Two Pseudogaps in the Cuprates
R.S. Markiewicz
dstrcom.tex
\documentstyle[twocolumn,epsf,aps,prb]{revtex}
\begin{document}
\title{Two Pseudogaps in the Cuprates}

\author{R.S. Markiewicz}
\address{Physics Department,
Northeastern U.,
Boston MA 02115}
\maketitle
%****
\narrowtext
%****

Meingast, et al.\cite{Mein} measured the thermal expansion anomaly in 
underdoped YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO), interpreting their data as 
suggesting a connection between the pseudogap and preformed pairs.  In so doing,
they neglected early evidence\cite{StoP,EmBatl,TS} 
that most cuprates have {\it two} pseudogaps, and that only the lower of the 
two, the `strong' pseudogap, $T_{fl}$, is associated with superconducting 
fluctuations.  More recent evidence in both YBCO and Bi$_2$Sr$_2$CaCu$_2$O$_{8+
\delta}$ (Bi2212), Fig.~\ref{fig:38a}, confirms that the measured pseudogap 
temperatures fall into two groups.  The higher, `weak' pseudogap temperature 
$T^*$ is associated with a gap opening, with no clear evidence for 
superconducting fluctuations: it is measured from transport and heat capacity 
(dashed line)\cite{pseud,pseudT}, photoemission leading edge (dotted 
line)\cite{Camp}, and tunneling (filled circles = `peak' feature)\cite{Miya}.  
[The latter feature is associated with the total gap near $(\pi ,0)$; it is 
found that $2\Delta /k_BT^*\simeq 6$, consistent with a ratio 4.3 found by Ido, 
et al.\cite{Ido}.]  At a considerably lower temperature, $T_{fl}$, a clear onset
of strong superconducting fluctuations is found in underdoped samples.   Early 
evidence was based mainly on magnetic measurements; newer evidence includes 
magnetic measurements (Cu NMR $1/T_{2G}$ reduction)\cite{TIYM},
onset of Kosterlitz-Thouless fluctuations\cite{Cor}, and interlayer Josephson
tunneling\cite{Bern}.  The fluctuations found by Meingast, et al. clearly fall
into this latter group, extending to only about 2/3 of $T^*$ at the lowest
doping.  

\begin{figure}
\leavevmode
   \epsfxsize=0.33\textwidth\epsfbox{fluctt1.eps}
\vskip0.5cm 
\caption{Doping dependence of T$_c$ (solid line) and superconducting 
fluctuations in Bi2212 (open circles) [\protect\onlinecite{Cor}] and (diamonds) 
[\protect\onlinecite{TIYM}], and YBCO (closed squares) 
[\protect\onlinecite{Bern}] and (open squares) [\protect\onlinecite{Mein}].  
Dotted line = leading-edge pseudogap from photoemission 
[\protect\onlinecite{Camp}] (arrows indicate temperatures are only lower 
limits); thick dashed line = weak pseudogap temperature $T^*$ in YBCO
[\protect\onlinecite{pseudT}]; closed circles = $\Delta /3$, where $\Delta$ =
peak position measured in tunneling [\protect\onlinecite{Miya}]; thick solid 
line = estimated superconducting temperature on a single stripe 
[\protect\onlinecite{MKI}].}
\label{fig:38a}
\end{figure}

Batlogg and Emery\cite{EmBatl} suggested that the weak pseudogap corresponds to 
the onset of electronic inhomogeneity (stripe fluctuations), the strong 
pseudogap to the onset of superconductivity on individual stripes, and the 
macroscopic superconducting transition $T_c$ is a signature of the
establishment of phase coherence between stripes.  It has also been 
postulated\cite{EmK} that superconducting pairing can be enhanced in the stripe
phase.  The data of Fig.~\ref{fig:38a} are very suggestive of this picture.  
First, the fact that there are no superconducting fluctuations near the weak 
pseudogap temperature $T^*$ makes it unlikely that $T^*$ is associated with 
preformed pairs.  Secondly, the strong pseudogap $T_{fl}$ is in good agreement
with a mean-field calculation\cite{MKI} (thick solid line) of the 
superconducting transition temperature on a charged stripe, assuming that 
$T_{fl}$ is proportional to the (pseudo-d-wave) gap along the stripe. 

Hence, the data of Meingast, et al., do not strongly support a model of 
preformed pairs, but are consistent with a stripe model for the pseudogap.

\begin{references}
\bibitem{Mein}C. Meingast, {\it et al.},
%V. Pasler, P. Nagel, A. Rykov, S. Tajima, and P. Olsson, 
Phys. Rev. Lett. {\bf 86}, 1606 (2001).
\bibitem{StoP}V. Barzykin and D. Pines, Phys. Rev. B{\bf 52},13585 (1995);
J. Schmalian, {\it et al.},
%D. Pines, and B. Stojkovic, 
Phys. Rev. Lett. {\bf 80}, 3839 (1998).
\bibitem{EmBatl}B. Batlogg and V.J. Emery, Nature {\bf 382}, 20 (1996); V.J. 
Emery, {\it et al.},
%S.A. Kivelson, and O. Zachar, 
Phys. Rev. B{\bf 56}, 6120 (1997)
\bibitem{TS}T. Timusk and B.W. Statt, Rep. Prog. Phys. 62 61 (1999). 
\bibitem{pseud}B. Batlogg, {\it et al.},
%H.Y. Hwang, H. Takagi, R.J. Cava, H.L. Kao, and J. Kwo, 
Physica C{\bf 235-240}, 130 (1994). 
\bibitem{pseudT}J.L. Tallon, {\it et al.},
%J.R. Cooper, P.S.I.P.N. de Silva, G.V.M. Williams, and J.W. Loram, 
Phys. Rev. Lett. {\bf 75}, 4114 (1995).
\bibitem{Camp}J.C. Campuzano, {\it et al.},
%H. Ding, M.R. Norman, H.M. Fretwell, M. Randeria, A. Kaminski, J. Mesot, T. 
%Takeuchi, T. Sato, T. Yokoya, T. Takahashi, T. Mochiku, K. Kadowaki, P. 
%Guptasarma, D.G. Hinks, Z. Konstantinovic, Z.Z. Li, and H. Raffy, 
Phys. Rev. Lett. {\bf 83}, 3709 (1999).
\bibitem{Miya}N. Miyakawa, {\it et al.},
%P. Guptasarma, J.F. Zasadzinski, D.G. Hinks, and K.E. Gray,
Phys. Rev. Lett. {\bf 80}, 157 (1998).
\bibitem{Ido}M. Ido, {\it et al.},
%N. Momono, and M. Oda, 
J. Low. T. Phys. {\bf 117}, 329 (1999).
\bibitem{TIYM}Y. Tokunaga, {\it et al.},
%K. Ishida, K. Yoshida, T. Mito, Y. Kitaoka, Y.
%Nakayama, J. Shimoyama, K. Kishio, O. Narikiyo, and K. Miyake, 
Physica B{\bf 284-88}, 663 (2000).
\bibitem{Cor}J. Corson, {\it et al.},
%R. Mallozzi, J. Orenstein, J.N. Eckstein, and I. Bozovic, 
Nature {\bf 398}, 221 (1999).
\bibitem{Bern}C. Bernhard, {\it et al.},
%D. Munzar, A. Golnik, C.T. Lin, A. Wittlin, J. Huml\'i\v cek, and M. Cardona, 
Phys. Rev. B{\bf 61}, 618 (2000).
\bibitem{EmK}V.J. Emery and S.A. Kivelson, Nature {\bf 374}, 6521 (1995).
\bibitem{MKI}R.S. Markiewicz and C. Kusko, cond-mat/0102438.
\end{references}
\end{document}

cond-mat0108075.extracted_bib
\bibitem{Mein}C. Meingast, {\it et al.},
%V. Pasler, P. Nagel, A. Rykov, S. Tajima, and P. Olsson, 
Phys. Rev. Lett. {\bf 86}, 1606 (2001).

\bibitem{StoP}V. Barzykin and D. Pines, Phys. Rev. B{\bf 52},13585 (1995);
J. Schmalian, {\it et al.},
%D. Pines, and B. Stojkovic, 
Phys. Rev. Lett. {\bf 80}, 3839 (1998).

\bibitem{EmBatl}B. Batlogg and V.J. Emery, Nature {\bf 382}, 20 (1996); V.J. 
Emery, {\it et al.},
%S.A. Kivelson, and O. Zachar, 
Phys. Rev. B{\bf 56}, 6120 (1997)

\bibitem{TS}T. Timusk and B.W. Statt, Rep. Prog. Phys. 62 61 (1999). 

\bibitem{pseud}B. Batlogg, {\it et al.},
%H.Y. Hwang, H. Takagi, R.J. Cava, H.L. Kao, and J. Kwo, 
Physica C{\bf 235-240}, 130 (1994). 

\bibitem{pseudT}J.L. Tallon, {\it et al.},
%J.R. Cooper, P.S.I.P.N. de Silva, G.V.M. Williams, and J.W. Loram, 
Phys. Rev. Lett. {\bf 75}, 4114 (1995).

\bibitem{Camp}J.C. Campuzano, {\it et al.},
%H. Ding, M.R. Norman, H.M. Fretwell, M. Randeria, A. Kaminski, J. Mesot, T. 
%Takeuchi, T. Sato, T. Yokoya, T. Takahashi, T. Mochiku, K. Kadowaki, P. 
%Guptasarma, D.G. Hinks, Z. Konstantinovic, Z.Z. Li, and H. Raffy, 
Phys. Rev. Lett. {\bf 83}, 3709 (1999).

\bibitem{Miya}N. Miyakawa, {\it et al.},
%P. Guptasarma, J.F. Zasadzinski, D.G. Hinks, and K.E. Gray,
Phys. Rev. Lett. {\bf 80}, 157 (1998).

\bibitem{Ido}M. Ido, {\it et al.},
%N. Momono, and M. Oda, 
J. Low. T. Phys. {\bf 117}, 329 (1999).

\bibitem{TIYM}Y. Tokunaga, {\it et al.},
%K. Ishida, K. Yoshida, T. Mito, Y. Kitaoka, Y.
%Nakayama, J. Shimoyama, K. Kishio, O. Narikiyo, and K. Miyake, 
Physica B{\bf 284-88}, 663 (2000).

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

\bibitem{Bern}C. Bernhard, {\it et al.},
%D. Munzar, A. Golnik, C.T. Lin, A. Wittlin, J. Huml\'i\v cek, and M. Cardona, 
Phys. Rev. B{\bf 61}, 618 (2000).

\bibitem{EmK}V.J. Emery and S.A. Kivelson, Nature {\bf 374}, 6521 (1995).

\bibitem{MKI}R.S. Markiewicz and C. Kusko, cond-mat/0102438.
