# Does a low-carrier density ferromagnet hold the key to understanding high
temperature superconductors?

Gabrielle Beaudin  Département de Physique, Université de Montréal, Montréal,
Canada  Regroupement Québécois sur les Matériaux de Pointe (RQMP)  Alexandre
Désilets-Benoit  Département de Physique, Université de Montréal, Montréal,
Canada  Regroupement Québécois sur les Matériaux de Pointe (RQMP)  Andrea
Daniele Bianchi<EMAIL_ADDRESS>Département de Physique,
Université de Montréal, Montréal, Canada  Regroupement Québécois sur les
Matériaux de Pointe (RQMP)  Robert Arnold  School of Metallurgy and Materials,
University of Birmingham, Elm Rd, Birmingham, B15 2TT, United Kingdom  Stavros
Samothrakitis  Laboratory for Neutron Scattering and Imaging, Paul Scherrer
Institut, Forschungsstrasse 111, Villigen, CH-5232, Switzerland  Kilian D.
Stenning  Blackett Laboratory, Imperial College London, Prince Consort Road,
London, SW7 2AZ, United Kingdom  Mark Laver  Department of Physics, University
of Warwick, Coventry, CV4 7AL, United Kingdom  Simon Gerber  Laboratory for
X-ray Nanoscience and Technologies, Paul Scherrer Institut, Forschungsstrasse
111, Villigen, CH-5232, Switzerland  Nikola Anna Galvan Leos  Laboratory for
Neutron Scattering and Imaging, Paul Scherrer Institut, Forschungsstrasse 111,
Villigen, CH-5232, Switzerland  Jorge L. Gavilano  Laboratory for Neutron
Scattering and Imaging, Paul Scherrer Institut, Forschungsstrasse 111,
Villigen, CH-5232, Switzerland  Michel Kenzelmann  Laboratory for Neutron
Scattering and Imaging, Paul Scherrer Institut, Forschungsstrasse 111,
Villigen, CH-5232, Switzerland  Michael Nicklas  Physics of Quantum Materials,
Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Strasse 40,
Dresden, 01187, Germany  Robert Cubitt  Large Scale Structures Group, Institut
Laue Langevin, 71 avenue des Martyrs, Grenoble, 38042, France  Charles
Dewhurst  Large Scale Structures Group, Institut Laue Langevin, 71 avenue des
Martyrs, Grenoble, 38042, France

###### Abstract

We conducted a small-angle neutron scattering experiments (SANS) on the
ferromagnetic semi-metal EuB6, where we observed direct evidence for the
presence of magnetic polarons. We carried out SANS experiments over a large
range of scattering vectors $\absolutevalue{\overrightarrow{\mathbf{q}}}$ from
0.006 to 0.140 Å-1 and 2 to 60 K. Just above $T_{\mathrm{C}}$ our experiments
show magnetic scattering intensity, which has a Lorentzian dependence on the
wave vector, which is characteristic for the presence of magnetic polarons.
Below $T_{\mathrm{C}}$ the polarons merge and most of the observed intensity
is due to scattering from domain walls. We were able to extract a correlation
length $\xi$ which ranges from $100$ to $300$ Å for the size of the magnetic
polarons. This size is much larger than one would expect for magnetic
fluctuations, demonstrating the influence the magnetic polarons on the phase
transition.

###### pacs:

71.27+a,75.30Kz,71.10.Hf,25.40.Dn,75.50Pp

††preprint: APS/123-QED

While we have a firm understanding of the physics of a high density electron
gas, we have only begun to understand the physics at low carrier densities [1,
2]. This is particularly true for the case, where the charge carriers are
added to a background of localized magnetic moments. Contrary to intuition,
the charge carriers will interact strongly, with the localized moments and
form so-called magnetic polarons. Magnetic polarons can exist below
$T_{\mathrm{N}}$ in antiferromagnets and in the paramagnetic phase of
ferromagnets, as long as the carrier density is lower than the size of a
magnetically correlated volume [3, 4, 5]. However, until now, they have not
been directly observed. Their presence is expected in high temperature
superconductors for example, where the parent compound is a strongly
correlated antiferromagnetic Mott insulator, which then is typically doped
with holes. These holes are mobile impurities which are moving in a background
of strongly fluctuating magnetic moments. Due to strong interactions, they
become dressed with a magnetic cloud, the magnetic polaron [6, 7, 8].
Furthermore, the distribution of these holes does not remain uniform, but they
attract each other, leading to an electronic phase separation [6]. This
competition lies at the heart of a doping-dependent transition from an
anomalous metal to a conventional Fermi liquid, and was recently observed in a
quantum simulator of the Fermi-Hubbard model [9]. It is this strong
interaction between the motion of the holes and antiferromagnetism that is
believed to be at the heart of understanding the superconductivity in the
cuprates [8].

Magnetic polarons were first proposed in the late sixties to describe the
colossal magnetoresistive (CMR) effects in the Europium chalcogenides. Here,
the magnetic polarons are thought to be formed by the spin polarization of the
mobile charge carriers by localized $4f$ moments. This leads to the formation
of magnetic “bubbles” and the localization of charge carriers within them [10,
11, 12, 3, 4]. It is this CMR effect that could lead to new spintronic
transistors [13].

Here we present a small-angle neutrons scattering (SANS) study in
ferromagnetic EuB6, which we will argue is the ideal model system for studying
magnetic polarons. Our results give for the first time direct evidence for
magnetic polarons, and show their out-sized enhancement of the ferromagnetic
spin fluctuations driving the electronic phase transition.

EuB6 has a simple cubic crystal structure $(Pm\bar{3}m)$ but displays a
complex interplay between the electronic and magnetic degrees of freedom due
to its low carrier density [4, 14, 5]. The insulator-to-semi-metal transition
in EuB6 is concomitant with a ferromagnetic phase transition [15]. A number of
experiments have given indirect evidence for the presence of magnetic polarons
in EuB6 [16, 17, 18, 19, 20]. A scanning tunnelling microscopy (STM) study
showed that EuB6 becomes electronically inhomogeneous for temperatures above
$T_{\mathrm{C}}$. Here, at 20 K the size of the inhomogeneities are of the
order of 3 to 4 nm [21]. At the same time, measurements with a micro Hall-bar
pointed to magnetic inhomogeneities at these temperatures [21], which are
pinned to defects at the surface [22]. This is the very electronic phase
separation expected for magnetic polarons.

Electronic inhomogeneity in EuB6 was also observed in an angle resolved
magnetoresistance experiment [23]. This concurrence of magnetic polarons and
electronic inhomogeneity, as seen here in EuB6 also manifests itself in the
high-$T_{c}$’s. Here, a quantum nematic was first theoretically predicted for
the doped two-dimensional Mott insulator [24], and was later observed [25, 26,
27, 28, 29]. In the high-$T_{\mathrm{C}}$’s the relation between nematic order
and superconductivity, and its relation to a close-by structural instability
are hotly debated. Such a coupling of the quantum nematic to the lattice is
absent in EuB6 [30], which makes EuB6 a clean experimental platform to study
magnetic polarons.

EuB6 is a magnetic semiconductor, which exhibits two phase transitions [31].
Upon cooling from an insulating state at high temperatures, it first becomes a
semi-metal, indicated by a drop in resistivity at $T_{M}$ of 14.5 K (see Fig.
3 of the Supplemental Material). At the Curie temperature $T_{\mathrm{C}}$ of
11.8 K, it orders ferromagnetically [15]. EuB6 displays CMR behaviour near
$T_{\mathrm{C}}$ [17]. Also, EuB6 has a very low carrier density [32] of
$\approx 10^{25}$ m-3 at 20 K, which coexists with localized pure spin $4f$ Eu
moments $(S=7/2)$. This puts EuB6 into the regime where magnetic polarons are
expected to strongly affect the electrical conductivity [3, 4, 5]. This
scenario, is supported by a number of experiments [18, 33, 19, 20].

In our SANS experiment (see Fig. 1 of the Supplemental Material for a
schematic), we probed the magnetic response of EuB6 for three different ranges
of scattering vectors $q$. We are able to distinguish three different sources
of scattering in our SANS experiments. Firstly, diffuse scattering which grows
in size with decreasing temperature, which is originating from magnetic
polarons. Secondly, we observe an incommensurate magnetic peak which appears
below the Curie temperature $T_{\mathrm{C}}$. Thirdly, below $T_{\mathrm{C}}$
we observe a second diffuse scattering signal from ferromagnetic domain walls.
An overview is presented in Fig. 1.

Figure 1:  a) The solid black diamonds (empty red diamond) shows
$I\quantity(\absolutevalue{\overrightarrow{\mathbf{q}}})$ of the
incommensurate magnetic peak at 0 T (0.1 T). The solid line is a fit to a
power law. The solid blue circles are the data from the high $q$-regime (HQ)
in ZF averaged over the $q$-range from 0.050 to 0.140 Å-1. The open red
circles are the average over the low $q$ regime (LQ) in zero field from 0.006
to 0.025 Å-1. b) The medium $q$-regime is from 0.020 to $0.055$ Å-1(MQ). The
full black circles show the ZF data, the open blue circles are taken in 250 mT
and the full red squares in 500 mT. The inset shows the anisotropy of the SANS
signal when $H$ is applied perpendicular to the neutron beam.

We start our discussion with the incommensurate magnetic peak which appears
below $T_{\mathrm{C}}$ as a spot on the detector. At 1.5 K, its position
corresponds to a $\absolutevalue{\overrightarrow{\mathbf{q}}}$ of $(18.4\pm
0.2)\times 10^{-3}$ Å-1. This
$\absolutevalue{\overrightarrow{\mathbf{q}}}$-value is at wavelengths above
those required for double scattering and thus the peak is due to an
incommensurate spin density wave (ISDW). An ISDW is expected for EuB6, as a
group theoretical analysis using ISOTROPY [34] for the crystallographic space
group $Pm\overline{3}m$ of EuB6 with a $(100)$ propagation vector [15] yields
magnetic space groups with a symmetry below that of the crystal [17, 23]. The
period of this ISDW ranges from 680 Å to 726 Å and its intensity corresponds
to a magnetic moment of 2.5 $\mu_{\mathrm{B}}$/Eu at 2 K. An ISDW is
characteristic for strongly correlated systems, as diverse as Ca-doped Sr2RuO4
[35], a ruthenate, the La2CuO4 family [36] of high-$T_{c}$’s, as well as
BiFeO4 [37], a multiferroic. What is common to all these systems is that their
main order parameter is always found in strong competition with other
competing order parameters [36]. This suggests a similar scenario for EuB6,
where ferromagnetic order competes with an electronic phase transition. We
then proceeded to measure the temperature dependence of the intensity of this
peak, which is shown in Fig. 1a. As function of the temperature, the peak
appears at $T_{\mathrm{C}}$. The solid line is a fit to a power law, resulting
in a $T_{\mathrm{C}}$ of $11\pm$ $\beta=0.6\pm 0.1$ for 0 T and a
$\beta=0.3\pm 0.1$ at 0.5 T.

To study the temperature dependence of the diffuse magnetic signal, we
integrated the contributions from a large area of the detector (see Fig. 2 of
the Supplemental Material). For large momentum transfers, the
$\absolutevalue{\overrightarrow{\mathbf{q}}}$-range is sensitive to magnetic
fluctuations, which are enhanced by the magnetic polarons. Indeed, as shown in
Fig. 1a, we see an increase in the integrated signal with $q$ ranging from
0.050 to 0.140 Å-1 at 18 K, and a small bump at $T_{M}$ (see Fig. 3 of the
Supplemental Material). Below $T_{\mathrm{C}}$, the magnetic fluctuations are
reduced since the number of polarons decreases. Also, below $T_{\mathrm{C}}$,
most of the intensity is due to scattering from ferromagnetic domain
boundaries which is best probed through low scattering vectors. This is why
the signal remains constant below $T_{\mathrm{C}}$ for large momentum
transfers (Fig. 1a, HQ).

Figure 2:  The solid yellow circles show the temperature at which the averaged
intensity in the medium $q$-range starts to increase due to the formation of
magnetic polarons. Their position agrees well with the region delineated by
the the open black squares from the Hall effect [33], as well as the position
of the peak in weak non-linear resistance measurements (red triangles) [19].
The region of the $H-T$-diagram where symmetry breaking observed in AMRO is
shown as a color plot [23]. This region matches with the anomaly in
magnetostriction [20], and the position of $T_{M}$ identified in $\chi"$
measurements [23]. The phase line $T_{\mathrm{C}}$ separating the
ferromagnetic region was determined from $C_{p}$ and is shown as solid black
diamonds.

We also tracked the onset of the signal from the magnetic polarons with
magnetic fields. The data taken on the instrument D11 at the Institute Laue
Langevin and averaged over the medium $q$-regime (0.020 to $0.055$ Å-1) is
shown in Fig. 1b, where we can see a shift in the uptake of the intensity
towards higher temperatures: the bump marking $T_{M}$ is shifted from 16 to 22
K for a field of 0.25 T, and to 25 K for 0.5 T. This shift in temperature
matches the boundary of the phase diagram of magnetic polarons extracted from
noise spectroscopy measurements [19], and Hall effect [33], which confirms
that the signal is due to magnetic polarons. The phase diagram is shown in
Fig. 2. To complete the phase diagram, we used specific heat $C_{p}$
measurements in magnetic fields. As the peak in $C_{p}$ is strongly deformed
by the application of a magnetic field, we used a fit to a mean field model
for a ferromagnet to extract $T_{\mathrm{C}}$ [38] (see Fig. 3 of the
Supplemental Material). Critical exponents for the phase transition can be
extracted from magnetic susceptibility measurements [17, 39]. Our analysis
yields the exponents $\gamma$ of $1.03\pm 0.02$ and $\beta=0.49\pm 0.02$ (see
Fig. 4 of the Supplemental Material). These values are close to those of
expected for a mean field model ($\gamma=1$ and $\beta=0.5$). This justifies
the use of a mean field model to fit our $C_{p}$ data to extract
$T_{\mathrm{C}}\quantity(H)$, which is only weakly suppressed by the magnetic
field (Fig. 2). We would also like to point out that there is no
$\sin^{2}\theta$ dependence visible in the SANS signal in the field-applied
data (inset of Fig. 1b), showing that the signal we see in EuB6 is different
from that expected in the case of random anisotropic ferromagnets [40], or
from superparamagnetic nanoparticles [41]. This suggests that the magnetic
bubbles which we observe in EuB6 with SANS are not due to local static
inhomogeneities, but they are rather dynamic fluctuations, as expected for
magnetic polarons.

Figure 3: a) The upper panel shows
$I\quantity(\absolutevalue{\overrightarrow{\mathbf{q}}})$ data at 15 K. The
data is described by a Lorentzian function (red line). The dashed black line
is the contribution from 3D Heisenberg ferromagnetic fluctuations. b) At 5 K,
below $T_{\mathrm{C}}$, the red line is a fit to a sum of a $q^{-4}$ and a
Lorentzian, the dotted line is a guide to the eye showing the change in slope
from $q^{-2}$ (small dots) to $q^{-4}$ (dash) at $q=0.024$ Å-1.

Instead of integrating the intensity, we can inspect how the scattered
intensity varies as a function of
$\absolutevalue{\overrightarrow{\mathbf{q}}}$. This allows us to distinguish
the contributions of the magnetic polarons from those stemming from
ferromagnetic domains. It also allows us to extract the correlation length for
each contribution. For magnetic polarons, we expect an intensity which varies
as a Lorentzian [42]:

$I_{\mathrm{pol}}(q)=\frac{A}{q^{2}+\kappa_{p}^{2}}\qquad,$ (1)

where $\kappa_{p}$ is the inverse correlation length $\xi$. This is also the
$q$-dependence of magnetic fluctuations [40] close to a magnetic phase
transition. This dichotomy can be resolved, as $\xi$ from magnetic
fluctuations is only large for temperatures close to $T_{M}$, and it can be
calculated using the critical exponents determined from the scaling plots. In
EuB6 at 15 K and 0 T, the data over the entire $q$-range is described by a
Lorentzian shown as the full line in Fig. 3a.

Figure 4: $\kappa=1/\xi$ vs. $T$ of a 3D Heisenberg ferromagnet is shown as
the solid blue (Dashed lines are the 10% range around the estimated value).
The open circles show $\kappa(T)$ obtained from fitting a Lorentzian. The
inset shows $\xi$ vs. $T$. Again, the solid line is the calculation.

$\xi(T)$ has a drastically different size and temperature dependence than what
is expected for critical fluctuations. From the scaling of the magnetization,
and the field dependence of $C_{p}$, we know that the critical behavior of
classical spin fluctuations in EuB6 should be well described by a mean-field
theory [43]. Thus, we can compute $\xi(T)$, which is shown in the inset of
Fig. 4. However, the experimental $\kappa(T)=1/\xi\quantity(T)$ values are
much smaller, and have a different temperature dependence, than the estimate
from the mean field model. This demonstrates that this signal is not due to
classic ferromagnetic fluctuations, but that the magnetic fluctuations in EuB6
are strongly enhanced by magnetic polarons. Moreover, the magnetic polarons,
which drive the insulator to semi-metal transition, are also present at much
higher temperatures than classical fluctuations. It can be seen in the inset
of Fig. 4 that the size of the magnetic polarons starts to peak at $T_{M}$,
while the classical magnetic fluctuations diverge at $T_{\mathrm{C}}$. The
size of these compound objects is comparable to the size of the electronic
inhomogeneities observed by STM [21]. We can also compare the size of the
magnetic polarons to the volume occupied by a charge carrier. Using the charge
carrier density $n$ at 20 K of $n\approx 10^{25}$m-3 [32], we can estimate the
volume per charge carrier to be $\sim 700$ unit cells or 50 nm3, a size which
again falls in the same range, as observed here by SANS, or by STM [21].

Below $T_{\mathrm{C}}$, the distribution of the size of the static regions of
local spin ordering is expected to follow a Log-normal distribution [44]. This
behavior has two possible origins: scattering from ferromagnetic domain walls
[45] or scattering from static ferromagnetic clusters [46]. Here, the
$q$-dependence is expected to follow a Lorentzian-squared function [40]:

$I_{\mathrm{ferro}}(q)=\quantity(\frac{B}{q^{2}+\kappa_{F}^{2}})^{2}\qquad,$
(2)

where, $\kappa_{F}$ is the inverse of the correlation length of the
ferromagnetic domains [45]. Below $T_{\mathrm{C}}$, the characteristic length
is set by the ferromagnetic domains. For very large domains the signal is
proportional to $q^{-4}$.

Such a squared Lorentzian feature has been seen for example in CeNi1-xCux,
which is close to a ferromagnetic quantum critical point [46]. Here, it was
determined that the ferromagnetic order in CeNi1-xCux arrives through the
percolation of static ferromagnetic clusters. While this scenario bears
similarity to what we observe in EuB6, in CeNi1-xCux the clusters are static
and observed below the freezing point $T_{f}$ of a cluster glass transition,
while in EuB6 the magnetic polarons are dynamic and they are observed above
$T_{\mathrm{C}}$ while showing a Lorentzian. The Lorentzian squared behavior
we see in EuB6 below $T_{\mathrm{C}}$ is due to scattering by ferromagnetic
domain walls [45]. In EuB6, the ferromagnetic domains below $T_{\mathrm{C}}$
lead to a additional $q^{-4}$ contribution in addition to the $q^{-2}$
contribution from the dynamic magnetic polarons. This is shown in Fig. 3b. The
size of domains is larger than 100 Å, which is about the largest size we can
probe with our experiment.

In summary, we have showed evidence of magnetic polarons enhancing
ferromagnetic fluctuations in EuB6 using SANS for temperatures above the
ferromagnetic transition. Below $T_{\mathrm{C}}$, the polarons merge and most
of the observed intensity stems from scattering off domain walls In addition,
we determined that the polarons merge at the transition temperature
$T_{M}=14.5$ K resulting in a drastic drop of the electrical resistivity.
Applying a magnetic field shifts this percolation temperature to higher
temperatures. The correlation length we see above the ordering temperature
clearly indicates that the magnetic polarons strongly couple to the magnetic
environment leading to strongly enhanced magnetic fluctuations. Our results
put the existence of magnetic polarons in a ferromagnetic model system on a
firm footing changing the status of these quasiparticles from the hypothetical
to the real. Given the importance of the electronic phase separation
associated with magnetic polarons to spintronics and to high temperature
superconductivity we believe that our discovery of magnetic polarons which
will lead to exciting new physics.

###### Acknowledgements.

The research at UdeM received support from the Natural Sciences and
Engineering Research Council of Canada (Canada), Fonds Québécois de la
Recherche sur la Nature et les Technologies (Québec), and the Canada Research
Chair Foundation. This work is based on experiments performed on the
instruments SANS-I and SANS-II at the Swiss spallation neutron source SINQ,
Paul Scherrer Institute, Villigen, Switzerland and on instruments D11 and D33
at Institut Laue-Langevin, Grenoble, France. The work at the MPI-CPFS was
enabled through a DAAD grant.

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