Critical end-point and metamagnetic quantum criticality in Sr$_3$Ru$_2$O$_7$
May Chiao
We report a metamagnetic critical point at ($B^*$,$T^*$) $\simeq$ (5.1\,T, 1.1\,K) for magnetic fields applied perpendicular to the (tetragonal) $c$-axis of \SrTwoLayer. % First-order behaviour well below $T^*$ indicates that $(B^*,T^*)$ marks a critical end-point that terminates a line of first-order metamagnetic transitions. % The absence of first-order behaviour in the metamagnetic transition with $B \parallel c$ confirms that the non-Fermi liquid behaviour for $B \parallel c$ is underpinned by a metamagnetic quantum critical end-point for which $T^* \to 0$. % Scaling behaviour of the resistivity under hydrostatic pressure yields the surprising result that although $B^*$ increases rapidly with pressure, $T^*$ has only a weak pressure dependence.
327-2col.tex
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\begin{document}

\newcommand{\SrTwoLayer}{Sr$_3$Ru$_2$O$_7$} 
\newcommand{\SrOneLayer}{Sr$_2$RuO$_4$} 
\newcommand{\SrInftyLayer}{SrRuO$_3$} 
\renewcommand{\floatpagefraction}{0.5}


\title{Critical end-point and metamagnetic 
quantum criticality in Sr$_3$Ru$_2$O$_7$}
  
\author{May Chiao}

\thanks{Present address: Laboratorium f\"ur Festk\"orperphysik, 
Eidgen\"ossische Technische Hochschule,
ETH-Z\"urich, CH-8093 Z\"urich, Switzerland.}

\affiliation{Cavendish Laboratory, University of Cambridge, Madingley
  Road, Cambridge CB3 0HE, United Kingdom}

\author{C. Pfleiderer}

\affiliation{Physikalisches Institut, Universit\"at Karlsruhe,
  Wolfgang-Gaede-Strasse 1, D-76128 Karlsruhe, Germany}
  
\author{R. Daou}

\affiliation{Cavendish Laboratory, University of Cambridge, Madingley
  Road, Cambridge CB3 0HE, United Kingdom}
  
\author{A. McCollam}

\affiliation{Cavendish Laboratory, University of Cambridge, Madingley
  Road, Cambridge CB3 0HE, United Kingdom}

\author{S. R. Julian}

\affiliation{Cavendish Laboratory, University of Cambridge, Madingley
  Road, Cambridge CB3 0HE, United Kingdom}

\author{G. G. Lonzarich}

\affiliation{Cavendish Laboratory, University of Cambridge, Madingley
  Road, Cambridge CB3 0HE, United Kingdom}

\author{R. S. Perry}

\affiliation{School of Physics and Astronomy, University of Birmingham,
Edgbaston, Birmingham BI5 2TT, United Kingdom}

\author{A. P. Mackenzie}

\affiliation{School of Physics and Astronomy, University of 
St. Andrews, North Haugh, St. Andrews, Fife KY16 9SS, United Kingdom}

\author{Y. Maeno}

\affiliation{Department of Physics, Kyoto University, Kyoto 606-8502, Japan}

\affiliation{CREST, Japan Science and Technology Corporation,
  Kawaguchi, Saitama 332-0012, Japan}


\date{\today}


\begin{abstract}
We report a metamagnetic critical point at ($B^*$,$T^*$) $\simeq$
(5.1\,T, 1.1\,K) for magnetic fields applied perpendicular to the
(tetragonal) $c$-axis of \SrTwoLayer.
%
First-order behaviour well below $T^*$ indicates that $(B^*,T^*)$
marks a critical end-point that terminates a line of first-order
metamagnetic transitions.
%
The absence of first-order behaviour in the metamagnetic transition
with $B \parallel c$ confirms that the non-Fermi liquid behaviour for
$B \parallel c$ is underpinned by a metamagnetic quantum critical
end-point for which $T^* \to 0$.
%
Scaling behaviour of the resistivity under hydrostatic pressure yields
the surprising result that although $B^*$ increases rapidly with
pressure, $T^*$ has only a weak pressure dependence.  
\end{abstract}

\pacs{72.15.-v, 75.40.-s }

%\vskip2pc]

\maketitle

The intense exploration of the perovskite ruthenates over the past few
years has revealed a wide range of challenges to our understanding of
the metallic state.
%
This family includes the unconventional superconductor \SrOneLayer\
\cite{Maeno} and the itinerant ferromagnet \SrInftyLayer\
\cite{Allen,Klein}; Ca-doping \SrOneLayer\ at the Sr site even
provides access to a metal-insulator transition \cite{Braden}.
%
We recently reported evidence of non-Fermi liquid behaviour in
\SrTwoLayer\ in the vicinity of a metamagnetic transition which occurs
at fields $B_M(T)$ which depend on field orientation, with
$B_{M\perp}(0)$ = 5.1\,T for $B\perp c$, and $B_{M\parallel}(0)$ =
7.8\,T for $B||c$ \cite{Perry,Grigera}.
%
There are two regimes of non-Fermi liquid behaviour.
%
Above 1\,K the metamagnetic transition broadens rapidly with
temperature indicating a fall in the susceptibility for $B \sim B_M$
and there is a quasi-linear temperature dependence of the resistivity
over a large region of the $(B,T)$ plane centered on $B_M$; also, in
measurements with $B\parallel c$, a logarithmic divergence has been
seen in the electronic specific heat, $C(T)/T \sim -\ln T$
\cite{Perry}.
%
Below 1\,K in contrast, non-Fermi liquid behaviour is confined to $B
\sim B_{M\parallel}$; for other values and orientations of $B$ the
susceptibility and resistivity cross-over to conventional Fermi-liquid
behaviour.
%
The resistivity in particular shows $T^2$ behaviour, but the range of
temperature in which this is observed collapses as $B\rightarrow
B_{M\parallel}$ while the coefficient $A$ in $\rho(T) = \rho_0 + A
T^2$ becomes divergent.
%
Concurrently, for $T \rightarrow 0$ (and $B \parallel c$) a novel
manifestation of non-Fermi liquid behaviour appears as a power law in
the resistivity that is {\em higher} than $T^2$, being closer to $T^3$
\cite{Grigera}.
%
It has been speculated that the $T^{3}$ behaviour signals the onset of a
qualitatively new state, driven by divergent fluctuations and
susceptibility associated with a so-called metamagnetic quantum
critical end-point \cite{Grigera}.

Historically, metamagnetism refers to magnetic field induced phase
transitions of local moment antiferromagnetic insulators.
%
Instead, the itinerant metamagnetism addressed in this Letter is
thought to arise from a magnetic field induced spin splitting of the
Fermi surface.
%
Metamagnetism of this kind has been detected in numerous metallic
magnets; in some systems the rapid rise in the magnetisation $M$
is continuous (\lq metamagnetic-like' behaviour \cite{Flouquet95,Puech88})
but in others the metamagnetic transition can be a true phase transition with
a discontinuous jump in $M$ as a function of $B$ \cite{Goto_review}.
%
In a true metamagnetic transition the amplitude of the metamagnetic jump
$\Delta M$ decreases as $T$ increases,
vanishing at a critical point $(B^*,T^*)$ in the $(B,T)$ plane.
%
Above $T^{*}$ one sees crossover behaviour.
%
The termination of the line of first-order transitions at an isolated 
critical end-point is analogous to the liquid--vapour co-existence curve.
%
If the critical end-point were to occur at $T^* = 0$ one would have
a metamagnetic quantum critical end-point.

Non-Fermi liquid behaviour is often seen in systems with metamagnetic or
metamagnetic-like transitions.
%
For instance, UCoAl \cite{Andreev85} has a first order metamagnetic
transition at $B\sim$0.65\,T, and a critical point at $(B^*,T^*)
\sim$(0.8\,T,13\,K) \cite{Mushnikov99}.
%
However, the reported non-Fermi liquid behaviour in the resistivity as
$T\to 0$ is assumed to arise from the proximity to a zero field
ferromagnetic quantum critical point, rather than a metamagnetic one
\cite{Kolomiets99}.
%
In MnSi, though, the vicinity to itinerant metamagnetism at high
pressures may be responsible for the observed non-Fermi liquid
behaviour over a very large region of $T$ and $p$, but the quantum
critical end-point does not appear to play a special role \cite{MnSi}.
%
Regarding the heavy fermion metamagnetic systems, such as
CeRu$_2$Si$_2$ \cite{Flouquet95,Puech88} and UPt$_3$
\cite{Kim00,Heuser00,Aoki98}, non-Fermi liquid properties have also
been observed, but none of the heavy fermion systems studied to date
have shown a first-order jump in $M$ for any combination of
parameters.
%
Thus the relevance of fluctuations associated with a quantum critical
end-point cannot be unambiguously demonstrated.

In this Letter we report the presence of a line of first order metamagnetic
phase transitions in \SrTwoLayer\ terminating in a critical end-point
for $B \perp c$. 
%
The absence of such first-order behaviour in the metamagnetic
transition with $B\parallel c$ implies that $T^*$ vanishes at some
intermediate angle, thus supporting the suggestion that a quantum
critical end-point underlies the non-Fermi liquid behavior for $B
\parallel c$ \cite{Grigera}.  
%
Moreover we investigate how the critical point evolves under pressure, 
showing that although $B^*$ rises rapidly with pressure, $T^*$ is 
comparatively unaffected. 
%
This unexpected pressure dependence of the critical end-point,
inferred from the resistivity, shows that quantum criticality in
\SrTwoLayer\ is not consistent with the standard model of itinerant 
metamagnetism \cite{Yamada93}.

The single crystals of Sr$_3$Ru$_2$O$_7$ examined here were grown from
high purity starting materials in an infrared image furnace in Kyoto.
%
The high sample quality was confirmed by measurements of the d.c.
magnetisation and resistivity ratio before and after pressurisation in
a miniature Cu:Be clamp cell.
%
The ambient pressure properties of the samples studied here correspond
to those observed and reported for around 50 other crystals from the
same batch \cite{Perry}.

A.c.\ susceptibility measurements were carried out on a 1\,mm$^{3}$
sample in an 18\,T cryomagnetic facility with a base temperature of
6\,mK.
%
A small modulation field of less than 0.01\,T was applied at a frequency 
of 77\,Hz.
%
Au wires were Ag-epoxied onto the samples and then soldered to 
Cu-clad NbTi leads inside the pressure cell.
%
A 1:4 methanol-ethanol mixture served as pressure transmitting medium
which ensured that the pressure was highly isotropic at low $T$.  
%
The low temperature resistivity was measured in a commercial 12\,T
variable temperature $^4$He cryostat in the range 2\,K to 300\,K using a
low frequency four terminal a.c. technique.

In Fig.~1 we show susceptibility curves $\chi \equiv (\partial
M/\partial H)_{T,p}$ vs $B$, applied perpendicular to $c$, at
temperatures below 2\,K.
%
The peak in the susceptibility at $B_{M\perp}$ grows with decreasing
temperature down to 1.1\,K, but thereafter the peak falls sharply in
size and becomes cusp-like.
%
A cusp in $\chi (B)$ indicates a discontinuity in $M(B)$, since the
continuous change in magnetisation across the transition is given by
the integral of $\chi$ with respect to $B$.
%
Torque magnetometry also shows a clear jump in the torque at the
transition \cite{Alix}.
%
These results unambiguously show that a first order jump develops in
$M$ below about 1\,K in this sample.  
%
Thus we interpret the peak in $\chi$ at (5.1\,T, 1.1\,K) to be 
some kind of critical point.  
%
The inset of Fig.~1 shows the position of the peak in the $(B,T)$
plane, with a solid line representing the first-order regime.
%
Nevertheless, the magnitude of the peak, given in absolute units in Fig.~1,
does not diverge as expected from a naive Landau-Ginzburg free
energy for a metamagnetic transition \cite{Yamada93}, nor from a more
sophisticated treatment more appropriate to a quantum critical
metamagnetic transition \cite{Millis02,GGL}.
%
The simplest explanation we can suggest is that slight inhomogeneities
in the sample smear out the transition, and indeed $T^{*}$ has shown
some variation between samples \cite{Grigera-next} though we believe that
this may reflect in-plane anisotropy of the metamagnetic transition.

We turn next to the basal plane resistivity, used as a probe of the
electronic behaviour.  Shown in Fig.~2 is the normalised, isothermal
magnetoresistance $\rho/\rho (B=0)$ measured at 1.5\,K at several
pressures up to 10\,kbar, for both $B~{||}~c$ and $B{\perp}c$.
%
At the lowest $T$ a double peak structure may be resolved at the fields 
$B_{M\perp 1}$ and $B_{M\perp 2}$ for $B \perp c$.
%
Together with $B_{M\parallel}$ (for $B~||~c$) there are three
identifiable anomalies.
%
An extrapolation of these three transition fields to negative
pressures, shown in Fig.~3, reveals that they all extrapolate to
$B_{M}$=0 at the {\em same} negative pressure $p_{c\perp} = p_{c ||} =
p_c \approx -14$ kbar.
%
This suggests that at a hypothetical negative pressure $p_{c}$ there
is a zero-field ferromagnetic transition.
%
The existence of a zero-field quantum critical point at negative 
pressure is reinforced by the fall in $A$, the $T^2$ coefficient of 
resistivity, measured at $B=0$, with increasing pressure (see  
Fig.~3, inset). 

Our resistivity curves as a function of pressure are very reminiscent
of those observed in CeRu$_2$Si$_2$, in which it was found that there
is scaling of the form
\begin{equation}
M(H,T,p)/\mu_B = \psi(H/H_s(p), T/T_s(p) ) \label{scaling_eqn}
\end{equation}
where $H_s(p)$ and $T_s(p)$ are scaling parameters 
\cite{Puech88,Mignot88}. 
%
We find similar scaling behaviour when we plot the relative change of
the magnetoresistance $\Delta\rho = \rho (B) - \rho (B=0)$, normalised
by $\Delta\rho (B_M)$, versus $B/B_{M\parallel}$ and $B/B_{M\perp}$
($B_{M\perp}$ corresponding to the average of $B_{M\perp 1}$ and
$B_{M\perp 2}$).
%
This quantity is seen to evolve in a universal manner, implying that
$\rho(B,T,p)$ scales with the critical field and may be described by a
function $\rho(B/B_c(p),T)$ above 2.5\,K.

It is surprising that there is no need for a scaling parameter
$T_s(p)$, i.e. $T_s(p)$, which would naturally be identified with
$T^*$, is in \SrTwoLayer\ independent of pressure.
%
Thus, if we consider a critical regime centered on $B_M$ as
illustrated for example in Perry et al.\ \cite{Perry}, the dominant
effect of pressure is to shift the critical regime along the $B$
axis, without a significant change of the temperature dependence.

In the standard model \cite{Millis02,GGL,MnSi,Yamada01}, pressure
dependence is built in via one parameter only, $\chi(T=0)$.
%
For the case of \SrTwoLayer\ both $B^*$ and $T^*$ are expected to be
equally strongly pressure dependent, in stark contrast with
experimental observations.
%
The actual strong pressure dependence of $B^*$ and the weak $p$
dependence of $T^*$ require instead an additional pressure dependence
of the other parameters, such as the mode-mode coupling term or the
frequency and momentum spread of the spin fluctuation spectrum.
%
The required cancellation of $p$-dependencies for $T^*$ would
be purely accidental.
%
The non-divergence of $\chi$ at the critical point, combined with the
pressure independence of $T^*$, suggests that the underlying physics
of the metamagnetic transition in \SrTwoLayer\ is beyond the standard
sixth-order Ginzburg-Landau theory in which $M$ is the expansion
parameter \cite{Yamada93}.

CeRu$_2$Si$_2$ might provide an interesting alternative model: there
the metamagnetic-like transition is a crossover associated with the
destruction of antiferromagnetic correlations \cite{RossatMignod88};
moreover magnetovolume effects provide positive feedback to
dramatically sharpen what would, at constant volume, be a rather broad
crossover \cite{Puech88,Lacerda89}.  
%
If the magnetovolume coupling were just a little bit stronger it
appears that the crossover could become first order.

Although bulk thermodynamic measurements show that \SrTwoLayer\ is
nearly ferromagnetic \cite{Ikeda00}, recent neutron scattering
experiments have found that a fall in $\chi(T)$ below 16\,K at $B=0$T is
associated with the growth of antiferromagnetic spin fluctuations
\cite{Capogna02}.
%
Moreover lattice distortions in the form of tilts and rotations of the
oxygen octahedra couple strongly to magnetism in the ruthenates
\cite{Shaked00,Leitus99,Friedt01}, and they are thought to be decisive
in stabilising ferro- vs.\ antiferromagnetism \cite{Mazin97,Fang01}.
%
\SrTwoLayer\ has a rotation-distortion of $\sim7^{\rm o}$ about the
$c$-axis \cite{Shaked00,Inoue97} that provides an obvious mechanism
for positive magnetoelastic feedback for the metamagnetic transition.
%
So it seems plausible that a drop in antiferromagnetic correlations,
made discontinuous by positive feedback from magnetoelastic coupling,
could produce the observed first-order metamagnetic behaviour in
\SrTwoLayer.

This model might explain why $\chi$ does not diverge at the
metamagnetic critical point, and critical fluctuations at finite $q$ would
contribute to the large peak in the resistivity seen at $B_M$ in our 
resistivity curves.
%
It is not clear that the CeRu$_2$Si$_2$ model explains the weak
pressure dependence of $T^*$, but some experimental tests suggest
themselves: in particular, neutron scattering could be used to track
the field dependence of both antiferromagnetic correlations (as was
done in CeRu$_2$Si$_2$ \cite{RossatMignod88}) and the rotation angle
of the octahedra through the metamagnetic transition.

In conclusion, we have found a metamagnetic critical end-point at finite 
temperature for fields applied in the basal plane $B \perp c$ of \SrTwoLayer, 
though the peak in the susceptibility at the critical point is much 
smaller than expected. 
%
We have in addition shown that the application of pressure drives the critical 
point up in field, but surprisingly does not produce a substantial 
shift of the critical temperature. 

We wish to thank A.J. Millis, D. Forsythe, S.-I. Ikeda, A. Rosch, S.S.
Saxena, M. Uhlarz and H. v. L\"ohneysen.  We gratefully acknowledge
financial support from the ESF (FERLIN), NSERC (Canada), EPSRC (UK),
the Royal Society and the Leverhulme Trust.


\begin{figure}
\includegraphics[scale=0.5]{fig1.eps}
\caption{A.c. susceptibility as a function of magnetic field for  
  $T = (0.06,0.4,0.8,0.9,1.0,1.1,1.3,1.5,1.8,2.5)$\,K .  The maximum
  peak corresponds to the critical point at
  $(B^{*},T^{*}$)=(5.08T,1.1K).  At the lowest temperatures (60\,mK
  and 400\,mK), the cusp-like features indicate first-order behaviour
  in $M$.  Hence for $T<T^*$ we have used solid lines to represent
  first-order behaviour; dashed lines represent crossover behaviour
  above $T^*$.  Inset: Temperature of the susceptibility peak as a
  function of magnetic field, with the solid line showing the line of
  first-order transitions, terminating in a metamagnetic critical
  end-point $(B^*,T^*)$. }
\end{figure}


\begin{figure}
\includegraphics*[scale=0.5]{fig2.eps}
\caption{Normalised magnetoresistance for $B~{||}~c$ (upper) and $B\perp c$ 
(lower) in Sr$_3$Ru$_2$O$_7$ for pressures up to 10\,kbar (increasing
pressure from left to right) at 2.5\,K.}
\end{figure}

\begin{figure}
\includegraphics[scale=0.4]{fig3.eps} 
\caption{Increase of the metamagnetic transition
fields $B_{||}$, $B_{\perp 1}$ and $B_{\perp 2}$ in Sr$_3$Ru$_2$O$_7$
as a function of pressure, where $B_{\perp 1}$ corresponds to the
lower field value of the double transition and $B_{\perp 2}$ the
higher.  In any case, all three critical fields extrapolate to a
unique negative critical pressure $p_c \approx -14$\,kbar.  }
\end{figure}

\begin{figure}
\includegraphics[scale=0.37]{fig4.eps}
\caption{Normalised magnetoresistance (refer to text) as a function of
the magnetic field scaled by the respective critical field $B_M$.  
Left: $B_M = (B_{\perp 1} + B_{\perp 2})/2$ is an average of the two 
critical fields for $B_{\perp c}$.  Right: $B_{M}=B_{||}$.}
\end{figure}


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\end{document}






cond-mat0207697.extracted_bib
\begin{thebibliography}{99}

\bibitem{Maeno} Y. Maeno {\it et al.}, Nature {\bf 372}, 532 (1994).

\bibitem{Allen} P.B. Allen {\it et al.}, Phys. Rev. B {\bf 53}, 4393
  (1996).

\bibitem{Klein} L. Klein {\it et al.}, J. Phys.: Cond. Matt. {\bf 8},
  10111 (1996).

\bibitem{Braden} M. Braden {\it et al.}, Phys. Rev. B {\bf 58}, 847
(1998).

\bibitem{Perry} R.S. Perry {\it et al.}, Phys. Rev. Lett. {\bf 86}, 2661
(2001).

\bibitem{Grigera} S.A. Grigera {\it et al.}, Science {\bf 294}, 329 (2001)

\bibitem{Flouquet95}    % review of CeRu2Si2
J.\ Flouquet {\it et al.},
Physica B {\bf 215}, 77 (1995).

\bibitem{Puech88}
L.\ Puech {\it et al.},
J.\ Low Temp.\ Phys.\ {\bf 70}, 237 (1988).

\bibitem{Goto_review} for a recent review see T.\ Goto {\it et al.},
Physica B {\bf 300} (2001) 167.

\bibitem{Andreev85}
A.V.\ Andreev {\it et al.},
Sov.\ Phys.\ Solid State {\bf 27}, 1145 (1985).

\bibitem{Mushnikov99}
N.V.\ Mushnikov {\it et al.},
Phys.\ Rev.\ B {\bf 59}, 6877 (1999).

\bibitem{Kolomiets99}
A.V.\ Kolomiets {\it et al.},
Physica B {\bf 261}, 415 (1999).

\bibitem{MnSi} C. Pfleiderer, S.R. Julian and G.G. Lonzarich, Nature
{\bf 414}, 427 (2001).

\bibitem{Kim00}
J.S. Kim {\it et al.}, Solid State Commun.\ {\bf 114}, 413 (2000).

\bibitem{Heuser00}
K.\ Heuser {\it et al.},
J.\ Low Temp.\ Phys.\ {\bf 118}, 235 (2000).

\bibitem{Aoki98}
H. Aoki {\it et al.}, J.\ Magn.\ Magn.\ Mater.\ {\bf 177-181}, 271 (1998).

\bibitem{Yamada93} H. Yamada, Phys. Rev. B {\bf 47}, 11211 (1993).

\bibitem{Alix} A. McCollam {\it et al.}, to be published.

\bibitem{Millis02}
A.J. Millis {\it et al.},
Phys.\ Rev.\ Lett.\ {\bf 88}, 217204 (2002).

\bibitem{GGL} G.G. Lonzarich, to be published.

\bibitem{Grigera-next} S.A. Grigera {\it et al.}, to be published.

\bibitem{Mignot88}
J.-M. Mignot {\it et al.},
J.\ Magn.\ Magn.\ Mater.\ {\bf 76\&77}, 97 (1988).

\bibitem{Yamada01}
H.\ Yamada {\it et al.}, Phys.\ Rev.\ B {\bf 65}, 024413 (2001).

\bibitem{RossatMignod88}
J.\ Rossat-Mignod {\it et al.}, J.\ Magn.\ Magn.\ Mater.\ {\bf 76\& 
77}, 376 (1988).

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\end{thebibliography}