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\begin{document}
\title{Local magnetic order in La$_2$CuO$_4$ seen via $\mu ^+$SR spectroscopy}

\author{V G Storchak$^1$, J H Brewer$^2$, D G Eshchenko$^3$, P W Mengyan$^4$, O E Parfenov$^1$, 
A M Tokmachev$^1$, P Dosanjh$^2$ and S N Barilo$^5$}

\address{$^1$ Kurchatov Institute, Kurchatov Sq.~1, Moscow 123182, Russia}
\address{$^2$ Department of Physics and Astronomy, 
 University of British Columbia, Vancouver, BC, Canada V6T 1Z1}
\address{$^3$ Bruker BioSpin AG, Industriestrasse 26, 8117 F{\"a}llanden, Switzerland}
\address{$^4$ Department of Physics, Texas Tech University, Lubbock, Texas 79409-1051, USA}
\address{$^5$ Institute of Solid State and Semiconductor Physics, 
 National Academy of Sciences of Belarus, Minsk 220072, Belarus}

\ead{mussr@triumf.ca}

\begin{abstract}

The origin of a pseudogap in the underdoped phase of cuprates has become one of the leading issues 
in understanding the mechanism(s) of high-$T_c$ superconductivity. Several experiments 
%, in particular, 
(i.e. polarized neutron diffraction studies) support %certain 
theoretical models based on the 
circulating current (CC) picture. These CC models suggest a novel ordered phase formed by orbital 
magnetic moments caused by spontaneous orbital currents. However to date, $\mu$SR experiments in 
cuprate superconductors have not revealed any magnetic field higher than 0.2~G associated with 
orbital moments. Here we present high magnetic field $\mu$SR spectroscopy of a stoichiometric 
La$_2$CuO$_4$ crystal which shows %that along with the 
antiferromagnetic splittings %we see 
and 
additional 45~G line splittings, consistent with orbital currents.
\end{abstract}

\section{Introduction}

The origin of the pseudogap (PG) and its relation to the nature of high-$T_c$ superconductivity 
%is hotly 
has been highly debated 
%already 
for %more than 
over 20 years. In the underdoped regime of the phase diagram, 
cuprate superconductors show features of a correlated metal exhibiting a remarkable departure from 
the standard Fermi liquid behavior. Their transport, magnetic and thermodynamic properties 
%point towards 
suggest strong reduction of the electronic density of states (DOS) below a temperature $T^*$, 
though %such 
the DOS does not reach zero at the lowest temperature and the system remains metallic 
\cite{Norman2003}. However, ARPES experiments \cite{Kaminski2002} 
suggest opening of a $q$-dependent real gap 
in a one-particle excitation spectrum, supported by other spectroscopies 
\cite{Timusk1999}.

Certain theoretical models consider the PG state as an ordered phase with a well defined order 
parameter and a related broken symmetry 
\cite{Varma1997,Varma2006,Chakravarty2001}. In this scenario, the PG 
temperature $T^*$ is a phase transition temperature to an ordered state formed by orbital magnetic moments caused by spontaneous orbital currents 
flowing in specific patterns within or around CuO$_2$ plaquettes. 
The fluctuations associated with the broken symmetry are considered to be 
responsible for both the superconductivity, playing role of a pairing glue, and the 
non-Fermi-liquid behavior below the $T^*$ phase transition line on the phase diagram.

The key experimental evidence of a novel ordered state comes from spin-polarized neutron scattering experiments which reported commensurate magnetic peaks below $T^*$ in YBCO and H1201 
\cite{Fauque2006,Bourges}, consistent with ARPES data, which point to time-reversal symmetry breaking (TRSB) that nevertheless preserves lattice translation invariance.
Sizeable magnetic moments on the order of 0.1-0.2 $\mu _B$ are reported in the PG state 
\cite{Fauque2006,Bourges}. 

However, $\mu$SR experiments which have convincingly demonstrated their 
sensitivity to TRSB fields in a number of weak magnetic systems, produced no evidence of a magnetic 
order in the PG state. At first, spontaneous static magnetic fields were reported in YBCO, which 
%later on 
were later reinterpreted as being due to spatial charge inhomogeneities \cite{Sonier2001}. No 
TRSB is reported in LSCO where $\mu$SR experiments set an upper limit of $\sim 0.2$~G for magnetic 
field at the muon site while the expected local field is estimated to amount about 40~G 
%is estimated for the smallest 
%reported moment in YBCO 
\cite{MacDougall2008}.

This discrepancy between neutron and muon experiments has been attributed to screening of the 
charge density in the metallic-plane unit cells in the vicinity of the muon \cite{Shekhter2008}. On 
the other hand, orbital currents and associated magnetic moments may %well 
be present in the 
limiting case of the underdoped regime --- in the insulating stoichiometric La$_2$CuO$_4$, should 
CC be an intimate feature of chemical bonding in the CuO$_2$ plane.

Here we present 
%zero and high magnetic field 
$\mu$SR spectroscopy of stoichiometric La$_2$CuO$_4$ 
crystal (accompanied by SQUID measurements), 
in which the magnetic field at the muon should not be 
affected by charge density screening.
 
\section{Results and discussion}

High-quality La$_2$CuO$_{4+x}$ single crystal grown 
%under quasi-equilibrium conditions 
from CuO flux have been used for current studies. 
%The crystal orientation, 
%lattice parameters, and mosaicity have been determined by X-ray single-crystal 
%diffractometry. It has been established that the sample mosaicity along the $c$-axis is 
%no more than 0.05$^\circ $. 
Oxygen over its stoichiometric content has been removed by annealing in 
vacuum.
% at a temperature less than 950~K and oxygen partial pressure less than 
%2$\cdot $10$^{-5}$ bar. 
The lattice 
parameters of the crystal correspond to the low temperature orthorhombic 
stoichiometric La$_2$CuO$_4$ 
%($Bmab$ space group) 
\cite{Parfenov2002}.
N\'eel temperature measured using SQUID amounts to 320~K. 

Time-differential $\mu$SR experiments, using 100\% spin-polarized positive muons implanted into 
these samples, were carried out on the M15 surface muon channel using the $\it HiTime$ 
spectrometer. At low temperature, zero field $\mu$SR spectra consist of two (small and large 
amplitude) components, well known from numerous previous studies and indicative of two 
inequivalent muon sites in antiferromagnetic La$_2$CuO$_4$. 
Figure \ref{fig:La2CuO4-ZF-Tdep} shows temperature dependences 
of the corresponding frequencies. 
Above 150~K, the low-frequency 
component (signal with a small amplitude) disappears below the background. 
The N\'eel temperature $T_N\approx 320$~K and 
magnetic fields at the muon sites $B=428.7$~G and $B=111.8$~G 
determined from the temperature dependences shown in Figure \ref{fig:La2CuO4-ZF-Tdep} are 
consistent with numerous earlier measurements of stoichiometric La$_2$CuO$_4$.

\begin{figure}[ht]
\includegraphics[width=0.55\textwidth]{La2CuO4-ZF-Tdep.eps} 
\hfill %% \hspace{2pc}
\begin{minipage}[b]{0.4\textwidth}\caption{\label{fig:La2CuO4-ZF-Tdep}
 Temperature dependences of the muon spin precession frequencies 
 in stoichiometric La$_2$CuO$_4$ in %magnetic field $H=0$. 
 zero applied magnetic field. 
 Red circles and 
 blue triangles: frequencies obtained via a 2-component fit of $\mu$SR spectra at low temperature. 
 Black squares: frequencies obtained via a 1-component fit of $\mu$SR spectra at higher temperature.  
 \\~\\~\\
}
\end{minipage}
\end{figure}
%\vspace{-4mm}

In high magnetic field $B$ transverse to the initial muon spin polarization and parallel to the 
$c$-axis of La$_2$CuO$_4$ crystal, muon spin precession spectra exhibit at least 7-component 
pattern. The characteristic Fourier transform is shown in Figure \ref{fig:La2CuO4-100K-1T-spec}. 
Central line at about 135.6~MHz coincides with the single line detected in 
%the reference 
%non-magnetic sample of CaCO$_3$. 
CaCO$_3$, a non-magnetic reference sample. 
We attribute this line to those muons (small portion of the muon 
flux) which missed the sample and stopped in a non-magnetic environment. 
Two small peaks positioned symmetrically around the central line 
correspond to the AFM splitting of the low-frequency (low-amplitude) signal 
observed in zero magnetic field. 
Our main interest is focused on large-amplitude signals 
again positioned symmetrically around the central line. 
We attribute these signals to high-frequency (large-amplitude) signal in 
Figure \ref{fig:La2CuO4-ZF-Tdep}.

\begin{figure}[ht]
\includegraphics[width=0.55\textwidth]{La2CuO4-100K-1T-spec.ps}
\hfill %% \hspace{1pc}
\begin{minipage}[b]{0.4\textwidth}
\caption{\label{fig:La2CuO4-100K-1T-spec}
 Frequency spectrum of muon spin precession in La$_2$CuO$_4$ 
 in a transverse magnetic field of $H=1$~T at $T=100$~K.  
 Central line frequency coincides with that of a single peak 
 detected in the reference non-magnetic sample (CaCO$_3$).  
 \\~\\~\\~\\~\\
}
\end{minipage}
\vspace{-4mm}
\end{figure}

In order to %find out 
uncover the origin(s) of the magnetic field %on 
at the muon site, 
which may come from different sources, we %have to find out exactly 
must exactly determine the muon positions 
in the La$_2$CuO$_4$ lattice. 

Stoichiometric La$_2$CuO$_4$ is a collinear antiferromagnetic material with four sublattices. 
Magnetic moment on Cu atoms is about 0.5~$\mu _B$ directed along the diagonal of the 
CuO$_2$ plaquette \cite{Kastner1998}. 
A peculiar feature of magnetic ordering  in 
La$_2$CuO$_4$ is the presence of a weak ($\sim $0.002 $\mu _B$) ferromagnetic coupling of spins
within CuO$_2$ layers \cite{Kastner1998,Thio1988} which is believed to originate from 
Dzyaloshinskii-Moriya exchange interaction. 
These moments are orthogonal to CuO$_2$ layers and have 
opposite directions in the neighbouring planes. 
They are coupled to in-plane magnetic moments on Cu 
via tilting of CuO$_6$ octahedra. 
%Thus 
Therefore, CuO$_2$ is an antiferromagnet with %a 
hidden, weak ferromagnetism. 
A spin-flop transition to this hidden phase (weak ferromagnetism) takes place 
in high magnetic field applied along the $c$-axis \cite{Kastner1998,Thio1988}. 
Within this phase, the direction of ferromagnetic moments in every CuO$_2$ plane 
becomes the same. 
Our SQUID measurements (Figure \ref{SQUID}) at 250~K show that the spin-flop 
transition is observed at magnetic field of about 3.5-4~T. 
The magnetization jump $\sim $0.002 $\mu _B$ is consistent with literature data
%corresponds well to the previously published values 
\cite{Kastner1998,Thio1988}.

\begin{figure}[ht]
\includegraphics[width=0.55\textwidth]{squid_spin_flop.eps}
\hfill
\begin{minipage}[b]{0.4\textwidth}
\caption{\label{SQUID}
 Magnetic field dependence of magnetic moment per Cu atom along the $c$-axis. 
%on magnetic field. 
 Blue color: measurements in increasing magnetic field; 
 red color: measurements in decreasing magnetic field.  
\\~\\~\\~\\~\\
}
\end{minipage}
\vspace{-4mm}
\end{figure}

%High magnetic field muon spin resonance can be used to study t
The spin-flop transition discussed above 
%in La$_2$CuO$_4$ 
is clearly seen in
%. The microscopic 
$\mu$SR measurements at 250 K (Figure \ref{fig:FFT-La2CuO4_250K}), where 
half of the lines in the spectra vanish %es 
at a magnetic field of about 4~T. 
%which means that 
A remarkable consequence of the spin-flop 
transition is that the magnetic unit cell becomes equivalent to the 
%atomic 
crystallographic unit cell; the number 
of magnetic sublattices decrease %s 
from 4 to 2. 
From symmetry considerations %it 
this means that %the 
tilting of the CuO$_6$ 
octahedra %makes 
forces the muon to occupy only half of the otherwise crystallographically equivalent positions. 
However, in order to go beyond general 
considerations, possible muon positions should be clearly established.

\begin{figure}[ht]
\includegraphics[width=0.55\textwidth]{FFT-La2CuO4_250K.eps}
\hfill
\begin{minipage}[b]{0.4\textwidth}
\caption{\label{fig:FFT-La2CuO4_250K}
 Fourier transforms of the muon spin precession signal in La$_2$CuO$_4$ 
 at $T=250$~K in different transverse external magnetic fields.  
\\~\\~\\~\\~\\~\\~\\
}
\end{minipage}
%\vspace{-2mm}
\end{figure}

The muon stopping site(s) in La$_2$CuO$_4$ have been discussed in several papers
%Many attempts to determine  made 
\cite{muon_site}. Information obtained from zero-field $\mu$SR spectra is insufficient and 
additional data are typically sought from electronic structure calculations. It is common to 
predict the muon position to be in the vicinity of the apical oxygen atom but the exact 
positions are quite different in a number of publications leading to much confusion. 
In contrast, high-field $\mu$SR measurements 
with different setups provide data on projections of the magnetic field on the muon 
which correspond perfectly to the frequencies determined from the zero-field experiments. 

To find muon positions based on this extended set of experimental data, we performed calculations 
based on the dipole-field approximation \cite{Yaouanc2011} assuming the periodic antiferromagnetic 
structure with the known magnetic moments on Cu atoms. There are only two types of 
%symmetry 
inequivalent muon positions for the 
high-frequency high-amplitude signal in Figure~\ref{fig:La2CuO4-ZF-Tdep}: one is close to the 
apical oxygen and the other is very close to La atom. 
From chemical bonding considerations the first type is far more stable. 
Similar analysis 
is possible for the low-frequency signal and only one 
of the two positions is chemically substantiated. 
Figure~\ref{structure} shows possible muon positions 
for both low- and high-frequency signals. The 
tilting of the CuO$_6$ octahedra produces 
two types of chemically inequivalent positions differing 
by the distance between the muon and the apical oxygen atom. 
Only one type of these positions is 
occupied by muons. Our calculations confirm that 
it is consistent with vanishing half of the spectral 
lines at the spin-flop transition (accompanied by 
10\% decrease of the line shift from the free muon signal).
%Only one type of these positions is 
%occupied by muons and our calculations show that it, indeed, 
%leads to vanishing half of the spectral lines 
%at the spin-flop transition with their separation from 
%the free muon signal being decreased by about 10\%.

\begin{figure}[ht]
\null \hfill
\includegraphics[width=12pc]{lco_muon_pos.eps}
\hfill %% \hspace{4pc}
\begin{minipage}[b]{20pc}
\caption{\label{structure}
 Unit cell for the low temperature orthorhombic phase of La$_2$CuO$_4$: 
 %and estimated muon positions: 
 green - Cu atoms, 
 grey - La atoms, 
 blue - O atoms, 
 red - muon positions 
 %(high-frequency signal is from the muon close to Cu 
 %and low-frequency signal comes from the muon close to La)
 (high- and low-frequency signals come from 
 the muon residing near Cu and La, respectively); 
 atomic positions are shown for pristine La2CuO4, 
 the perturbation caused by the muon is not taken into account.
\\~\\~\\~\\~\\
}
\end{minipage}
\end{figure}

One should expect 4 distinct signals in the high-field $\mu$SR spectra from two independent muon 
positions known from zero-field $\mu$SR experiments and the antiferromagnetic nature of the 
sample. 
Instead, at least 6 peaks 
have been observed in high magnetic fields of different directions. 
That means that apart from the AFM field, there is 
{\sl an additional source of magnetic field on the muon\/} %:
where  
some muons experience magnetic 
field larger than that coming from AFM while the other feel 
smaller magnetic field which causes 
the characteristic splitting of the large-amplitude signals 
(see Figure~\ref{fig:FFT-La2CuO4_250K}). 
The apparent absence of splittings %of
in low-frequency signals can be 
explained by the increased distance from the CuO$_2$ planes 
leading to much smaller splittings 
which are not well resolved in the experiments.

There are two 
types of models explaining intra-unit-cell magnetic order in cuprates 
which may reveal the nature of 
the additional source of magnetic field on the muon.  
One relies upon spins being ascribed to oxygen 
atoms while the other is based on circulating 
orbital currents. 
Polarized neutron experiments 
have not been able to differentiate 
the existing models beyond establishing TRSB 
\cite{Fauque2006,Bourges}. 
It seems to be possible using high-field $\mu$SR experiments.  
While the strength of additional magnetic fields on the muon 
(which causes the observed splitting in the first place) 
is not known {\it a priori}, the ratio of signal splittings for 
different directions of the external magnetic field 
can be used to rule out certain models; 
although, a lot can be inferred from simple symmetry considerations.

The first class of models is based on the antiferromagnetic ordering of spins on in-plane oxygen 
atoms. When the spins are directed along the $b$- or $c$-axis of the crystal \cite{Moskvin} 
the spectra are not described even qualitatively due to absence of splitting along the $c$-axis. 
The case of oxygen spins directed along the $a$-axis \cite{Fauque2006} is less obvious as the 
symmetry is correct but the ratio of the splittings along the $b$- and $c$-axes is about 1.6, 
while our experiments determine it to be close to 1.
%This model cannot be rejected right away but it is not very probable. 

Magnetic orders based on CC provide a happy alternative. The model dubbed 
$CC$-$\Theta _I$ and formed by 4 orbital current loops for each CuO$_2$ plaquette \cite{Varma1997} 
is not consistent with polarized neutron experiments 
as well as with 
%or 
our $\mu$SR data complemented by symmetry considerations. 
The most widely used model is $CC$-$\Theta _{II}$ with 2 opposite current 
loops O-Cu-O per CuO$_2$ plaquette \cite{Varma2006}. 
This model has correct symmetry properties and 
predicts the ratio of the splittings along the $b$- and $c$-axes to be about 1.1, which is close 
to the observed value. The orbital magnetic moment extracted from our data amounts 
about 0.04~$\mu _B$, which is comparable with the theoretical value 
on the order 0.1~$\mu _B$ \cite{Varma1997}. 
An alternative staggered orbital current 
phase with the same symmetry as $CC$-$\Theta _{II}$ but Cu-Cu-Cu currents \cite{Stanescu}
is 
%possible but our 
%calculations show that it is 
also unlikely because the ratio of the splittings along the $b$- and 
$c$-axes is too large. 

In summary, high magnetic field $\mu$SR studies 
in stoichiometric La$_2$CuO$_4$ 
reveal additional source of magnetic field 
on the muon consistent with Varma model of circulating currents 
%loops
\cite{Varma2006}.

\ack
This work was supported by the Kurchatov Institute, NSERC of 
Canada and the U.S. DOE, Basic Energy Sciences (Grant DE-SC0001769).


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\end{document}
