\documentclass[a4paper]{jpconf}
\usepackage{graphicx}
\begin{document}
\title{\boldmath 
 Search for ALCR-$\mu^+$SR spin polaron resonances 
 in Cd$_2$Re$_2$O$_7$ and FeGa$_3$ 
\unboldmath}

\author{\underline{J.H.~Brewer}}
\address{Department of Physics and Astronomy, 
 University of British Columbia, Vancouver, BC, Canada V6T 1Z1}
\ead{jess@triumf.ca}
\author{V.G.~Storchak}
\address{Kurchatov Institute, 
 Kurchatov Sq.~1, Moscow 123182, Russia}
\author{D.G.~Eshchenko}
\address{Bruker BioSpin AG, 
 Industriestrasse 26, 8117 F{\"a}llanden, Switzerland}
\author{R.L.~Lichti and P.W.~Mengyan}
\address{Department of Physics, 
 Texas Tech University, Lubbock, Texas 79409-1051, USA}
\author{D.J.~Arseneau}
\address{TRIUMF, 4004 Wesbrook Mall, 
 Vancouver, BC, Canada V6T 2A3}

\begin{abstract}
Certain magnetic materials may contain spin polarons (SP) --- 
tiny, stable ferromagnetic ``droplets'' 
in a paramagnetic or weakly ordered ``sea'' of more random spins.  
The positive muon may become associated with SP, 
giving a unique window on their properties.  
In this experiment the $\mu^+$SR spectroscopy of 
``avoided level crossing resonance'' ($\mu$ALCR) was used 
on several candidate materials to further explore this model.  
The results leave many questions unanswered.  
\end{abstract}

\section{Introduction}

Numerous semiconducting\cite{Storchak2009A1,Storchak2009L1,Storchak2012A1} 
and metallic\cite{Storchak2010L1} magnetic materials 
have been found to exhibit characteristic 
two-frequency $\mu^+$SR precession signals 
in high transverse magnetic field (HTF-$\mu^+$SR), 
but skepticism remains over the assignment of these spectra 
to muons associated with de Gennes' legendary 
magnetic polarons (MP)\cite{deGennes1960A1} 
or spin polarons (SP).  
This is understandable, since the SP picture is 
a radical departure from ``conventional wisdom'' 
about both muonium and magnetism.  
It is therefore incumbent upon both advocates 
and adversaries of this interpretation 
to present as much spectroscopic evidence as possible 
in support or contradiction of the SP picture.  

\begin{figure}[htb]
%% \begin{center}
%% \vspace*{-5mm}
\includegraphics[width=0.6\columnwidth]{PaschenBach.ps}
\hfill 
\begin{minipage}[b]{0.35\textwidth}
\caption{\label{fig:PaschenBach} \sf 
 Muonium energy level diagram in the 
 limit of high magnetic field (muon Larmor 
 frequency $\nu_\mu \gg A$, the hyperfine frequency).  
\\~\\~\\~
}
\end{minipage}
%% \end{center} 
%% \vspace*{-4mm}
\end{figure}

In studies of muonium (Mu = $\mu^+ e^-$) in solids 
and muonated radicals in liquids, 
the hyperfine frequency of the species 
containing the muon is routinely measured 
in high transverse magnetic field (HTF-$\mu^+$SR) 
as the splitting $A$ between two precession frequencies 
on opposite sides of the muon Larmor frequency $\nu_\mu$, 
as shown in Fig.~\ref{fig:PaschenBach}.  
In a SP, the electron's spin is locked to 
that of the local ferromagnetic (FM) ``droplet'' 
by a very strong exchange interaction, 
but the same sort of splitting still appears.  
Of course, muons in an antiferromagnetic (AFM) crystal 
with local fields parallel or antiparallel to the applied field 
will produce a similar pattern, 
so other considerations must help determine 
which scenario applies.  

\begin{figure}[htb]
%% \begin{center}
%% \vspace*{-5mm}
\includegraphics[width=0.5\columnwidth]{ALCR.ps}
\hfill 
\begin{minipage}[b]{0.45\textwidth}
\caption{\label{fig:ALCR} \sf 
 Avoided level crossings occur when 
 ``flipping'' a third spin $X$ that couples to the 
 muon and/or electron causes the same change 
 in energy as ``flopping'' the muon spin at a 
 certain applied field $B_0$.  A resonant loss of 
 muon polarization results at $B_0$.  This method is 
 widely used to identify the partner $X$, its 
 interaction strength $J$ and, from that, the site 
 and environment of the muon. 
\\~\\~\\~
}
\end{minipage}
%% \end{center} 
%% \vspace*{-4mm}
\end{figure}

The basic mechanism of 
muon avoided level crossing resonance ($\mu$ALCR) 
is depicted in Fig.~\ref{fig:ALCR}.  
Usually it is assumed that $J \ll A$.  
The opposite is the case in a SP, 
if $J$ is the exchange interaction between 
the SP electron and the paramagnetic ions 
comprising the SP.  
However, if $J$ is the {\sl nuclear\/} 
hyperfine (NHF) coupling between the SP 
electron and a nearby {\sl nuclear\/} spin, 
the same situation may apply as in ordinary 
semiconductors\cite{Kiefl1987L2} or radicals \cite{Kiefl1986L1}, 
for which $\mu$ALCR spectroscopy \cite{Kiefl1987L2,Kiefl1986L1,Kreitzman1986L1} 
has yielded a plethora of information about the location, 
structure and effective spin hamiltonian 
of the paramagnetic center.  

It is therefore natural (even mandatory) to explore 
the $\mu$ALCR spectroscopy of candidate SP systems in the same way.  
We performed an initial study of 
several nominally dissimilar materials 
with nearly identical SP-like HTF-$\mu^+$SR spectra.  

\section{\boldmath SP$\mu$ \unboldmath Spectroscopy}

The MP or SP is formed when a free electron 
in a paramagnetic or weakly magnetic host 
has a huge exchange interaction $J$ with neighboring magnetic ions, 
causing their spins to be locked together 
(along with that of said electron) 
into a small ($\sim 1$ unit cell) 
ferromagnetic (FM) ``droplet'' --- the SP --- 
widely believed to be the root cause of 
various important phenomena such as 
metal-insulator transitions 
in magnetic semiconductors (MS)\cite{Nagaev2002B1}.  
The energy decrease due to this local FM ordering 
compensates for the increased kinetic energy 
of the SP electron caused by its localization.  
In some cases the additional electrostatic binding energy 
of said electron to the $\mu^+$ assists in the localization; 
in other cases a SP population exists independent of the muon 
but the negatively charged SP attracts 
(and is bound to) the $\mu^+$ in a neutral ``SP$\mu$'' complex, 
analogous to muonium or muonated radicals.  

The ability of HTF-$\mu^+$SR to probe the SP 
depends upon the muon's incorporation 
into the SP in a consistent way; 
thus the system in question is often described as 
a {\sl bound\/} magnetic polaron (BMP).  
However, this designation is usually taken to imply 
a SP that is not free to move through the lattice, 
which may be a poor picture of the SP$\mu$ system 
in cases where the muon is ``bound'' to the SP 
only by the latter's negative charge, 
providing only a small extra impedance to 
the mobility of this exotic charge carrier.  

The two SP candidate materials chosen for this $\mu$ALCR study 
are the metallic, magnetically frustrated pyrochlore 
Cd$_2$Re$_2$O$_7$ and the magnetic semiconductor FeGa$_3$, 
both of whose HTF-$\mu^+$SR frequency spectra are shown 
in Fig.~\ref{fig:FFTs}.  

\begin{figure}[htb]
\begin{center}
\vspace*{-5mm}
\includegraphics[width=0.475\columnwidth]{Cd2Re2O7-FFT.ps}
\hfill 
\includegraphics[width=0.475\columnwidth]{FeGa3-FFT.ps}
\vspace*{-1mm}
\caption{\label{fig:FFTs} \sf
 LEFT: 
 Frequency spectra of HTF-$\mu^+$SR signals in 
 Cd$_2$Re$_2$O$_7$ at $H=1$~T\cite{Storchak2010L1}.
 The splittings do not change appreciably with $H$
 (not shown).  
%
 RIGHT: HTF-$\mu^+$SR frequency spectra at $H=1$~T 
 in the magnetic semiconductor FeGa$_3$.  
 Again the splittings do not change significantly with $H$.  
}
\end{center} 
\vspace*{-4mm}
\end{figure}

Cd$_2$Re$_2$O$_7$ is a weakly metallic pyrochlore with 
geometrically frustrated magnetism, a heavy electron 
effective mass and a superconducting transition at $\sim 1$~K.  
In the region below $\sim 60$~K, where its resistivity follows 
a $T^2$ dependence (characteristic of a Fermi liquid regime), 
the HTF-$\mu^+$SR frequency spectrum at $H = 5$~T shows the 
splittings characteristic of a SP with a large hyperfine 
coupling\cite{Storchak2010L1}.  

FeGa$_3$ is a narrow-gap diamagnetic semiconductor 
for which the gap formation is attributed to 
strong electron correlations within a narrow $3d$ band.  
Below about 10~K an extremely narrow SP band is 
thought to form; in the same range, characteristic SP 
splittings are observed in the HTF-$\mu^+$SR frequency
spectra, with splittings independent of field 
up to 5~T\cite{Storchak2012A1}.

Both materials have plentiful nuclear spins which
might couple to the SP electron, possibly producing a
rich $\mu$ALCR spectrum.  

The HTF-$\mu^+$SR ``signature'' of SP$\mu$, 
like that of Mu or a muonated radical, 
is a pair of frequencies split about the free muon Larmor frequency 
($\nu_\mu \hbox{\rm [MHz]}= 135.54 B_\mu \hbox{\rm [T]}$, 
where $B_\mu$ is the local magnetic field at the muon site) 
by $+\Delta\nu_+$ and $-\Delta\nu_-$ with 
$\Delta\nu_+ - \Delta\nu_- = A$, 
the effective frequency of the $\mu$-$e$ HF interaction.  
This similarity, plus the fact that 
SP$\mu$ spectra were first discovered in MS, 
may tempt one to attribute such spectra to normal Mu atoms; 
but this cannot be the case, first because 
(as in Cd$_2$Re$_2$O$_7$, shown in Fig.~\ref{fig:FFTs}[left]) 
SP$\mu$ is seen in {\sl metallic\/} hosts 
where the Coulomb attraction of the $\mu^+$ 
is screened by conduction electrons, 
and second because the fields of large nearby moments 
in the paramagnetic phase would instantly relax the Mu electron 
if it were not tightly locked to the spins in the FM ``droplet''.  
%
Very similar spectra are seen in 
the MS FeGa$_3$ (Fig.~2) %% \ref{fig:FFTs}[right]) 
at low temperature.  
It is remarkable that the signature of SP$\mu$ 
should be observed in such different magnetic systems; 
usually one does not expect a hitherto undiscovered phenomenon 
to be quite so ubiquitous.  
In each instance one can 
often %% , with sufficient determination, 
find alternative explanations for the observed 
pairs of lines --- 
two sites with different Knight shifts, 
or one site in an antiferromagnetic environment, 
%% (even if the sample is known to be magnetically disordered) 
or other more exotic interpretations 
invoking modification by the $\mu^+$ 
of the local magnetic properties of the host.  
%% As long as the alternative is not perceived as 
%% more ``outlandish'' than the SP$\mu$ model, 
%% it seems to be deemed preferable, 
%% using some emotional version of Ockham's Razor.  
%% Even if this were a valid argument 
%% in each individual case, however, it 
However, such alternative-hunting 
fails to explain the remarkable consistency of 
the HTF-$\mu^+$SR spectra observed in such diverse 
magnetic metals, semiconductors and insulators.  

\section{\boldmath SP-$\mu$ALCR \unboldmath Experiments}

\begin{figure}[htb]
%% \vspace*{-1mm}
\includegraphics[width=0.475\columnwidth, angle=0]{Cd2Re2O7-ALCR.eps}
\hfill %% \\[-0.425\columnwidth] \null \hfill
\includegraphics[width=0.475\columnwidth, angle=0]{Ag_vs_FeGa3-ALCR+inset.eps}
%% \vspace*{3mm}
\caption{\label{fig:ALCRs} \sf 
%% \begin{minipage}{0.45\textwidth}
 LEFT: 
 $\mu$ALCR spectrum from 0 to 7~T 
 in metallic Cd$_2$Re$_2$O$_7$ at 30~K, 
 taken on the M15 muon channel at TRIUMF 
 with the {\it HiTime\/} $\mu$SR spectrometer.  
 Upper right inset: residuals from a polynomial fit 
 (red line) to the systematic decrease at high field 
 due to curling up of electron orbits.  
 Lower left inset: detail of 
 a broad resonance at $\sim 0.75$~T 
 ($\nu_\mu \approx 100$~MHz $\approx A_{SP}/2$) 
 and a hint of a smaller resonance at $\sim 1.3$~T.  
%% \end{minipage} ~ \begin{minipage}{0.45\textwidth}
 RIGHT: 
 $\mu$ALCR spectra from 0 to 3~T at 5~K 
 in FeGa$_3$ (upper curve, red circles) 
 and Ag (lower curve, black squares), 
 taken on the new M20 muon channel at TRIUMF 
 with the {\it Helios\/} $\mu$SR spectrometer.  
 If there is any resonant ``dip'' 
 in the FeGa$_3$ polarization curve, it is either 
 extremely broad or extremely narrow.  
 The decrease in the FeGa$_3$ curve at very low field 
 (see inset, lower right) 
 is presumably just decoupling from local dipolar fields.  
%% The difference can be attributed to target geometry effects.  
%% \end{minipage}
}
%% \vspace*{-3mm}
\end{figure}

We therefore measured $\mu$ALCR spectra over a very large field range 
in the correlated metallic pyrochlore Cd$_2$Re$_2$O$_7$ at 30~K
and in the semiconductor FeGa$_3$ at 10~K.  
The results are shown in Fig.~\ref{fig:ALCRs}.  
In each case the measurements were made (as usual) 
by scaling the Forward (F) and backward (B) 
raw positron count rates and calculating the asymmetry 
as $(B-F)/(B+F)$.  As is often the case, 
this ``raw'' asymmetry is susceptible to 
huge systematic longitudinal field (LF) dependences 
as positron orbits ``curl up'' in high fields.  
Another common problem is stochastic variations 
due to instabilities in beamline power supplies 
or the proton beam drifting on the production target; 
that effect requires use of differential method 
where the applied LF is rapidly ``toggled'' 
between $H - \delta H$ and $H + \delta H$.  
This latter method was not required for these experiments 
on M15 and the newly-rebuilt M20, 
whose power supplies are very stable.  
Therefore the huge systematic changes 
in the ``baseline'' asymmetry are evident 
in Fig.~\ref{fig:ALCRs}.  

The large, broad resonance in Cd$_2$Re$_2$O$_7$ 
at $\sim 0.75$~T 
(corresponding to a frequency $\nu_\mu \approx 100$~MHz)  
is consistent with a zero-crossing resonance (ZCR) 
for muons at a site with a local field of that magnitude 
antiparallel to the applied field, 
as in MnF$_2$ \cite{Kiefl1987L1}.  
This would imply that the SP hyperfine splitting $A_{SP}$ 
in Cd$_2$Re$_2$O$_7$ at low $T$ is not the large 
($\Delta\nu_{\rm big} \approx 200$~MHz) splitting 
shown in Fig.~\ref{fig:FFTs} but the smaller 
($A_{SP} = \Delta\nu_{\rm small} \sim 30$~MHz) 
splitting between blue and red peaks 
in each of the groups on either side of $\nu_\mu$, 
while $\Delta\nu_{\rm big}$ comes from 
local magnetic ordering at the muon site.  
If there is actually a smaller resonance 
at $\sim 1.3$~T, it might be a true $\mu$ALCR, 
presumably with $X =$ a Re nuclear moment.   

Presumably the $H$ dependence of the asymmetry in Ag 
is due entirely to systematics and represents 
a calibration for full asymmetry, except for a possible 
offset due to slight geometrical differences.  
It is therefore unsurprising that the FeGa$_3$ asymmetry 
has the same shape at high LF (above about 1.6~T) 
except for a geometrical offset.  
At lower fields, however, considerable asymmetry is lost 
in FeGa$_3$ (relative to Ag), in a monotonic drop of 
about 0.066, nearly half of a typical full asymmetry 
in such LF measurements.  
This is consistent with Mu decoupling 
in a semiconductor with nearly 100\% Mu formation 
and a nearly vacuum-like Mu HF coupling.  
The additional drop-off as $H \to 0$ 
is presumably due to decoupling from local 
nuclear dipolar fields.  
The HTF-$\mu^+$SR splittings observed for FeGa$_3$ 
in Fig.~\ref{fig:FFTs} are certainly not consistent 
with a HF coupling of close to $A_0 = 4463$~MHz 
for a majority of muons forming vacuum-like Mu atoms.  

\section{Conclusions}

Our search for rich $\mu$ALCR structure 
in these SP candidate materials 
has been generally unsuccessful.  
The NHF couplings between the SP electron and 
neighboring nuclei may be either too weak or 
too anisotropic to engender resonances.  
There might be extremely {\sl narrow\/} resonances 
that would require scanning the appropriate 
field range with much smaller step sizes.  
(We used steps of 4, 10, 50 and 100 G in low, 
medium and high field regions.) 
Such a survey would benefit enormously from 
a good guess of where such resonances 
might be expected to appear.

\section*{References}

%% {\raggedright
%% \bibliographystyle{iopart-num} 
%% \bibliography{LIT} 
%% \end{document}
%% }

\begin{thebibliography}{9}

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\bibitem{Storchak2009L1}
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Magnetic polaron bound to the positive muon in SmS: 
Exchange-drive formation of a mixed-valence state 
 {\it Phys. Rev.} B {\bf 79} 193205-193208 

\bibitem{Storchak2012A1}
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Thermal destruction of spin-polaron bands in
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\bibitem{Storchak2010L1}
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Spin Polarons in the Correlated Metallic Pyrochlore Cd$_2$Re$_2$O$_7$ 
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\end{thebibliography}

\end{document}
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