# Mapping out the spin fluctuations in Co-doped LaFeAsO single crystals by NMR

Piotr Lepucki1, Raphael Havemann1, Adam P. Dioguardi1, Francesco Scaravaggi1,
Anja U. B. Wolter-Giraud1, Rhea Kappenberger1, Sai Aswartham1, Sabine
Wurmehl1,2, Bernd Büchner1,2 and Hans-Joachim Grafe1 1IFW Dresden, Institut
für Festkörperforschung, Helmholtzstraße 20, D-01069 Dresden, Germany
2Institut für Festkörper- und Materialphysik und Würzburg-Dresden Cluster of
Excellence ct.qmat, Technische Universität Dresden, D-01062 Dresden, Germany

###### Abstract

We determine the phase diagram of LaFe1-xCoxAsO single crystals by using
nuclear magnetic resonance (NMR). Up to a nominal doping of $x=0.03$, it
follows the phase diagram for F-doped polycrystals. Above $x=0.03$, the
F-doped samples become superconducting, whereas for Co-doping the structural
and magnetic transitions can be observed up to $x=0.042$, and
superconductivity occurs only for higher doping levels and with reduced
transition temperatures. For dopings up to $x=0.056$, we find evidence for
short-range magnetic order. By means of relaxation-rate measurements, we map
out the magnetic fluctuations that reveal the interplay of nematicity and
magnetism. Above the nematic ordering, the spin fluctuations in LaFe1-xCoxAsO
are identical to those in Ba(Fe1-xCox)2As2, suggesting that nematicity in
LaFeAsO is a result of the fluctuating spin density wave as well.

The interplay of nematicity and magnetism is one of the most discussed topics
in iron pnictide research. Two main directions have crystallized, (i) spin
fluctuations lead to a breaking of the tetragonal symmetry and induce the
nematic order below $T_{S}$ [1, 2, 3, 4, 5, 6, 7, 8, 9], and (ii) charge
density fluctuations on the Fe $d_{xz}$ and $d_{yz}$ orbitals increase with
decreasing temperature and lead to an unequal charge density distribution
below $T_{S}$, i.e. to an orbital order [10, 11, 12, 13, 14, 15]. One way to
shed light on this problem is to compare orbital and spin-fluctuations above
$T_{S}$, as well as orbital order below $T_{S}$ and spin density wave (SDW)
order below the magnetic transition temperature, $T_{N}$, in different
families of iron pnictides. However, the lack of doped single crystals of the
LaFeAsO family has prevented a full comparison with other iron pnictides or
chalcogenides until recently [15].

In this Letter, we present NMR results on LaFe1-xCoxAsO single crystals, and
compare our data to results published for Co-doped ${\rm Ba}{\rm Fe}_{2}{\rm
As}_{2}$, FeSe, and polycrystalline F-doped LaFeAsO. We find that the phase
diagram of LaFe1-xCoxAsO agrees well with that of F-doped samples up to a
nominal doping of $x=0.03$. Above this level, the F-doped samples become
superconducting, while the Co-doped samples still exhibit nematic and SDW
order up to $x=0.042$. Superconductivity is suppressed for these doping
levels, and sets in only at $x=0.056$. This sample shows remnants of nematic
order and local static magnetism at low temperatures, but no bulk coexistence
of superconductivity and magnetism. Finally, $x=0.06$ is the only
concentration with bulk superconductivity, albeit with a strongly reduced
$T_{c}$ compared to F-doping. Our results show that in-plane doping pins the
magnetism in LaFeAsO and suppresses superconductivity. Contrasting the spin
fluctuations measured by the spin-lattice relaxation rate, $(T_{1}T)^{-1}$, in
LaFe1-xCoxAsO with those of ${\rm Ba}({\rm Fe}_{1-x}{\rm Co}_{x})_{2}{\rm
As}_{2}$ shows that above $T_{S}$, $(T_{1}T)^{-1}$ is identical for samples
with the same $T_{S}$. This indicates that the leading instability for low
doping levels is towards a SDW in both compounds, in agreement with recent
theoretical predictions [8, 9].

Co-doped single crystals of LaFeAsO were prepared by solid state crystal
growth as described elsewhere [16]. The sample sizes were smaller than 1 x 1 x
0.1 mm, and the maximal weight about 0.3 mg. NMR has been measured on the 75As
nucleus for magnetic fields $H$ oriented along [100]ortho (or [010]ortho), and
along [001] (called $a$, $b$, and $c$). Due to the particular position of As
in the crystal structure, 75As NMR is an excellent probe of stripe-type spin
fluctuations in iron pnictides, which can be measured by the spin-lattice
relaxation rate and its anisotropy [17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28]. Furthermore, the NMR spectra split at the nematic ordering
temperature due to an anisotropy of the Knight shift and of the electric
quadrupole interaction, and can thus distinguish the $a$ and $b$ directions in
the orbital ordered state [29, 13, 23, 24, 14, 30, 31, 27].

Figure 1: (Color online) Upper row: $(T_{1}T)^{-1}$ versus temperature with
the magnetic field applied along the $ab$-plane (parent compound from [14]).
In the last panel, $(T_{1}T)^{-1}$ of $x=0.06$ F-doping is included for
comparison (dashed line indicates $T_{c}=21$ K) [28]. The scale for $x=0.056$
and $x=0.06$ is 10 times zoomed in compared to the magnetically ordered
samples. The green lines are Curie-Weiss fits [32]. Middle row: Stretching
parameter $\beta$ of the spin-lattice relaxation rate $T_{1}^{-1}$, open
squares: stretching of the spin-spin relaxation rate $T_{2}^{-1}$. Open
triangles: stretching of $T_{1}^{-1}$ of $x=0.06$ F-doping. Lower row: Ratio
of the spin-lattice relaxation rates, $R=T_{1c}/T_{1ab}$, the solid lines are
$R=1.5$. Blue shading indicates bulk antiferromagnetism, orange nematic order,
and yellow superconductivity.

The spin-lattice relaxation rate divided by temperature is shown in the upper
row of Fig. 1. $(T_{1}T)^{-1}$ increases at $T_{S}$ even more rapidly than a
Curie-Weiss fit ($(T_{1}T)^{-1}=a+C/(T-T_{N})$) to the data above $T_{S}$
[32], and reaches a maximum at $T_{N}$. This behavior is the same for all
samples that order long-range magnetically, i.e. up to $x=0.042$, and
indicates a critical slowing of stripe-type spin fluctuations towards the
magnetically ordered SDW phase. It is also found in magnetically ordered
F-doped polycrystals and in ${\rm Ba}({\rm Fe}_{1-x}{\rm Co}_{x})_{2}{\rm
As}_{2}$ single crystals [28, 20, 22, 33]. The maximum of $(T_{1}T)^{-1}$ has
been taken as $T_{N}$, and the temperature where $(T_{1}T)^{-1}$ increases
more rapidly than the Curie-Weiss fit defines $T_{S}$. These temperatures
agree well with bulk measurements [34, 15].

For $x=0.056$, $(T_{1}T)^{-1}$ is strongly reduced, and does not reach a
maximum anymore, i.e. long-range magnetic order is absent. Note that the scale
in Fig. 1 is zoomed in by a factor of 10 for $x=0.056$ and $0.06$. Critical
spin fluctuations above a potential magnetically ordered phase are definitely
absent for these two doping levels. For $x=0.056$, $(T_{1}T)^{-1}$ increases
monotonically down to the lowest measured temperature. Yet, a change in slope
is still visible at a temperature that would agree with a potential structural
transition.

Finally, for $x=0.06$, $(T_{1}T)^{-1}$ goes through a broad maximum consistent
with a glassy freezing of spin fluctuations at low temperatures without long-
range magnetic order [35, 28]. Below $T_{c}$, $(T_{1}T)^{-1}$ decreases
rapidly due to the opening of the superconducting gap. The slope of
$T_{1}^{-1}$ below $T_{c}$ can be described by a power-law with $T^{2}$, which
is smaller than in F-doped samples with $T^{3}$ or an even higher exponent
[36, 28]. This is most likely due to the increased disorder by in-plane Co-
doping compared to out-of-plane F-doping [37].

The middle row of Fig. 1 shows the stretching exponent, $\beta$, of the decay
of the nuclear magnetization, which indicates a distribution of spin-lattice
relaxation rates, where $T_{1}^{-1}$ is the median of this distribution [38,
39]. Such a distribution is present for all doping levels except for $x=0$.
Stretched exponential relaxation often occurs in disordered systems, and is
present also for F-doping [35, 28], and in ${\rm Ba}({\rm Fe}_{1-x}{\rm
Co}_{x})_{2}{\rm As}_{2}$ for approximately $0.048\leq x\leq 0.066$, i.e. for
a much smaller doping range [38, 20, 33]. $\beta$ reaches a minimum of about
0.4, indicating that the relaxation rate varies by several orders of magnitude
within a sample [39]. More sophisticated analyses of stretched relaxation to
inspect the true distribution of $T_{1}^{-1}$ [22, 40] could not be performed
here owing to the limited signal intensity of the small single crystals.

The lowermost row of Fig. 1 shows the ratio of $(T_{1}T)^{-1}$ measured for
$H||ab$ and $H||c$, $R=T_{1ab}^{-1}/T_{1c}^{-1}$. Due to the peculiar
hyperfine coupling of the 75As nucleus, $R$ is about 1.5 for isotropic spin
fluctuations above $T_{S}$, and becomes larger than 1.5 below $T_{S}$, where
the spin fluctuations become anisotropic [17, 18, 19, 21]. All samples that
order long-range magnetically also show a clear change of $R$ at the
structural phase transition. The change of $R$ for $x=0.056$ is smoothed,
indicating that a possible structural phase transition does not occur in the
bulk of this sample. Finally, for $x=0.06$, $R$ is constant with temperature,
i.e. this sample is tetragonal down to the lowest $T$.

The NMR spectra are shown in Fig. 2 exemplarily for $x=0.03$, 0.042, and 0.056
for $H||[100]_{ortho}$ or $[010]_{ortho}$. The spectra for $x=0.035$ and 0.06
are shown in the supplement [41]. For all doping levels that exhibit a
structural phase transition, the spectra broaden notably below $T_{S}$.
Surprisingly, the spectra broaden at $T=40$ K for $x=0.056$, too, as can be
seen in Fig. 3(a), where the full width at half maximum (FWHM) is plotted
against temperature. The spectra of $x=0.03$ even show remnants of a
splitting, which indicates the anisotropy between the $a$ and $b$ directions
due to the nematic transition [29, 14]. For higher doping levels, the reduced
orthorhombicity, $\epsilon=\Delta L/L$, [42] prevents resolving the splitting.
Undoped LaFeAsO exhibits a splitting of about 75 kHz, which is already on the
order of the FWHM for $x=0.03$. The steep increase of the linewidth at $T_{S}$
is another determination method of the structural transition temperatures,
which agree well with those determined by $(T_{1}T)^{-1}$. Note that for field
orientation $H||[110]_{ortho}$, the FWHM increases only moderately with
decreasing temperature, see Fig. 3(a). For this orientation, the spectra of
undoped LaFeAsO do not split below $T_{S}$ [14]. The angular dependence of the
linewidth below $T_{S}$ can be used to confirm the correct orientation of the
samples [41].

Figure 2: (Color online) NMR spectra for $x=0.03$, 0.042, and 0.056 for
$H||[100]_{ortho}$ or $[010]_{ortho}$. Solid lines mark $T_{N}$, dashed lines
mark $T_{S}$. For better visibility, spectra at low temperatures are enlarged.
Figure 3: (Color online) (a) FWHM for all doping levels for $H||[100]$ or
$[010]$. Open symbols are for $H||[110]$. (b) Spin-spin decay of the nuclear
magnetization, $M_{xy}$, for $x=0.042$. (d) Spin-spin decay for $x=0.056$. (c)
$M_{xy}$ at $t=0$ s versus temperature for $x=0.042$ and $x=0.056$. (e)
$(T_{1}T)^{-1}$ for ${\rm La}{\rm Fe}_{1-x}{\rm Co}_{x}{\rm As}{\rm O}$ with
$x=0.035$ and $x=0.06$, for ${\rm Ba}({\rm Fe}_{1-x}{\rm Co}_{x})_{2}{\rm
As}_{2}$ with $x=0.04$ and $x=0.08$ [43], and for FeSe [13]. Down arrows
indicate $T_{N}$, dashed-down $T_{c}$, and up arrows $T_{S}$.

Below the SDW transition, the spectra shift to higher frequency and further
broaden, indicating the existence of internal static hyperfine fields at the
75As nuclei. The shift and broadening is typical for the field orientation
$H||ab$ [17]. With increasing doping, the hyperfine field quickly decreases
[20]. The shift and broadening is therefore best visible for the lowest doping
level of $x=0.03$ in Fig. 2. In addition, the spectra develop a shoulder at
even higher frequency, indicating that some nuclei feel higher internal
magnetic fields. Interestingly, such a shoulder is also visible for $x=0.056$
below about 20 K, and indicates that at least parts of this sample develop
static magnetism as well.

Another feature of the spectra is a loss of signal intensity about 15-20 K
above the magnetic ordering temperature. Such a wipeout of signal intensity is
also present in ${\rm Ba}({\rm Fe}_{1-x}{\rm Co}_{x})_{2}{\rm As}_{2}$, and
has been ascribed to a shortening of the spin-lattice and spin-spin relaxation
rates and their distribution over a few orders of magnitude [38, 22]. This is
known as a dynamic wipeout, where the nuclei relax so fast that they cannot be
observed anymore. Our $(T_{1}T)^{-1}$ and $\beta$ values for $x=0.03$, 0.035,
and 0.042 are very close to those observed in underdoped ${\rm Ba}{\rm
Fe}_{2}{\rm As}_{2}$, and therefore, a shortening of relaxation times seems to
be the main reason for wipeout in underdoped LaFe1-xCoxAsO. However, the
intensity decreases for $x=0.056$ below about 20 K, too, despite the fact that
$(T_{1}T)^{-1}$ is about 10 times smaller for this sample.

To further investigate the development of magnetism in the nematic phase, we
measured the spin-spin relaxation rate for $x=0.042$ and $x=0.056$, i.e. at
the border between long-range magnetic order and superconductivity. The decay
of the in-plane nuclear magnetization, $M_{xy}$, is plotted in Fig. 3(b) and
(d). $M_{xy}$ was fit globally at all temperatures for a given sample to a
combined exponential and Gaussian form:
$M_{xy}(t)=M_{0}\cdot\exp(-(2t/T_{2})^{\beta})\cdot\exp(-(2t)^{2}/2T_{2G}^{2})$,
where $M_{0}$ is the initial nuclear magnetization, $t$ the time between the
$90^{\circ}$ and $180^{\circ}$ pulses. $T_{2}$ is the exponential spin-spin
relaxation time, and $T_{2G}$ the Gaussian relaxation time, which turned out
not to be temperature dependent [44, 22, 28, 41].

$\beta\leq 1$ accounts for a distribution of spin-spin relaxation times
similar to the distribution of the spin-lattice relaxation times. Yet, in
contrast to $T_{1}$, $\beta$ affects mostly the first few tens of microseconds
of the $T_{2}$-relaxation, which is the time slot that is most difficult to
measure due to ringing of the resonance circuit, especially for such small
samples as have been measured here [45]. $\beta$ of $T_{2}$ is shown in the
middle row of Fig. 1 in comparison to $\beta$ of $T_{1}$. For $x=0.042$, both
$\beta$ show approximately the same temperature dependence, whereas for
$x=0.056$, $\beta(T_{1})$ is notably smaller than $\beta(T_{2})$. However,
both data sets in Fig. 3(b) and (d) clearly show that, even considering a
stretching, $M_{xy}(t=0)$ decreases with decreasing temperature (see Fig.
3(c)). This is evidence for the presence of local static magnetic fields below
about $T_{SRO}=42$ K for $x=0.042$ and below about $T_{SRO}=23$ K for
$x=0.056$, in contrast to the mostly dynamic wipeout observed in ${\rm
Ba}({\rm Fe}_{1-x}{\rm Co}_{x})_{2}{\rm As}_{2}$. Static short range magnetism
has also been found in F-doped LaFeAsO polycrystals below $T_{S}$, and has
been attributed to a nanoscale phase separation in doped LaFeAsO [28], which
we found for Co-doped polycrystals as well [37].

Comparing the low-frequency spin fluctuations above $T_{S}$ in ${\rm La}{\rm
Fe}_{1-x}{\rm Co}_{x}{\rm As}{\rm O}$ with those in ${\rm Ba}({\rm
Fe}_{1-x}{\rm Co}_{x})_{2}{\rm As}_{2}$ indicates whether the nematic order is
driven by magnetic fluctuations or by orbital order [4, 9]. Fig. 3(e) shows
$(T_{1}T)^{-1}$ for two samples for each of the different families, which have
similar structural transition temperatures, namely for ${\rm Ba}({\rm
Fe}_{1-x}{\rm Co}_{x})_{2}{\rm As}_{2}$ with $x=0.04$ from Ref. 20 and for
${\rm La}{\rm Fe}_{1-x}{\rm Co}_{x}{\rm As}{\rm O}$ with $x=0.035$ from this
work. The relaxation rates for these two samples are identical above $T_{S}$.
Below $T_{S}$, $(T_{1}T)^{-1}$ increases much faster with decreasing
temperature for ${\rm Ba}({\rm Fe}_{1-x}{\rm Co}_{x})_{2}{\rm As}_{2}$. This
is also true for the undoped samples except for a constant offset, if the
temperature is normalized by $T_{S}$ [41]. Differences in spin fluctuations
therefore appear only below $T_{S}$. The reason for this is possibly the
stronger coupling of nematicity and magnetism in ${\rm Ba}{\rm Fe}_{2}{\rm
As}_{2}$ compared to LaFeAsO, in agreement with the smaller difference of
$T_{S}$ and $T_{N}$ in ${\rm Ba}{\rm Fe}_{2}{\rm As}_{2}$. For optimal doping
levels which do not exhibit a structural transition, $(T_{1}T)^{-1}$ is again
identical for both compounds down to the superconducting transition
temperature $T_{c}$ (see Fig. 3(e)). In contrast, $(T_{1}T)^{-1}$ of FeSe is
even smaller than that of optimally doped LaFeAsO and ${\rm Ba}{\rm
Fe}_{2}{\rm As}_{2}$, even below $T_{S}$ [46]. Here, the coupling of
nematicity and magnetism is even weaker, so that no strong enhancement of low-
frequency spin fluctuations appears, neither above nor below $T_{S}$, and no
static magnetism develops despite of the nematic order at $T_{S}=90$ K. These
results suggest that the leading instability for LaFeAsO and ${\rm Ba}{\rm
Fe}_{2}{\rm As}_{2}$ is the SDW, whereas FeSe shows orbital order without
magnetic fluctuations [13], in agreement with recent theoretical
considerations [9, 8].

Figure 4: (Color online) Phase diagram resulting from NMR and SQUID
susceptibility on Co-doped LaFeAsO single crystals compared to F-doped
polycrystals [28]. Closed symbols are from NMR, open symbols are from SQUID
and dilatometry [34]. $T^{nem}$ is determined by elastoresistivity [15].

All results can be combined in the phase diagram in Fig. 4. Up to $x=0.03$ the
phase diagram of Co-doped LaFeAsO coincides with that of F-doped polycrystals,
and $T_{S}$ and $T_{N}$ decrease almost linearly with doping. Above $x=0.03$,
F-doped samples become superconducting, and nematicity and magnetism disappear
abruptly, whereas for Co-doping, $T_{S}$ and $T_{N}$ can still be detected up
to $x=0.042$. This indicates that in-plane Co-doping pins the magnetism to
some extend. The sample with $x=0.056$ shows remnants of a structural
transition (see Fig. 1 and 3(a)) as well as short-range magnetic order below
$T_{SRO}\approx 23$ K. This is also the first doping that exhibits
superconductivity, albeit with a strongly reduced volume fraction. Finally,
$x=0.06$ is the only doping with bulk superconductivity and absence of static
nematicity and magnetism [34]. Nevertheless, compared to F-doping, the
superconducting transition temperature is greatly reduced, and the weaker
slope of $T_{1}^{-1}$ below $T_{c}$ indicates scattering at impurities, which
are most likely the in-plane Co-dopants. Interestingly, the nematic transition
temperature, $T^{nem}$, determined by elastoresistivity measurements [15]
follows the short-range magnetic order temperature determined here, leaving
the possibility open that nematicity is related to magnetism in Co-doped
LaFeAsO as well.

Compared to Co-doped ${\rm Ba}{\rm Fe}_{2}{\rm As}_{2}$, $T_{N}$ and $T_{S}$
decrease much faster with doping in LaFeAsO, which is surprising considering
the much larger superconducting range for ${\rm Ba}{\rm Fe}_{2}{\rm As}_{2}$
and the significantly higher $T_{c}$’s. This could be a consequence of the
nanoscale phase separation in LaFeAsO [28], where doping leads to a
percolation of the magnetic square lattice, and therefore to a more effective
suppression of nematicity and magnetism. In contrast, superconductivity and
magnetism can coexist in Co-doped ${\rm Ba}{\rm Fe}_{2}{\rm As}_{2}$, and no
phase separation occurs. However, the normal state spin fluctuations in both
compounds are very similar, indicating that nematicity is a result of the
fluctuating SDW order in LaFeAsO, as has been predicted by theory recently [9,
8].

We would like to thank Rüdiger Klingeler, Christian Hess and Xiaochen Hong for
helpful discussions. This work has been supported by the Deutsche
Forschungsgemeinschaft (DFG) through Grant No. AS 523/4-1, Grant No.
DI2538/1-1, research training group GRK1621, through the SFB 1143 (project id
247310070), and the Würzburg-Dresden Cluster of Excellence on Complexity and
Topology in Quantum Matter ct.qmat (EXC 2147, project id 390858490).

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