefficient due to charge carriers alone
\begin{equation}
\nu_N = \frac{E_y}{|\partial_x T|B} = 
\left[\frac{\alpha_{xy}}{\sigma} - S \tan\theta\right]\frac1B.
\label{nuN}
\end{equation}

At first glance, it may seem that $\nu_N$ is just the thermopower $S$ reduced by the Hall angle $\tan\theta$ (since it %%@
derives from a current transverse to the applied gradient).  However, by writing Eq. \ref {nuN} as $\nu = %%@
S[\tan\theta_{\alpha}-\tan\theta]/B$, with $\tan\theta_{\alpha}\equiv \alpha_{xy}/\alpha$, we see that the reduction %%@
factor involves a cancellation between the 2 angles $\theta$ and $\theta_{\alpha}$ (see Fig. \ref{Expt}(b)).  As shown %%@
by Sondheimer \cite{Sondheimer}, within Boltzmann theory the cancellation is exact if $\theta$ is independent of energy %%@
$\epsilon$ (see Appendix).  [Because of this cancellation, the ratio $E_y/E_x$ in a Nernst experiment does not %%@
represent a `Hall angle' for entropy currents.]

A useful order-of-magnitude estimate of $|\nu_N|$ is $|S\tan\theta/B|$ reduced by a factor of 10 (to account for the %%@
Sondheimer cancellation).  For LSCO in the range $0.1<x<0.17$, $S\simeq 10\;\mu$V/K and $\tan\theta/B \leq %%@
10^{-2}$ T$^{-1}$, we estimate $|\nu_N|\leq$ 10 nV/KT, which is what is generally observed.  This rule-of-thumb %%@
anticipates that when $S$ is of the order of $100\;\mu$V/K, $|\nu_N|$ may become as large as 50 nV/KT (see below).

{\em Diagonal and off-diagonal response}  Even if we disregard the Sondheimer cancellation, the carrier Nernst signal %%@
(being off-diagonal) is `small' compared with the (diagonal) thermopower signal whenever $\tan\theta\ll 1$.  By contrast, %%@
the vortex-Nernst signal represents the `large' response of the vortices to an applied $-\nabla T$.  If the flux-flow Hall %%@
angle $\tan\theta_f\ll 1$ (as in the cuprates), the vortex velocity $\bf v$ is very nearly $\parallel(-\nabla T)$.  Hence the %%@
Josephson field $\bf E_J$ represents the diagonal response of the flux motion (even though it appears as a transverse %%@
Hall-type signal).  On the other hand, the small velocity component $v_y=v_x\tan\theta$ transverse to $-\nabla T$ leads %%@
to a signal detected as the flux-flow  thermopower $S_f$, which {\em is the small off-diagonal signal} in the vortex %%@
liquid.  This reversal of roles for the Nernst and thermopower signals in going from the normal to the vortex-liquid state %%@
reflects the well-known duality between vortex and charge currents, and is the primary reason why the Nernst %%@
experiment is so useful for detecting vortex motion. The vortex Nernst signal reflects the {\em primary} response of %%@
vortices to an applied gradient, while the carrier Nernst effect is a relatively feeble off-diagonal response that is further %%@
attenuated by the Sondheimer cancellation.

For our present purpose, we are interested in the temperature range when the diagonal vortex signal has fallen to values %%@
comparable with the carrier signal.  The contribution of the vortices to the observed Nernst signal may be written as an %%@
off-diagonal term $\alpha^s_{xy}$ that adds to the normal-state term $\alpha^n_{xy}$ (now relabelled with `$n$').  %%@
Repeating the steps above, we find that the combined Nernst signal is comprised of 3 terms, viz.
\begin{equation}
\nu = \frac{E_y}{|\partial_xT | B}  = 
\left[\frac{\alpha^s_{xy}}{\sigma} + 
\frac{\alpha^n_{xy}}{\sigma} - S \tan\theta\right]\frac1B. 			
\label{nuall}
\end{equation}

In Xu et al. \cite{Xu}, we assumed that the Sondheimer cancellation was sufficiently complete to reduce $\nu_N$ below %%@
a threshold of $\sim$4 nV/KT.  Hence, any increase of $\nu$ above this threshold was identified with the vortex term %%@
$\alpha^s_{xy}/\sigma$.  This `threshold criterion' is valid for moderately large $x$ where $|S|$ is small.  However, %%@
when $x\ll 1$, a new approach is needed.  For samples with $x\le$ 0.07, the thermopower $S$ (which sets the scale for %%@
$\alpha_{ij}$) rises to values in the range 100-300 $\mu$V/K.  This strong enhancement means that $\nu_N$ may %%@
significantly exceed the threshold, the Sondheimer cancellation notwithstanding.  

Our new method is based on measuring separately $S$ and $\tan\theta$ at each $T$.  By subtracting the product %%@
$-S\tan\theta$ from $\nu$ in Eq. \ref{nuall}, we obtain the total off-diagonal Peltier term 

\begin{equation}
\left[\frac{\alpha_{xy}}{\sigma}\right]_{obs} = \frac{\alpha^s_{xy}}{\sigma} + \frac{\alpha^n_{xy}}{\sigma}.
\label{axy}
\end{equation}

An important point is that $\alpha^n_{xy}$ must decrease to zero as $T\rightarrow 0$ because it is a carrier-entropy %%@
current, just like $\alpha$ (the same is true of $\nu_N$).  By contrast, the vortex-like contribution $\alpha^s_{xy}$ %%@
strongly diverges as $T\rightarrow T_c$ from above, as the phase stiffness of the superconducting condensate increases.  %%@
If $T_c$ is very small (as in L2), the vortex term appears as a divergent signal as $T$ decreases.  This is a key point in %%@
what follows. 

\section{Experimental Results}\label{results}
The Nernst experiments are carried out with a thermal gradient (typically 5 K/cm) applied along $\bf\hat{x}$ (the %%@
longitudinal thermal gradient $-\partial_x T$ is measured with a chromel-alumel thermocouple).  The Nernst voltage %%@
$E_y$ is measured with a nanovoltmeter (Keithley 2000 with preamp 2001) while the field is slowly ramped at a rate of %%@
0.4 T/min between -8 to 8 T (or $\pm$ 14 T in high-resolution runs).  Prior to ramping $H$, we regulate $T$ to a %%@
stability of $\pm$1 mK (this takes 30 min.).  The drift of the voltmeter in zero $H$ ($\sim$10 nV over a 1-hour period) %%@
is sufficiently small to guarantee a reproducibility in the Nernst coefficient of $\pm$1 nV/KT (about 4 times higher in %%@
resolution than attained in Ref. \cite{Xu}).  

The need to measure nV signals reproducibly in a temperature gradient introduces specific experimental constraints for %%@
single crystals.  The present method of sweeping the field (in both directions) at {\em fixed} temperature is quite %%@
necessary to attain accuracies better than 100 nV/KT.  The alternative, faster, method of sweeping $T$ in a fixed field %%@
(with a repeat in reversed field) is error-prone and unreliable even if $T$ is swept at the rate of 1 K/min.  The reason is %%@
apparently the very long relaxation time of the temperature gradient within the crystal.  In our experience, such %%@
constant-$H$ traces versus $T$, though useful for a qualitative overview of $\nu$, are of limited accuracy.

\subsection{LSCO}
With the higher resolution, we have re-measured $\nu$ in a series of LSCO samples [L1, L2, L3, L4 and L5 with $x$ %%@
($T_c$) = 0.03 (0 K), 0.05 ($<$ 4 K), 0.07 (12 K), 0.12 (28.9 K), and 0.20 (32.5 K), respectively] ($S$ and %%@
$\tan\theta$ were also measured in L1-L3, but not in L4 and L5).  The variation of the Nernst signal $E_y/|\nabla T|$ %%@
with $H$ displayed in Fig. \ref{LSCO} (for Sample L4) is quite representative of LSCO.  At 20 K, for instance, $E_y$ %%@
rises steeply from zero when $H$ exceeds the melting field $H_m\simeq$ 2 T, approaching saturation at values 6-7 %%@
$\mu$V/K at high fields.  
%Figure 2
\begin{figure}[htb]
\centerline{\psfig{file=NvsH-L12.eps,width=2.8in}}
\caption{The field dependence of the Nernst signal $E_y/|\nabla T|$ in $\rm La_{2-x}Sr_xCuO_4$ (Sample L4, %%@
$x=0.12$) at fixed $T$ from 5 to 60 K.  $E_y$ rises steeply when $H$ exceeds $H_m$ ($\sim$ 5 T at 10 K).  For %%@
$T$ just below $T_c$ (= 28.9 K), $H_m\simeq 0$, and $E_y$ vs. $H$ shows very pronounced negative curvature.  %%@
The curves continue to display curvature up to 50 K.  Note the relatively slow decay of the signal at 14 T above $T_c$.
}
\label{LSCO}
\end{figure}
On warming across $T_c$ (= 28.9 K), the overall magnitude of the signal decreases (as does $H_m$).  Remarkably, %%@
instead of dropping to zero, the Nernst signal remains large as $T$ rises to 60 K, high above $T_c$.  We note that a %%@
negative curvature is present even at 40 K.  This would not be expected from the normal-state Nernst signal in a system %%@
with such short carrier lifetimes.  From such curves, $\nu$ is determined from the initial slope at each selected $T$.  The %%@
$T$ dependence of $\nu$ is displayed in Fig. \ref{NuLSCO} for the 3 most underdoped LSCO samples.  [For later %%@
discussion (sec. \ref{pseudo}), we note that $\nu$ continues to increase up to 14 T in the curves at 27 and 30 K.]

As previously reported \cite{Xu}, $\nu$ in Sample L3 ($x= 0.07$) is very small and nearly $T$-independent (-5 to -10 %%@
nV/KT between 300 and 130 K).  Below $\sim$130 K, it begins an inexorable increase that ultimately reaches 2 %%@
$\mu$V/KT at 12 K.  Using the threshold method, we previously identified 130 K as the onset temperature.  However, %%@
a gradual increase starting near 130 K is also apparent in both Samples L2 ($x$ = 0.05) and L1 ($x$ = 0.03).  In L2, %%@
$\nu$ continues to increase at lower $T$.  Significantly, however, in the sample with the lowest doping L1, $\nu$ attains %%@
a broad peak and then {\em decreases} towards zero .  As discussed above, the latter behavior is characteristic of %%@
$\nu_N$.  

In very underdoped samples, the thermopower is so large ($S\sim 300 \;\mu$V/K at 100 K in L1) that the maximum %%@
value of $\nu_N$ (40 nV/KT) greatly exceeds the 4 nV/KT threshold.  The juxtaposition of the last 2 curves shows %%@
especially clearly that the threshold method should be supplanted with another technique more appropriate for samples %%@
with very large $S$. 
%Figure 3
\begin{figure}[htb]
\centerline{\psfig{file=NuLSCO.eps,width=3.3in}}
\caption{(a) The $T$ dependence of $\nu$ in very underdoped LSCO (Samples L1, L2, L3, with $x$ = 0.03, 0.05 and %%@
0.07, respectively).  At low $T$, the different behaviors of $\nu$ in L1 (compared with L2 and L3) indicate distinct %%@
origins of the Nernst signal. In L1 (open squares), $\nu$ is entirely from the carriers, while in L2 and L3, it is from %%@
vortex-like excitations.  The lower panel (b) displays $S\tan\theta/B$ measured in L1-L3 (symbol association same as in %%@
(a)).  Above 150 K, cancellation is nearly complete between the two normal-state currents, $S\tan\theta/B$ and %%@
$\alpha^n_{xy}$.  At lower $T$, however, a significant residual $\nu_N$ is observed especially in L1.
}
\label{NuLSCO}
\end{figure}

Using the measured thermopower and Hall-angle, we have obtained the curves $S\tan\theta$ displayed in Fig. %%@
\ref{NuLSCO}(a) for L1-L3.  The 2 panels in Fig. \ref{NuLSCO} illustrates the Sondheimer cancellation at %%@
temperatures above 150 K.  In L3 ($x = 0.07$), $|\nu|$ is $\sim$10 nV/KT at 150 K whereas $S\tan\theta/B\simeq $ %%@
110 nV/KT.  Hence, both the normal-state off-diagonal currents $S\tan\theta$ and $\alpha^n_{xy}/\sigma$ must be %%@
closely matched in magnitude (see Eq. \ref{nuN}).  This cancellation is much less effective at low $T$, especially in L1.

Subtracting $S\tan\theta/B$ from $\nu$, we obtain the Peltier curves $\alpha_{xy}/\sigma B$ shown in Fig. %%@
\ref{alphaLSCO}.  In L1, $\alpha_{xy}/\sigma B$ (open squares) rises to a broad maximum near 100 K before falling %%@
towards zero as $T\rightarrow$ 0.  This behavior, characteristic of normal carriers, allows us to identify $\alpha_{xy}$ %%@
with $\alpha^n_{xy}$ at all $T$ in L1.  Sample L2 displays a closely similar profile (solid triangles) except that, below %%@
50 K, the decrease is interrupted by the onset of the vortex term at 40 K (arrow).  
%Figure 4
\begin{figure}[htb]
\centerline{\psfig{file=alphaLSCOred.eps,width=3.0in}}
\caption{The $T$ dependence of the Peltier off-diagonal term $\alpha_{xy}$ in L1-L3 obtained by subtracting %%@
$-S\tan\theta/B$ from $\nu$ (see Eq. \ref{nuall}).  In L1 ($x=0.03$), the profile of $\alpha_{xy}$ (decreasing to zero %%@
as $T\rightarrow 0$, open squares) identifies it as arising entirely from the carriers.  In L2 ($x=0.05$) and L3 %%@
($x=0.07$), however, the onset of the vortex term $\alpha^s_{xy}$ is apparent as an inflection point (arrows). The %%@
shaded regions are estimates of the vortex-like term $\alpha^s_{xy}/\sigma B$ in L2 and L3.
}
\label{alphaLSCO}
\end{figure}
Hence, removal of the contribution $S\tan\theta$ has made the onset of the vortex term quite unambiguous.  The new %%@
results show that even at $x=$ 0.05, fluctuation effects arising from pair formation are observed starting at 40 K.  In L3, %%@
the onset of the vortex-like term at 90 K may also be identified by the inflexion point (arrow).  To estimate the vortex %%@
contribution $\alpha^s_{xy}/\sigma$ in L2 and L3, we assume their normal contribution $\alpha^n_{xy}/\sigma$ has %%@
nearly the same form as $\alpha_{xy}/\sigma B$ in L1 apart from a slight re-scaling (as suggested by the similarities in %%@
the profiles of $S\tan\theta/B$ in Fig. \ref{NuLSCO}(b)).  In L2, we believe this procedure is quite reliable since the %%@
total $\alpha_{xy}$ is so closely matched to that in L1 above 40 K (and $\alpha^n_{xy}$ must approach zero as %%@
$T\rightarrow 0$).  In L3, the assumption is less objective.  However, errors incurred (of magnitude $\pm$10 nV/KT) %%@
are relatively insignificant because $\alpha^s_{xy}/\sigma$ rapidly inflates to values 1-2 $\mu$V/KT.  

The steep increase in the vortex-like contributions in L2 and L3, displayed as the shaded regions in Fig. %%@
\ref{alphaLSCO},  is rather striking.  In L2, we note that the anomalous signal starts near 40 K even though $T_c$ is %%@
nominally zero (below  4 K).  These plots represent one of our key findings.

%FIGURE 5
\begin{figure}[htb]
\centerline{\psfig{file=NvsH-B2.eps,width=2.5in}}
\caption{Variation of $E_y/|\nabla T|$ vs. $H$ in $\rm Bi_2Sr_{2-y}La_yCuO_6$ (Sample B2 with $y$ = 0.5) at %%@
temperatures 25 to 120 K.  Compared with LSCO, the curves display more pronounced curvature at temperatures high %%@
above $T_c$. 
}
\label{Bisweep}
\end{figure}

\subsection{Bi 2201}
The systematic variation of transport properties and hole concentration with La content $y$ in single-layer Bi 2201 has %%@
been investigated by Ando and co-workers \cite{Ando}.  With increasing hole density (decreasing $y$), $T_c$ goes %%@
through the familiar dome-shaped curve, attaining its maximum value $\sim$35 K at $y$ = 0.4 \cite{Ono}. 

We have investigated 3 samples B1, B2 and B3 in which $y$ ($T_c$) = 0.6 (17 K), 0.5 (29 K) and 0.4 (32 K), %%@
respectively.  Magnetization measurements show a relatively sharp Meissner transition at $T_c$ and the virtual absence %%@
of diamagnetic screening above (see below).  The field dependence of the Nernst signal in B2 (Fig. \ref{Bisweep}) is %%@
representative of the 3 Bi 2201 samples.  Below $T_c$, the trace of $E_y$ vs. $H$ displays highly pronounced %%@
curvature.  In comparison with LSCO, the negative curvature persists to higher temperatures above $T_c$.  As the hole %%@
mean-free-path (mfp) is very short (see below), it is clear that the curvature cannot arise from the normal-state %%@
contribution $\nu_N$.  Rather, the strong sensitivity to these moderate fields is consistent with a superconducting %%@
fluctuation origin. 

The values of $\nu$ measured in a field of 1 T are displayed in Fig. \ref{NuBi}.  The gradual decrease of $\nu$ over an %%@
extended range of $T$ above $T_c$ bears a striking resemblance to the profiles in LSCO (see Fig. 2 of Xu {\it et al.} %%@
\cite{Xu}).  The values of $T_{\nu}$ are also quite similar.  
%Figure 6
\begin{figure}[htb]
\centerline{\psfig{file=NuBi.eps,width=2.7in}}
\caption{The $T$ dependence of $\nu$ in Bi 2201 (Samples B1, B2 and B3 with $y$ = 0.6 and 0.5 and 0.4, %%@
respectively). At all $T$, we determined $\nu$ from the values of $E_y/|\nabla T|$ observed at 1 Tesla (this slightly %%@
underestimates $\nu$ when curvature is pronounced near $T_c$; see Fig. \ref{Bisweep}).  $T_{\nu}$ decreases %%@
systematically as the hole concentration increases (from B1 to B3).
}
\label{NuBi}
\end{figure}

%Figure 7
\begin{figure}[htb]
\centerline{\psfig{file=alphaB1.eps,width=2.5in}}
\caption{The $T$ dependence of $\nu$ (solid circles), $S\tan\theta/B$ (open triangles) measured in Sample B1.  The %%@
Peltier term $\alpha_{xy}/\sigma B$ (solid triangles) is obtained as the sum of these 2 curves (all refer to left scale).  %%@
$T_{\nu}$ is estimated from $\nu$ using the threshold criterion (vertical arrow) ($\nu_N$ is less than 2 nV/KT up to %%@
300 K). The magnetization curve (from zero field cooling curve measured in a 10 Oe field) shows a sharp Meissner %%@
transition at $T_c\simeq 17 K$ (solid squares).
}
\label{alphaB1}
\end{figure}
We have also measured $S$ and $\tan\theta$ in B1 and B2 (but not in B3).  In contrast to very underdoped LSCO, %%@
$S\tan\theta/B$ is generally quite small in these samples.  As seen in Figs. \ref{alphaB1} and \ref{alphaB2}, the largest %%@
values attained by $S\tan\theta/B$, 10 and 6 nV/KT in B1 and B2, respectively, are an order of magnitude smaller than %%@
in Samples L1-L3.  Most of the suppression comes from the shorter mfp (as determined from $\tan\theta$) in Bi 2201, %%@
which leads to greatly reduced normal-state off-diagonal currents $S\tan\theta$ and $\alpha^n_{xy}$.

Applying the new method to B1 and B2, we now find that the curves of $\alpha_{xy}/\sigma B$ and $\nu$ differ only %%@
slightly (by $\sim$10 nV/KT in the range 70-150 K) (Figs.  \ref{alphaB1} and \ref{alphaB2}).  Thus, there is little %%@
difference (given the measurement uncertainties) whether we use $\nu$ or $\alpha_{xy}/\sigma B$ to estimate %%@
$T_{\nu}$ (this could not have been anticipated, however, without isolating $\alpha_{xy}$).  The strongly suppressed %%@
$S\tan\theta$ provides very strong {\em quantitative} arguments against identifying the increase in $\nu$ below %%@
$T_{\nu}$ with the normal state carriers. [From Fig. \ref{alphaB2}, the observed $\alpha_{xy}$ in B2 increases by %%@
$\sim$50 nV/KT between 80 and 50 K.  It would be difficult to imagine a scenario in which this increase comes from the %%@
hole carriers when $S\tan\theta$ is actually decreasing from 6 to $\sim$3 nV/KT over the same temperature range.]  
 
%Figure 8
\begin{figure}[htb]
\centerline{\psfig{file=alphaB2.eps,width=2.5in}}
\caption{The $T$ dependence of $\nu$ (solid circles), $S\tan\theta/B$ (open triangles) and $\alpha_{xy}/\sigma B$ %%@
(solid triangles) in Sample B2 (left scale).  As in Fig. \ref{alphaB1}, $T_{\nu}$ is indicated by the arrow ($\nu_N$ is %%@
$\sim-10$ nV/KT up to 300 K).  The magnetization curve shows a sharp Meissner transition at $T_c\simeq 29 K$ %%@
(solid squares).
}
\label{alphaB2}
\end{figure}
In addition, as mentioned above, the Nernst signal $E_y$ vs. $H$ develops increasingly pronounced negative curvature %%@
below 50 K.  Curvature in fields less than 8 T cannot arise from hole carriers with such short lifetimes.  We find these %%@
arguments supporting vortex-like excitations in the range $T_c$ to $\sim$ 80 K especially compelling in Bi 2201.  In %%@
Figs. \ref{alphaB1} and \ref{alphaB2}, we have also displayed the relatively sharp Meissner transition determined from %%@
the diamagnetic susceptibility in B1 and B2, respectively.  The comparison emphasizes that in the large interval between %%@
$T_{\nu}$ and $T_c$, $\nu$ rapidly diverges, but the Meisnner response is essentially non-observable.

\section{Phase Diagram of fluctuations}\label{phase}
With the new results in L1-L3, we have revised our previous phase diagram for LSCO \cite{Xu}.  Figure %%@
\ref{phaseLSCO} shows the new values of $T_{\nu}$ together with the contours of $\alpha^s_{xy}/\sigma$ estimated %%@
in Samples L2 and L3 (shaded regions in Fig. \ref{alphaLSCO}).  For samples at higher doping ($x\ge$ 0.10), we %%@
display the contours corresponding to values of $\nu-\nu_N$, since, as explained above, the new procedure is %%@
unnecessary if $x>$ 0.07.  

In the revised diagram, vortex-like excitations are absent at all $T$ in the sample with  $x= 0.03$.  Between 0.03 and %%@
0.07, $T_{\nu}$ increases very steeply from 0 to 90 K with a slope $dT_{\nu}/dx$ of at least 2,400 K (the slope is %%@
larger if $T_{\nu}\simeq$ 0 at $x=0.04$ as well).  $T_{\nu}$ peaks at $\sim$128 K at $x = 0.11$, and then decreases %%@
nearly linearly with $x$, but at the slower rate ($dT_{\nu}/dx \sim$ -510 K). 
 
% Figure 9
\begin{figure}[htb]
\centerline{\psfig{file=PhaseDiaLSCOblue.eps,width=2.8in}}
\caption{The $x$ dependence of $T_{\nu}$ and the contours of the vortex-like Nernst signal in LSCO.  For the very %%@
underdoped samples L1-L3, $T_{\nu}$ is determined from $\alpha_{xy}$ as explained in the text.  The contours are %%@
those of $\alpha^s_{xy}/\sigma B$ (Fig. \ref{alphaLSCO}).  For samples with $x>$0.07, we have used the procedure %%@
of Xu {\it et al.} \protect\cite{Xu} to determine $T_{\nu}$ (the contours are the magnitudes of $\nu-\nu_N$).  Data for %%@
the 2 samples with $x$ = 0.10 and 0.17 are taken from Xu {\it et al.}.  All other data represent new measurements.
}
\label{phaseLSCO}
\end{figure}

We have also plotted $T_{\nu}$ and the contours of the anomalous Nernst signal $\nu-\nu_N$ for B1, B2 and B3 in the %%@
phase diagram of Bi 2201 (Fig. \ref{phaseBi}). $T_{\nu}$ decreases with increasing hole content (decreasing $y$), as %%@
observed in LSCO for $x>$0.12.  The scale of $T_{\nu}$ is similar to that in LSCO (as is $T_c$).

The phase diagrams in these single-layer cuprates brings out several interesting features.  The fluctuation regime extends %%@
to a maximum temperature $100-130$ K that is considerably higher than the maximum $T_c$ in either cuprate.  In this %%@
large interval, vortex-like excitations are readily detected, but no Meissner signal appears until $T_c$ is reached.  The %%@
contrast is especially dramatic in B1 and B2 (Figs. \ref{alphaB1} and \ref{alphaB2}).  

The onset temperature has a different dependence on doping than $T_c$.  Instead of mimicking the shape of $T_c$ vs. %%@
$x$, $T_{\nu}$ peaks at 0.11 in LSCO.  Hence, the fluctuation regime is noticeably skewed towards the underdoped %%@
side (it also extends more deeply into the small-$x$ region than the superconducting phase).  We note that the %%@
skewedness is evident in all the contours up to 100 nV/KT in Fig. \ref{phaseLSCO}, not just in $T_{\nu}$.  In Bi 2201, %%@
the less complete data set also show that the fluctuation regime extends to higher $T$ at lower hole concentration (larger %%@
$y$).  The asymmetry suggests that the strength of the fluctuation regime, as measured by the magnitude of $\nu-\nu_N$, %%@
tends to increase monotonically with decreasing $x$ (until it suddenly collapses when $x$ is too small).
% Figure 10
\begin{figure}[htb]
\centerline{\psfig{file=PhaseDiaBi2201.eps,width=2.8in}}
\caption{The $y$ dependence of $T_{\nu}$ and the contours of the vortex-like Nernst signal in $\rm %%@
Bi_2Sr_{2-y}La_yCuO_6$ determined from $\nu-\nu_N$ using the procedure of Xu {\it et al.} \protect\cite{Xu}.  The %%@
solid line for $T_c$ is from Ono {\it et al.} \protect\cite{Ono} (open circles are $T_c$ measured in B1-B3).  The %%@
contours bear a strong resemblance to those in LSCO (Fig. \ref{phaseLSCO}). 
}
\label{phaseBi}
\end{figure}
This general trend suggests that the energy scale associated with pair formation is largest at small $x$, falling nearly %%@
linearly as $x$ increases.  Because of strong fluctuations, long-range phase coherence -- necessary to sustain a Meissner %%@
effect -- occurs at a $T_c$ that is much lower than $T_{\nu}$.  

\section{Pseudogap and strong fluctuations} \label{pseudo} 
A key point of interest is the relation of the vortex-like excitations to the pseudogap state \cite{Timusk, Tallon}.  In %%@
LSCO, information on $T^*$  is largely derived from heat capacity.  The shoulder in $\gamma(T) = c_e(T)/T$ (where %%@
$c_e$ is the electroinc heat capacity) provides the estimate that $T^*$ equals $\sim$350 K at $x=0.08$ and decreases %%@
linearly to $\sim$90 K at $x=0.22$ \cite{Tallon,Loram}.  However, there appears to be a lack of corroboration for the %%@
estimated $T^*$ from nuclear magnetic resonance (NMR) relaxation, infra-red reflectivity, in-plane resistivity $\rho$ or %%@
tunneling.  The NMR relaxation rate $1/T_1T$ does not show the broad maximum that is a key signature of the %%@
pseudogap, while $\rho$ does not display the shoulder seen in other underdoped cuprates.

In Bi 2201, Fischer's group recently reported that the gap in the density-of-states observed by scanning-probe tunneling %%@
spectroscopy \cite{Fischer} is observed up to $\sim$68 K (but the sharp `coherence' peaks at the gap maxima collapse %%@
at $T_c\simeq$11 K).  The $c$-axis resistivity profile, however, suggests a much higher $T^*$ (100 K, 125 K, and %%@
250 K for La content $y$ = 0.4, 0.5, and 0.6, respectively) \cite{AndoT}.

The available evidence implies that $T_{\nu}$ is roughly a factor of 2 lower than $T^*$ in LSCO and Bi 2201 (or much %%@
closer, if we use the tunneling data in Bi 2201).  With the improved resolution for $T_{\nu}$ achieved here, the %%@
uncertainties now lie chiefly in $T^*$.  [We note, however, that estimates of $T^*$ in any cuprate vary considerably, %%@
depending on the particular experimental probe used \cite{Timusk}.  The differences may arise from the highly %%@
anisotropic nature of the pseudogap magnitude and the fact that a particular experimental technique is sensitive to %%@
excitations at a particular wavevector $\bf q$.]  

With these caveats stated, we note that the contours in Figs. \ref{phaseLSCO} and \ref{phaseBi} are consistent with the %%@
$x$ dependence of $T^*$ (away from the lightly doped regime).  This suggests that the energy scale for destroying the %%@
vortex excitations depends on $x$ in the same way as $T^*$.  This trend, suggestive of strong pairing tendencies in the %%@
lightly-doped regime, recalls early theories of cuprate superconductivity \cite{Anderson} which propose that, in the %%@
single-layer parent material, pairing correlations are already `built-in' in the spin-$\frac12$ antiferromagnet.  

\emph{Fluctuation regime}    The physical picture suggested is that the pseudogap state, while distinct from $d$-wave %%@
superconductivity, is nevertheless closely similar in important aspects.  Pairing correlations seem to be already intrinsic at %%@
high temperatures in the pseudogap state, and fluctuations between the two states become ever stronger as we cool %%@
away from $T^*$.  Even at temperatures 50-100 K above $T_c$, these pairing correlations begin to support %%@
vortex-like excitations that are detectable as an anomalous Nernst signal.  Closer to $T_c$ (within 10-20 K), the %%@
phase-rigidity length inflates dramatically to reach macroscopic length-scales.  This increase is reflected in the rapid %%@
growth of $\nu$ as reported by Xu {\it et al.} \cite{Xu} (the concomitant increase in kinetic inductance also becomes %%@
observable at Terahertz frequencies \cite{Corson} in this interval).  In the conventional picture of strong phase %%@
fluctuations, the case for an extended fluctuation regime above $T_c$ has been made by several theorists %%@
\cite{Emery,fluc}.  In theories of strongly interacting systems, a discussion of fluctuations between the pseudogap state %%@
and $d$-wave superconductivity in the SU(2) theory has been published recently by Lee and Wen \cite{Lee}.   

Perhaps the most interesting question raised by these experiments is whether the vortex-like excitations are the familiar %%@
vortices in a superconducting condensate or novel electronic excitations specific to the pseudogap state.  Is it possible %%@
that, closer to $T_{\nu}$, the excitations are more properly regarded as vortex-like defects of the pseudogap state %%@
rather than Abrikosov vortices?  We elaborate further on two points mentioned above. The first is the {\em continuity} %%@
between the anomalous Nernst signal above $T_c$ and that in the Abrikosov state below.  If we view the Nernst signal %%@
$E_y/|\nabla T|$ as a function of both $H$ and $T$, it is apparent that the fluctuation regime covers a very wide region %%@
in the $H$-$T$ plane.  In this paper, we have traced the large fluctuating regime as we move along the $T$ axis in zero %%@
field.  The fluctuation regime uncovered is actually part of a very large region of the $H$-$T$ plane (as may be seen by %%@
scanning $H$ at fixed $T$).  As noted in the curves in Fig. \ref{LSCO}, the vortex Nernst signal at temperatures near %%@
$T_c$ continues to increase with $H$ (up to 14 T) .  We do not observe a decrease of the Nernst signal that might flag %%@
the crossing of an `$H_{c2}$' line regardless of how close we get to $T_c$.  Detailed analysis of the low-$T$ data %%@
show that the high temperature fluctuation regime connects continuously with the the high-field fluctuation regime below %%@
$T_c$ (the extended phase diagram of the fluctuations will be reported elsewhere).  By the continuity argument, the %%@
vortex-like excitations -- if distinct -- must evolve smoothly into Abrikosov vortices as $T$ decreases towards $T_c$.

The interesting counterpoint is that the onset of the Meissner response is relatively sharp (Figs. \ref{alphaB1} and %%@
\ref{alphaB2}).  The resistivity profile also implies that conventional amplitude fluctuations in the sense of %%@
Aslamasov-Larkin \cite{Larkin} occur in a fairly narrow interval.  Unlike in conventional superconductors, the %%@
diagmagnetic response and resistivity do not `see' the large fluctuation regime uncovered by the Nernst signal above %%@
$T_c$.  These 2 experimental points are seemingly at odds from the viewpoint of conventional theories of fluctuations, %%@
but we believe they provide very important hints.

The existence of stable vortex-like defects of the Schwinger-boson condensate in the disordered antiferromagnetic state, %%@
and their binding to holes, has been discussed by Ng \cite{Ng}.  Resurgent interest in this interesting regime seems likely, %%@
and we may expect the Nernst effect to play a key role in elucidating its properties (for e.g. its sensitivity to Zn and its %%@
extension into the overdoped regime).  

{\em Summary}  We have applied the Nernst effect to investigate vortex motion at elevated temperatures in crystals of %%@
$\rm La_{2-x}Sr_xCuO_4$ and $\rm Bi_2Sr_{2-y}La_yCuO_6$.  To address specifically the lightly doped regime, %%@
we have adopted a new experimental procedure to sharpen the difference between the vortex-like Nernst signal and the %%@
carrier Nernst signal near the former's onset temperature $T_{\nu}$.  The total Nernst signal is comprised of a term in %%@
the off-diagonal Peltier current $\alpha_{xy}$ and a term involving the Hall-angle $S\tan\theta$.  Combining %%@
measurements of the latter with the Nernst effect, we may back out the $\alpha_{xy}$ current.  We show that in LSCO %%@
samples with $x\le 0.07$ (in which $S$ is very large), the profile of $\alpha_{xy}/\sigma B$ shows very clearly the onset %%@
of the vortex-like terms (arrows in Fig. \ref{alphaLSCO}).  The new analysis allows the phase diagram in LSCO to be %%@
revised to accomodate the lightly doped regime.  The fluctuation regime (which harbors these vortex-like excitations) is %%@
observed to extend to a maximum temperature of $130$ K and to be skewed towards the underdoped side.  Applying %%@
the same procedure to 3 samples of La-doped Bi 2201, we show that the carrier contribution to $\nu$ is essentially %%@
negligible.  Its (partial) phase diagram shares many similar features with that of LSCO.  We emphasize the deep %%@
penetration of the vortex-like regime into the pseudogap state, and interpret the results in terms of strong fluctuations %%@
between the pseudogap state and $d$-wave superconducting state.  We discuss whether the vortex-like excitations are %%@
identical with vortices of the superconducting condensate or novel electronic excitations of the pseudogap state.

\section*{Appendix: Carrier Nernst coefficient in Boltzmann approach}\noindent
We summarize here the standard expressions for the Nernst coefficient in conventional metals. Carriers diffusing in a %%@
thermal gradient $-\nabla T$ in the presence of $\bf E$ and a weak field $\bf B$ satisfy the Boltzmann equation  
\begin{eqnarray}
{\bf v_k}\cdot \frac{\partial f^0}{\partial \epsilon} 
\frac{(\epsilon_{\bf k} - \mu)}{T}(-\nabla T) 
+ \frac{e{\bf E}}{\hbar}\cdot {\bf v_k} 
\frac{\partial f^0}{\partial \epsilon} \nonumber \\
+ \frac{e{\bf v_k\times B}}{\hbar} \cdot \frac{\partial f_{\bf k}}{\partial {\bf k }} 
= -\frac{g_{\bf k}}{\tau_{\bf k}},
\label{Boltz}
\end{eqnarray}
where $ g_{\bf k} = f_{\bf k}- f^0_{\bf k}$ is the difference between the perturbed distribution function $f_{\bf k}$ %%@
and its value $f^0_{\bf k}$ at equilibrium ($\bf v_k$ and $\tau_{\bf k}$ are the velocity and lifetime in state $\bf k$).  %%@
To find the Peltier conductivity elements, we may set $\bf E$ = 0, and expand $g_{\bf k} = g^{(0)}_{\bf k} + %%@
g^{(1)}_{\bf k} + \cdots$, where $g^{(0)}$ (linear in $-\nabla T$) gives $\alpha$ while $g^{(1)}$ (linear in $-\nabla %%@
T$ and $B$) gives the off-diagonal term $\alpha_{xy}$. By iteration of Eq. \ref{Boltz}, we have
\begin{eqnarray}
g^{(0)}_{\bf k} & = & -\tau_{\bf k} \frac{\partial f^0}{\partial \epsilon}
{\bf v_k}\cdot \frac{(\epsilon_{\bf k} - \mu)}{T} (-\nabla T),\\
g^{(1)}_{\bf k} & = & \frac{e{\bf v_k\times B}}{\hbar} \cdot 
\frac{\partial g^{(0)}_{\bf k}}{\partial {\bf k }}.
%\left[ {\bf v_k}\tau_{\bf k} \frac{(\epsilon_{\bf k} - \mu)}{T} \nabla T 
%\frac{\partial f^0}{\partial \epsilon} \right].
\label{g}
\end{eqnarray}
The off-diagonal current $J_y$ is just $eg^{(1)}{\bf v_k}$ integrated over the Fermi Surface.  Hence, the off-diagonal %%@
conductivity is (in terms of the mean-free-path $\vec{\ell}{\bf (k) \equiv v_k}\tau_{\bf k}$)
\begin{equation}
\alpha_{yx} = e^2\sum_{\bf k}  
\frac{(\epsilon_{\bf k} - \mu)}{T}
\left(-\frac{\partial f^0}{\partial \epsilon}\right) \ell_y 
\frac{{\bf v_k\times B}} {\hbar} \cdot
\frac{\partial \ell_x}{\partial {\bf k}}.
\label{alphaxy}
\end{equation}
In 2D systems (with arbitrary dependence of $\ell({\bf k})$ on $\bf k$), we may use the swept-area representation %%@
\cite{Ong} to reduce this complicated expression to 

\begin{equation}
\alpha_{xy} = \frac{2e^2B}{(2\pi)^2T\hbar^2} 
\int d\epsilon \left(-\frac{\partial f^0}{\partial \epsilon}\right)
(\epsilon -\mu) {\cal A}_{\ell}(\epsilon),
\label{alpha2D}
\end{equation}
where ${\cal A}_\ell(\epsilon) = \oint d\ell_x \ell_y $ is the area swept out by $\vec{\ell}({\bf k})$ as $\bf k$ goes %%@
around a contour at energy $\epsilon$.  As $\sigma_{xy}$ is proportional to ${\cal A}_\ell(\mu)$ \cite{Ong}, we find
\begin{equation}
\alpha_{xy} = \frac{\pi^2}{3}\frac{k_B^2T}{e}
\left[\frac{\partial\sigma_{xy}}{\partial\epsilon} \right]_{\mu}.
\label{alphaxy}
\end{equation}
This recalls the familiar relation between $\alpha$ and $\sigma$
\begin{equation} 
\alpha = \frac{\pi^2}{3}\frac{k_B^2T}{e} \left[\frac{\partial \sigma}{\partial \epsilon}\right]_\mu.
\label{alpha}
\end{equation}

Substituting Eqs. \ref{alphaxy} and \ref{alpha} into Eq. \ref{nuN}, and using the small Hall-angle approximation %%@
($\sigma_{xy}/\sigma\simeq \theta$), we have 
\begin{equation}
\nu_N  =  \left[ \frac{\alpha_{xy}}{\sigma} - \frac{\sigma_{xy}}{\sigma}\frac{\alpha}{\sigma} \right]\frac1B 
   =  \frac{\pi^2}{3}\frac{k_B^2T}{e} \frac{\theta}{B} 
\left[ \frac{\partial \ln\theta}{\partial \epsilon} \right]_\mu.
\label{nuN2}
\end{equation} 
If $\theta$ is only weakly dependent on energy at $\epsilon_F$ (a good approximation in conventional metals), we have %%@
nearly exact cancellation of the two contributions and $\nu_N$ is very small \cite{Sondheimer}.  

To estimate $\nu_N$ in case $\theta(\epsilon)$ has an anomalously strong power-law dependence $\epsilon^p$, we %%@
write $\nu_N \simeq (86\;\mu{\rm V/K})[k_BT/\epsilon_F] (\theta/B)p$. Using the values $\epsilon_F\sim$ 0.5 eV, %%@
and $(\theta/B)\simeq 4 \times 10^{-3}$ at 100 K (to approximate optimum-doped LSCO), we find that $\nu_N %%@
\simeq 6p$ nV/KT, which is still small.

{\em Isothermal versus adiabatic conditions}  In Nernst experiments on metals in which the electronic thermal %%@
conductivity $\kappa^e$ dominates the phonon conductivity $\kappa^{ph}$, the (electronic) thermal Hall conductivity %%@
$\kappa_{xy}$ generates a transverse gradient $-\partial_y T$.  This leads to a transverse current comparable to the %%@
other transverse currents in Eq. \ref{Jy}, which now reads
\begin{eqnarray}
J_y &= & \alpha_{yx}(-\partial_x T)  + \sigma_{yx}E_x + \alpha(-\partial_y T)+ \sigma E_y\nonumber\\
& = & \left[\alpha_{yx} - \sigma_{yx}\frac{\alpha}{\sigma}\right](-\partial_x T) + \alpha(-\partial_y T) + \sigma E_y = %%@
0.
\label{Jyk}
\end{eqnarray}
Under adiabatic conditions (the transverse edges are free to `float' to different temperatures), we have $-\partial_y T = %%@
(\partial_x T)\kappa_{yx}/[\kappa^{ph}+\kappa^e]$.  Assuming $\kappa^{e}\gg\kappa^{ph}$, and writing %%@
$\kappa_{yx}/\kappa^e \equiv \eta\sigma_{yx}/\sigma$, where $\eta$ is a positive number of order 1, we get the %%@
equation  
\begin{equation}
\left[\alpha_{yx} - (1+\eta)\sigma_{yx}\frac{\alpha}{\sigma}\right](-\partial_x T) + \sigma E_y = 0.
\end{equation}
Physically, the transverse thermal gradient produces a current that augments the Hall current $\sigma_{yx}E_x$.

Technically, under adiabatic conditions the measured $E_y$ does not give the Nernst coefficient.  Heroic efforts are %%@
required to `short out' the transverse gradient (for example, by thermally anchoring the transverse edges to each other %%@
with a piece of thick wire).  Fortunately, in the cuprates, the phonon conductivity $\kappa_{ph}$ is more than 10 times %%@
larger than $\kappa^e$.  Hence, the phonons act as a shorting fluid that keeps $\partial_y T$ negligibly small.  %%@
Measurements of $\kappa_{xy}$ in YBCO are reported by Zhang {\it et al.} \cite{Zhang}.  In LSCO, $\kappa_{xy}$ %%@
is much smaller, and barely detectable \cite{Krishana}.  The standard geometry in Fig. \ref{Expt} is effectively in the %%@
isothermal limit for conductors in which $\kappa^e\ll \kappa^{ph}$.\\
\\
\centerline {* \quad * \quad *}\\

The research at Princeton is supported by the U.S. National Science Foundation (MRSEC grant NSF-DMR %%@
98-09483).  N.P.O. and S. U. acknowledge support from a grant from the New Energy and Industrial Tech. Develop. %%@
Org., Japan (NEDO).  We wish to thank P. W. Anderson, J. Clayhold, S. Kivelson, P. A. Lee, A. J. Millis, V. N.  %%@
Muthukumar, J. Orenstein and Z. Weng for helpful dicussions.

\bigskip
$^\dagger${\it Present address of ZAX. Department of Physics, Zhejiang University, Hangzhou 310027, China.}
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D. Parks (Dekker, NY, 1969), p. 1107.
\bibitem{Palstra} T.T.M. Palstra, B. Batlogg, L.F. Schneemeyer and J.V. Waszezak, Phys. Rev. Lett. {\bf 64}, 3090 %%@
(1990).
\bibitem{Hagen} S. J. Hagen {\it et al.}, Phys. Rev. B {\bf 42}, 6777 (1990).
\bibitem{Ri} H. C. Ri {\it et al.}, Phys. Rev. B {\bf 50}, 3312 (1994);  Zeh {\it et al.}, Phys. Rev. Lett. 
   {\bf 64}, 3195 (1990).
\bibitem{Clayhold} J. A. Clayhold, {\it et al.}, Phys. Rev. B {\bf 50}, 4252 (1994);  J. A. Clayhold,  Phys. Rev. B {\bf %%@
54}, 6103 (1996).   
\bibitem{Corson} J. Corson, R. Mallozzi, J. Orenstein, J. N. Eckstein, and I. Bozovic,  Nature {\bf 398}, 221 (1999).
\bibitem{Xu} Z. A. Xu, N. P. Ong, Y. Wang, T. Kageshita, and S. Uchida, Nature {\bf 406}, 486 (2000). 
\bibitem{Wang} Yayu Wang, Z. A. Xu, N. P. Ong and S. Uchida, {\em submitted for publication}.
\bibitem{Sondheimer} E. H. Sondheimer, Proc. Roy. Soc. A {\bf 193}, 484 (1948).
\bibitem{Ando} Y. Ando, Y. Hanaki, S. Ono, T. Murayama, K. Segawa, N. Miyamoto, and Seiki Komiya, Phys. Rev. %%@
B {\bf 61}, R14956 (2000).
\bibitem{Ono} S. Ono, Y. Ando, T. Murayama, F. F. Balakirev, J. B. Betts, and G. S. Boebinger, Phys. Rev. Lett. {\bf %%@
85}, 638 (2000).
\bibitem{Timusk} For a review, see T. Timusk and B. Statt, Rep. Prog. Phys. {\bf 62}(1) 61-122, (1999).
\bibitem{Tallon}  J. L. Tallon and J. W. Loram, Physica C {\bf 349}, 53 (2001).   
\bibitem{Loram} Loram, J. W., Mirza, K.A., Cooper, J. R, Athanassopoulou, N. $\&$ Liang, W.Y.  in {\it Proc. of the %%@
10th HTS Anniversary Workshop on Physics, Materials and Applications}, ed. Batlogg, B., Chu, C.W., Chu, W.K., %%@
Gubser, D.U. and Muller, K.A. (World Scientific, Singapore, 1996), 341.
\bibitem{Fischer} M. Kugler, \O. Fischer, C. Renner, S. Ono, Y. Ando, Phys. Rev. Lett. {\bf 86}, 4911 (2001).
\bibitem{AndoT} A. N. Lavrov, Yoichi Ando and S. Ono, {\em preprint}.
\bibitem{Anderson} P.W. Anderson, Science {\bf 235}, 1196 (1987).
\bibitem{Larkin} L. G. Aslamazov and A.I. Larkin, Fiz. Tverd. Tela {\bf 10}, 1104 (1968) [Sov. Phys. -Solid State %%@
{\bf 10}, 875 (1968)].
\bibitem{Emery} V. J. Emery, and S. A. Kivelson, Nature {\bf 374}, 434 (1995).
\bibitem{fluc} M. Norman, M. Randeria, H. Ding, and J. C. Campuzano, Phys. Rev. B {\bf 57}, 11093 (1998).
\bibitem{Lee}  Patrick A. Lee and Xiao-Gang Wen, Phys Rev B {\bf 63}, 224517 (2001).
\bibitem{Ng} Tai Kai Ng, Phys. Rev. Lett. {\bf 82}, 3504 (1999); Phys. Rev. B {\bf 52}, 9491 (1995). 
\bibitem{Ong} N. P. Ong, Phys. Rev. B {\bf 43}, 193 (1991).
\bibitem{Zhang} Y. Zhang, N.P. Ong, Z.A. Xu, K. Krishana, R. Gagnon, and L. Taillefer, Phys. Rev. Lett. {\bf 84}, %%@
2219 (2000); Y. Zhang, N.P. Ong, P. W. Anderson, D. A. Bonn, R. X. Liang, and W. N. Hardy, Phys. Rev. Lett., {\bf %%@
86}, 890 (2001).
\bibitem{Krishana} K. Krishana and N. P. Ong, {\it unpublished}.
\end{references}

\end{document}


%%%%%%






 

cond-mat0108242.extracted_bib
\bibitem{Stephen} See for e.g. Y. B. Kim and M. J. Stephen and Y. B. Kim, in {\em Superconductivity}, v. 2, ed. R. %%@
D. Parks (Dekker, NY, 1969), p. 1107.

\bibitem{Palstra} T.T.M. Palstra, B. Batlogg, L.F. Schneemeyer and J.V. Waszezak, Phys. Rev. Lett. {\bf 64}, 3090 %%@
(1990).

\bibitem{Hagen} S. J. Hagen {\it et al.}, Phys. Rev. B {\bf 42}, 6777 (1990).

\bibitem{Ri} H. C. Ri {\it et al.}, Phys. Rev. B {\bf 50}, 3312 (1994);  Zeh {\it et al.}, Phys. Rev. Lett. 
   {\bf 64}, 3195 (1990).

\bibitem{Clayhold} J. A. Clayhold, {\it et al.}, Phys. Rev. B {\bf 50}, 4252 (1994);  J. A. Clayhold,  Phys. Rev. B {\bf %%@
54}, 6103 (1996).   

\bibitem{Corson} J. Corson, R. Mallozzi, J. Orenstein, J. N. Eckstein, and I. Bozovic,  Nature {\bf 398}, 221 (1999).

\bibitem{Xu} Z. A. Xu, N. P. Ong, Y. Wang, T. Kageshita, and S. Uchida, Nature {\bf 406}, 486 (2000). 

\bibitem{Wang} Yayu Wang, Z. A. Xu, N. P. Ong and S. Uchida, {\em submitted for publication}.

\bibitem{Sondheimer} E. H. Sondheimer, Proc. Roy. Soc. A {\bf 193}, 484 (1948).

\bibitem{Ando} Y. Ando, Y. Hanaki, S. Ono, T. Murayama, K. Segawa, N. Miyamoto, and Seiki Komiya, Phys. Rev. %%@
B {\bf 61}, R14956 (2000).

\bibitem{Ono} S. Ono, Y. Ando, T. Murayama, F. F. Balakirev, J. B. Betts, and G. S. Boebinger, Phys. Rev. Lett. {\bf %%@
85}, 638 (2000).

\bibitem{Timusk} For a review, see T. Timusk and B. Statt, Rep. Prog. Phys. {\bf 62}(1) 61-122, (1999).

\bibitem{Tallon}  J. L. Tallon and J. W. Loram, Physica C {\bf 349}, 53 (2001).   

\bibitem{Loram} Loram, J. W., Mirza, K.A., Cooper, J. R, Athanassopoulou, N. $\&$ Liang, W.Y.  in {\it Proc. of the %%@
10th HTS Anniversary Workshop on Physics, Materials and Applications}, ed. Batlogg, B., Chu, C.W., Chu, W.K., %%@
Gubser, D.U. and Muller, K.A. (World Scientific, Singapore, 1996), 341.

\bibitem{Fischer} M. Kugler, \O. Fischer, C. Renner, S. Ono, Y. Ando, Phys. Rev. Lett. {\bf 86}, 4911 (2001).

\bibitem{AndoT} A. N. Lavrov, Yoichi Ando and S. Ono, {\em preprint}.

\bibitem{Anderson} P.W. Anderson, Science {\bf 235}, 1196 (1987).

\bibitem{Larkin} L. G. Aslamazov and A.I. Larkin, Fiz. Tverd. Tela {\bf 10}, 1104 (1968) [Sov. Phys. -Solid State %%@
{\bf 10}, 875 (1968)].

\bibitem{Emery} V. J. Emery, and S. A. Kivelson, Nature {\bf 374}, 434 (1995).

\bibitem{fluc} M. Norman, M. Randeria, H. Ding, and J. C. Campuzano, Phys. Rev. B {\bf 57}, 11093 (1998).

\bibitem{Lee}  Patrick A. Lee and Xiao-Gang Wen, Phys Rev B {\bf 63}, 224517 (2001).

\bibitem{Ng} Tai Kai Ng, Phys. Rev. Lett. {\bf 82}, 3504 (1999); Phys. Rev. B {\bf 52}, 9491 (1995). 

\bibitem{Ong} N. P. Ong, Phys. Rev. B {\bf 43}, 193 (1991).

\bibitem{Zhang} Y. Zhang, N.P. Ong, Z.A. Xu, K. Krishana, R. Gagnon, and L. Taillefer, Phys. Rev. Lett. {\bf 84}, %%@
2219 (2000); Y. Zhang, N.P. Ong, P. W. Anderson, D. A. Bonn, R. X. Liang, and W. N. Hardy, Phys. Rev. Lett., {\bf %%@
86}, 890 (2001).

\bibitem{Krishana} K. Krishana and N. P. Ong, {\it unpublished}.
