Anisotropic manifestation of short-range magnetic correlations in Ce$_{0.6}$La$_{0.4}$RhIn$_5$
V. F. Correa
We report thermal-expansion and magnetostriction results on Ce$_{0.6}$La$_{0.4}$RhIn$_5$ single crystals. This particular La concentration ($x_c \sim$ 0.4) corresponds to the critical one at which the long-range magnetic order vanishes ($T_N =$ 0). However, as also observed in specific heat and magnetic susceptibility measurements, a large ``hump'' is seen in thermal-expansion at $T_{SR} \sim$ 4.5 K. This anomaly, claimed to be related to short-range correlations, is observed only along the c-axis, confirming that anisotropy plays an important role in these 115 compounds.No particular feature is associated with the magnetic correlations in the magnetostriction measurements. The magnetic field dependence of the volume is quadratic in field as expected for a paramagnetic system, above and below $T_{SR}$. Finally, the magnetic field dependence of the crystal electric field contribution to the thermal-expansion seems to reinforce the idea that the La doping leads to a $| \pm 5/2 \rangle$-rich ground state doublet.
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\title{Anisotropic manifestation of short-range magnetic correlations in 
Ce$_{0.6}$La$_{0.4}$RhIn$_5$}

\author{V. F. Correa}
\affiliation{Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA}

\author{L. Tung}
\affiliation{Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA}

\author{S. M. Hollen}
\affiliation{Occidental College, Los Angeles, California 90041, USA}

\author{P. G. Pagliuso}
\affiliation{Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA}

\author{N. O. Moreno}
\affiliation{Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA}

\author{J. C. Lashley}
\affiliation{Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA}

\author{J. L. Sarrao}
\affiliation{Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA}

\author{A. H. Lacerda}
\affiliation{Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA}

\date{\today}

\pacs{71.27.+a, 65.40.De, 75.80.+q, 75.40.-s}


\begin{abstract}

We report thermal-expansion and magnetostriction results on 
Ce$_{0.6}$La$_{0.4}$RhIn$_5$ single crystals. This particular La concentration 
($x_c \sim$ 0.4) corresponds to the critical one at which the long-range 
magnetic order vanishes ($T_N =$ 0).
However, as also observed in specific heat and magnetic susceptibility measurements, 
a large ``\emph{hump}'' is seen in thermal-expansion at $T_{SR} \sim$ 4.5 K. This 
anomaly, claimed to be related to short-range correlations, is observed only along 
the c-axis, confirming that anisotropy plays an important role in these 115 
compounds.No particular feature is associated with the magnetic correlations 
in the magnetostriction measurements. The magnetic field dependence of 
the volume is quadratic in field as expected for a paramagnetic system, above and 
below $T_{SR}$.
Finally, the magnetic field dependence of the crystal electric field 
contribution to the thermal-expansion seems to reinforce the idea that the La 
doping leads to a $| \pm 5/2 \rangle$-rich ground state doublet.
    

\end{abstract}


\maketitle


\section{introduction}

The physics of heavy fermions (HF) is dominated by the competition between 
different microscopic mechanisms (and related energy scales) and this results in 
a wide spectrum of different ground states including unconventional superconductivity 
and magnetic order.\cite{Grewe}
The magnetic properties (resulting from the interaction between the f-localized 
magnetic moments and the free electron spins) are determined by the 
competition between the long range RKKY interaction and the short range 
Kondo effect, both of which depend on the magnetic exchange parameter \textit{J}. 
This gives rise to magnetic order or a non magnetic Kondo singlet state, respectively.
Moreover, changing macroscopic variables such as pressure, magnetic field 
or doping can lead to significant changes in the microscopic parameter \textit{J}
producing new ground states.\cite{Hewson}

These features are exemplified in the recently discovered CeMIn$_5$ (M = Co, Rh, Ir) 
family.\cite{Hegger,Petrovic1,Petrovic2} These compounds 
crystallize in the tetragonal HoCoGa$_5$-like structure consisting of alternating 
layers of magnetic CeIn$_3$ and non-magnetic MIn$_2$ along the c-axis.
Long-range antiferromagnetism as well as hybridization and mass enhancement due 
to the Kondo effect have been observed in the 115 family.
\cite{Hegger}
Also, magnetically mediated superconductivity and non-Fermi liquid (NFL) 
behavior have been reported.\cite{Hegger,Petrovic2,Zapf,Light} 

In particular, CeRhIn$_5$ shows ambient pressure antiferromagnetism 
\cite{Hegger} ($T_N$ = 3.8 K) and pressure induced superconductivity 
\cite{Hegger,Fisher} ($T_c$ = 2.1 K at P = 16 kbar). 
Neutron diffraction experiments \cite{Bao1} reveal a reduced magnetic moment 
associated with the Ce ions (0.75$\mu_B$ compared to 
0.92$\mu_B$ expected for the ground state doublet) suggesting an 
important Kondo compensation. 
This is corroborated by specific heat measurements \cite{Hegger} that 
show only 30\% of $R\ln2$ entropy released up to $T_N$, as well as by the 
logarithmic temperature dependence of the resistivity, \cite{Christianson1} 
characteristic of Kondo lattice compounds.

The magnetic ordered state consists of an anisotropic spin density wave, that 
is inconmensurate with the lattice \cite{Bao1,Curro} and opens a gap 
of the order of 8 K in the Fermi surface, below $T_N$.\cite{Curro,Cornelius1,Cornelius2}
Nuclear quadropolar resonance \cite{Curro} and neutron diffraction \cite{Bao1}  
experiments indicate that the Ce magnetic moments, antiferromagnetically ordered,  
lie completely in the basal plane devoloping a helicoidal structure along 
the c-axis. Specific heat measurements \cite{Cornelius1,Cornelius2} 
show an abrupt drop of the Sommefeld coefficient $\gamma$ below $T_N$ confirming 
the gapping of the Fermi surface. 

The layered structure suggests an important 2D character supported by the spiral 
magnetic structure: magnetic order in the planes, weakly coupled between them. 
However, transport \cite{Christianson1} and susceptibility \cite{Takeuchi} 
measurements show very little anisotropy. Inelastic neutron scattering \cite{Bao2}, 
as well, indicates that three dimensional magnetic fluctuations are relevant. 
de Haas van Alphen measurements reveal a multi-band Fermi surface that might
explain the varying role of the anisotropy.\cite{Hall}

The ground state of CeRhIn$_5$ can also be tuned by chemical sustitution. 
\emph{Out-of-plane} doping in CeRh$_{1-y}$Ir$_y$In$_5$ reveals an interesting 
coexistance of antiferromagnetism and susperconductivity in the range 
0.25 $ < x <$ 0.60. \cite{Pagliuso1} \emph{In-plane} doping in Ce$_{1-x}$La$_x$RhIn$_5$ 
also exhibits interesting features. 
The inconmensurate SDW is seen up to $x \approx$ 0.4, where $T_N$ goes 
to zero.\cite{Pagliuso2} NFL behavior is observed around this critical concentration, 
perhaps related to the development of an expected quantum critical point.\cite{Light} 
A ``\emph{hump}'' also appears around $T_{SR} =$ 4 K in specific heat and magnetic 
susceptibility measurements that are claimed to be associated with short-range 
magnetic correlations, \cite{Light,Pagliuso2,Kim} as inferred also by NQR and NMR
experiments. \cite{Iwamoto}  

In this work, we present thermal-expansion and magnetostriction results on 
Ce$_{0.6}$La$_{0.4}$RhIn$_5$ single crystals. As in the other free energy second 
derivatives techniques, namely specific heat and magnetic susceptibility, 
the short-range correlations are unambiguosly detected by thermal expansion. 
A large \emph{peak} is observed at $T_{SR} \sim$ 4.5 K but, interestingly, this feature 
shows up only along the c-axis, confirming that the dimensionality plays a relevant role 
in these 115 compounds.
As in the pure compound, crystal electric field (CEF) effects are important. The 
magnetic field dependence of the CEF contribution to the thermal-expansion seems to 
reinforce the idea of a $| \pm 5/2 \rangle$-rich ground state doublet.  
On the other hand, magnetostriction measurements do not show any particular behavior 
associated with magnetic correlations. The field dependence of the volume is quadratic 
in field as expected for a paramagnetic system, above and below $T_{SR}$.  
 

\section{Results}

Ce$_{0.6}$La$_{0.4}$RhIn$_5$ single crystals were grown by the self flux technique.
For thermal-expansion and magnetostriction measurements we use a 2.6$\times$3.6$\times$7 
mm$^3$ high quality sample (as evidenced by the observation of quantum oscillations up 
to 30 K\cite{Correa}). 
The thermal-expansion experiments were performed using a capacitance dilatometer.

\begin{figure}[b]
\includegraphics[width=\columnwidth]{fig1.EPS}
\caption[]{Linear thermal-expansion vs temperature along the c-axis ($\alpha_c$) 
and perpendicular to it ($\alpha_{ab}$). Inset: Volume thermal-expansion and the 
estimated lattice contribution ($\Theta_D$ = 245 K).}
\label{fig1}
\end{figure}

Fig. \ref{fig1} displays the linear thermal-expansion coefficient $\alpha = 1/L(dL/dT)$ 
along the c-axis of the crystal and perpendicularly to it (i.e., along the ab-basal plane), 
$\alpha_c$ and $\alpha_{ab}$ respectively. 
The peak observed at low temperatures in $\alpha_c$ is a magnetic effect while the negative 
contribution around 25 K is associated with crystal electric field (CEF) effects, as reported 
by Takeuchi \textit{et. al.} \cite{Takeuchi} in the pure CeRhIn$_5$ and by Malinowski 
\textit{et. al.} \cite{Malinowski} in the related Ce$_2$RhIn$_8$.

The volume thermal-expansion $\beta = 1/V(dV/dT)$, which for a tegragonal symmetry can be 
calculated as $\beta = \alpha_c + 2 \alpha_{ab}$, is shown in the inset of Fig. 
\ref{fig1}. Also shown in this inset is the Debye curve corresponding to $\theta_D =$ 
245 K, claimed to be the Debye temperature for the non-doped CeRhIn$_5$ as inferred from 
specific heat measurements in the non-magnetic analog LaRhIn$_5$. \cite{Hegger} 
The lattice contribution fits quite well the experimental data above 15 K due to a partial 
cancellation of the CEF contributions that is negative along the c-axis and positive in the 
basal plane as shown below.


\section{Discussion}

The deviation of the total thermal-expansion from the lattice and the CEF calculated 
contributions becomes strong below 15 K, as reported also by Takeuchi \textit{et al.} 
\cite{Takeuchi} in CeRhIn$_5$ and Malinowski \textit{et al.} \cite{Malinowski} in 
Ce$_2$RhIn$_8$. This temperature marks the onset of the magnetic correlations.
Fig. \ref{fig2} shows the low temperature behavior of both $\alpha_c$ and $\alpha_{ab}$. 
A peak is observed in $\alpha_c$ at $T_{SR} \sim $ 4.5 K, but only a very tiny 
``\emph{bump}'' is seen in $\alpha_{ab}$ associated with this feature.
The antiferromagnetic transition vanishes at $x =$ 0.4.\cite{Pagliuso2}
Thus, this peak may be related to short-range magnetic correlations as was observed 
in specific heat and susceptibility experiments in $x =$ 0.5 single crystals.
\cite{Light,Pagliuso2,Kim} 
The inset of Fig. \ref{fig2} shows the temperature dependence of the electronic 
contribution to the specific heat ($C^e$) for a $x =$ 0.4 sample, which is quite similar 
to that reported for $x =$ 0.5. \cite{Light} The subtracted lattice contribution was 
inferred from an estimate of the Debye tempererature of the non-magnetic analog LaRhIn$_5$.
The dashed line is an estimate of the contribution from crystal electric field effects 
using the CEF scheme obtained through 
neutron scattering experiments \cite{Christianson2}, with 80 K and 274 K the first and 
second excited energy levels, respectively. 
The important conclusion to be drawn from the thermal-expansion data is the anisotropic 
manifestation of the short-range magnetic correlations, being observed only along the 
c-axis.

This anisotropic behavior is an indication of the anisotropic nature of the interaction 
between the spins and the lattice, that is, of the spin-orbit coupling. A full 
understanding of the magnetoelastic tensor would provide a comprehensive explanation of 
the thermal-expansion results.
Although theoretically this is a very difficult enterprise, from magnetostriction 
experiments we can obtain information about the magnetoelastic tensor in some special 
directions, as well as further information regarding the magnetic correlations. 

\begin{figure}[t]
\includegraphics[width=\columnwidth]{fig2.EPS}
\caption[]{Low temperature linear thermal-expansion showing the short-range 
correlations associated peak. Inset: Electronic specific heat vs. temperature 
showing also a feature aroung 4.5 K (symbols), and the calculated CEF contribution 
using the same scheme as in Fig. \ref{fig2} (dashed line).}
\label{fig2}
\end{figure}

Linear magnetostriction data (for fields applied in the basal plane) are shown in Fig. 
\ref{fig3} for two different temperatures: one below and the other slightly above 
$T_{SR}$, 1.8 K and 5 K. The field is applied parallel to the sample dimension being tested.
As inferred from the inset, the field dependence of $\Delta L_{ab} / L_{ab}$ is quadratic 
below and above $T_{SR}$, as expected for a paramagnetic system.\cite{Malinowski,Zieglowski} 
This fact strengthens the conclusion that features observed at $T_{SR}$ in 
thermal-expansion and specific heat measurements are due to short-range correlations.
If long-range order were present, $\Delta L / L$ would have different field dependencies 
above and below $T_{SR}$. 
The temperature dependence of $\Delta L_{ab} / L_{ab}$ is very small in this low temperarure 
region (in any field configuration). Moreover, the field dependence of $\Delta L_{c} / L_{c}$
(not shown here) increases with temperature. This behavior was already reported by Malinowski 
\textit{et al.} \cite{Malinowski} in the related Ce$_2$RhIn$_8$, and was ascribed to the 
magnetic correlations.          
The same quadratic behavior is observed along the c-axis and in any field direction. However, 
the field depedence of $\Delta L_{c} / L_{c}$ (with $B$ along the basal plane) is one order 
of magnitude larger than the result observed in Fig. \ref{fig4}.

These observations offer a natural explanation for the anisotropic manifestation of the 
magnetic correlations observed in $\alpha$. Using a 2D anisotropic Ising model, 
Light \textit{et al.} \cite{Light} suggested that the disorder 
introduced by La doping cancels out the interplane magnetic correlations. At $x \sim$ 0.5 
only short range in-plane correlations remain, and these become evident around $T_{SR}$.
As the Ce magnetic moments lie in the plane, one can naively expect that the local magnetic 
field associated with these moments is mainly directed in the basal plane.
Our magnetostriction results tell us that in this case (field in the plane) the 
magnetovolume effects are much larger in the c-axis than in the plane.
Similarly, thermal-expansion measurements in pure CeRhIn$_5$ \cite{Takeuchi} reveal  
that the peak associated with $T_N$ is larger in $\alpha_c$ than in $\alpha_{ab}$. 

\begin{figure}[t]
\includegraphics[width=\columnwidth]{fig3.EPS}
\caption[]{In-plane linear magnetostriction for fields applied parallel to the dimension 
that is being tested, at two different temperatures. Inset: magnetostriction as a 
function of $B^2$.}
\label{fig3}
\end{figure}

\begin{figure}[t]
\includegraphics[width=\columnwidth]{fig4.EPS}
\caption[]{Electronic contribution to the thermal-expansion (symbols). Solid line:
CEF fit using parameters from neutron scattering measurements (see text). Dashed 
line: Idem but with mixing parameter $\eta$ = 0. Upper panel: $\alpha_c$. Lower 
panel: $\alpha_{ab}$.}
\label{fig4}
\end{figure}

Finally, we consider in greater detail the contribution of crystal electric field spliting 
to thermal-expansion. Of particular interest is the extent to which the CEF spliting 
evolves with $x$ in Ce$_{1-x}$La$_{x}$RhIn$_5$. The Ce$^{3+}$ J = 5/2 
multiplet splits in three doublets in the presence of tetragonal point symmetry: \cite{Ohama}


\begin{equation} \label{CEF}
	\begin{array}{lll}
	\Gamma_7^{(2)} & = & \sqrt{(1 - \eta^2)} \,\, | \pm 5/2 \rangle \, - \, \eta \,\, | \mp 3/2 \rangle, \\
	\Gamma_7^{(1)} & = & \eta \,\, | \pm 5/2 \rangle \, + \, \sqrt{(1 - \eta^2)} \,\, | \mp 3/2 \rangle, \\
	\Gamma_6 & = & | \pm 1/2 \rangle.
	\end{array} 
\end{equation}


\noindent Neutron scattering results \cite{Christianson2} on pure CeRhIn$_5$ reveal that 
$\Gamma_7^{(2)}$ is the ground state doublet, while $\Gamma_7^{(1)}$ at $\Delta_1$ = 80 K 
and $\Gamma_6$ at $\Delta_2$ = 274 K are the first and second excited states, respectively.
A value of 0.6 is obtained for the mixing paremeter $\eta$.

Within the formalism of Morin \textit{et al.} \cite{Morin} that takes into account CEF, 
quadrupolar and magnetoelastic effects, the contribution to linear thermal-expansion for a 
tetragonal solid can be estimated as


\begin{equation} \label{alfa}
	\begin{array}{lll}
	\alpha_c & = & A \, \frac {\displaystyle d \langle O_2^0 \rangle} {\displaystyle dT}, \\
	\\
	\alpha_{ab} & = & B \, \frac {\displaystyle d \langle O_2^0 \rangle} {\displaystyle dT},
	\end{array}
\end{equation}

\noindent where $A$ and $B$ are temperature independent functions of the elastic constants 
and magnetoelastic coefficients and $O_2^0$ is the relevant Stevens equivalent operator. 
The thermodynamic expectation value of this operator $\langle O_2^0 \rangle$ can be 
calculated through (\ref{CEF}).

\begin{figure}[b]
\includegraphics[width=\columnwidth]{fig5.EPS}
\caption[]{Linear thermal-expansion $\alpha_c$ for different magnetic fields applied along 
the c-axis.}
\label{fig5}
\end{figure}

In Fig. \ref{fig4} we show the electronic linear thermal-expansion along the c-axis 
($\alpha_c^e$, upper panel) and perpendicular to it ($\alpha_{ab}^e$, lower panel) after 
subtracting the lattice contribution, estimated from the $\theta_D$ = 245 K Debye curve.
The solid lines are the CEF contributions derived from Eq. (\ref{alfa}), using the values 
for $\eta$, $\Delta_1$ and $\Delta_2$ obtained from neutron 
scattering experiments in the pure compound.\cite{Christianson2} The only free parameters 
are the $T$-independent coefficients $A$ and $B$, that are set such that the maxima coincide. 
The detailed agreement is rather poor, especially for $\alpha_c^e$, although the overall 
shape is correct. 

In Fig. \ref{fig5} we show $\alpha_c$ as a function of temperature for different 
magnetic fields applied along the c-axis. Experimentally, 
the minimum in $\alpha_c$ associated with CEF effects clearly shifts to 
higher T with increasing field. The magnetic field splits the three doublets and, according 
to our CEF scheme, mixes together $\Gamma_7^{(2)}$ and $\Gamma_7^{(1)}$. So, the full 
diagonalization of the complete Hamiltonian (CEF plus Zeeman) results in six singlet states. 
An estimate of $\alpha_c^{CEF}$(18 T) according to this new scheme can be observed in 
the upper panel of Fig. \ref{fig6}. In the same figure $\alpha_c^{CEF}$(0) is shown for 
comparison.
In our calculation the absolute value of the c-axis thermal-expansion has a large increase 
and the minimum moves down from $T \sim$ 42 K to 12 K when applying 18 T (parallel  
to the c-axis).
This huge change comes from the Zeeman splitting ($\sim$ 40 K for $B =$ 18 T) of the ground 
state doublet that must be compared with the original $\Delta_1$ = 80 K separation to the 
first excited state in zero field. As a result, the first excited state 
starts to be populated at much lower T when a field is applied and, thus, the 
associated peak occurs at lower T as well. This predicted behavior is in direct 
contradiction with what we observe experimentally. 

\begin{figure}[t]
\includegraphics[width=\columnwidth]{fig6.EPS}
\caption[]{Calculation of the CEF contribution to the c-axis thermal-expansion for 
zero field and 18 T. Upper panel: using a mixing parameter $\eta$ = 0.6. Lower 
panel: $\eta$ = 0.}
\label{fig6}
\end{figure}


According to Eq. (\ref{alfa}), there would be no maxima in thermal-expansion associated 
with the population of the first excited singlet if the expectation value of $O_2^0$ is 
the same in both the ground and the first excited singlet. This occurs when the mixing 
parameter $\eta$ vanishes.
The lower panel of Fig. \ref{fig6} shows the calculated $\alpha_c^{CEF}$(0) and  
$\alpha_c^{CEF}$(18 T) for $\eta$ = 0. The absolute value of the minimum slightly decreases 
and moves weakly toward higher T. This behavior is much more consistent with our 
experimental observation. 
In the case that $\eta =$ 0 , the ground state doublet $\Gamma_7^{(2)}$ is now a pure 
$| \pm 5/2 \rangle$ state while the first excited $\Gamma_7^{(1)}$ is a pure 
$| \pm 3/2 \rangle$. Using this approach, we estimate the tempearature depedent 
thermal-expansion. 
This result is shown as the dashed curves in Fig. \ref{fig4} using the same $\Delta_1$ 
and $\Delta_2$ as for the solid lines. At least along the c-axis, the agreement with the 
experimental results is much better.
On the other hand, if $\eta =$ 0 in undoped CeRhIn$_5$, then the peak observed in 
inelastic neutron scattering experiments \cite{Christianson2} at 23 meV would be 
forbidden by symmetry.        
Thus, from these observations we hypothesize that substituting La for Ce leads to a 
decrease in the mixing parameter $\eta$, due presumibly to chemical pressure effects 
and changes in hybridization strength. However, 
as confirmed by the good agreement between theory and experiment in the specific heat 
(inset of fig. \ref{fig2}), La doping has no major effect on the level spliting.


\section{Conclusions}

Thermal expansion measurements were performed in Ce$_{0.6}$La$_{0.4}$RhIn$_5$ single 
crystals. Short-range magnetic correlations are clearly detected as a 
``\emph{hump}'' at $T_{SR} \sim$ 4.5 K, as observed also in specific heat and magnetic 
susceptibility experiments. 
This feature is observed only along the c-axis confirming 
the anisotropic nature of this compound.
As expected for a paramagnetic system, magnetostriction experiments show a quadratic 
field dependence of the volume above and below $T_{SR}$ indicating that this 
magnetic-related anomaly can not be associated with a long-range order. 
Finally, La doping appears to increase the $| \pm 5/2 \rangle$ character of the crystal 
electric field ground state doublet in Ce$_{1-x}$La$_{x}$RhIn$_5$.


\section{Acknowledgments}

We gratefully acknowledge C. D. Batista for helpful discussions and F. Drymiotis for 
his collaboration in the specific heat experiments. Work at the NHMFL was performed 
under the auspices of the National Science Foundation, the State of Florida and the 
US Department of Energy.
  

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