#  75As NMR study of the antiferromagnetic Kondo lattice CeNiAsO

Fangjun Lu1 Xiaobo He1 Kangqiao Cheng1 Zhuo Wang1 Jian Zhang1 Yongkang Luo1
<EMAIL_ADDRESS>1Wuhan National High Magnetic Field Center and School of
Physics, Huazhong University of Science and Technology, Wuhan 430074, China;

###### Abstract

We revisit the magnetic properties of the antiferromagnetic Kondo lattice
CeNiAsO by 75As nuclear magnetic resonance measurements. Our results confirm
two successive antiferromagnetic transitions of Ce moments at $T_{N1}=9.0(3)$
K and $T_{N2}=7.0(3)$ K. A Knight shift anomaly, characterized by the failure
of $K(T)-\chi(T)$ scaling, is observed below $T^{*}\sim 15$ K, which gives a
measure of the onset of coherent $c-f$ correlations. This energy scale is
further confirmed by the spin-lattice relaxation rate ($1/T_{1}$). The
analysis of spin dynamics also reveals a quasi-two-dimensional character of
spin fluctuations in CeNiAsO. This work paves the way for further 75As nuclear
magnetic resonance studies under pressure.

##  I<EMAIL_ADDRESS>Introduction

The ZrCuSiAs-structured rare-earth oxypnictides not only boosted a new wave of
research boom on high-$T_{c}$ superconductorsKamihara  _et al._ (2008); Chen
_et al._ (2008a, b); Wang  _et al._ (2008), but also provided a novel
playground for investigating strongly correlated electronic effects and
quantum critical phenomena. Depending on the specific chemical element in the
$Tm$ and $Pn$ sites, the family of Ce$TmPn$O ($Tm$=transition metal, $Pn$=P,
As) displays a wide spectrum of physical properties. The member CeFeAsO shows
a spin-density-wave (SDW) type antiferromagnetic (AFM) transition of Fe-$3d$
electrons near 140 K, and superconductivity can be induced by appropriate F-
doping when the SDW order is suppressedChen  _et al._ (2008a). At a much lower
temperature $\sim 4$ K, the Ce local moments also order
antiferromagneticallyChen  _et al._ (2008a); Zhao  _et al._ (2008); de la Cruz
_et al._ (2010); Luo  _et al._ (2010). In contrast, the analogue compound
CeFePO is a non-superconducting heavy-fermion metal with pronounced
ferromagnetic correlationsBrüning  _et al._ (2008); Luo  _et al._ (2010);
Kitagawa  _et al._ (2011); Lausberg  _et al._ (2012); Jesche  _et al._ (2017).
For $Tm$=Co, strong interplay between Co-$3d$ and Ce-$4f$ magnetism was
observed in both CeCoPOKrellner  _et al._ (2009) and CeCoAsOSarkar  _et al._
(2010), where the Co-$3d$ electrons show ferromagnetic orders near 75 K, while
Ce-$4f$ electrons are on the border to magnetism with enhanced Sommerfeld
coefficient. The appearance of $3d$ magnetism in these compounds complicates
the study of the $4f$ physics.

Unlike the cases for $Tm$=Fe or Co, the $3d$ electrons in the Ni counterpart
were shown to be nonmagneticXu  _et al._ (2008); Luo  _et al._ (2011). This
makes CeNiAsO an ideal candidate to explore the heavy-fermion properties of
the $4f$ electrons. Our previous studies based on magnetic susceptibility and
specific heat measurements revealed that the Ce moments in CeNiAsO undergo two
successive AFM transitions at $T_{N1}\approx 9.3$ K and $T_{N2}\approx 7.3$ K
[see also in Fig. 1(c)]. The Sommerfeld coefficient was found to be $\sim 203$
mJ/mol$\cdot$K, indicative of substantial $3d$-$4f$ hybridization. Applying
physical pressure to CeNiAsO or substituting As with smaller P ions suppresses
the AFM orders and leads to a quantum critical point (QCP) at $p_{c}\approx
0.65$ GPa or $x_{c}\approx 0.4$, respectivelyLuo  _et al._ (2014). Around the
QCP, Fermi surface structure reconstruction was indicated by the abrupt change
in the Hall coefficient, characteristic of local (or Kondo-breakdown) quantum
criticalitySi  _et al._ (2001); Coleman and Schofield (2005); Gegenwart  _et
al._ (2008); Paschen  _et al._ (2012). Recently, magnetic structures of the
two AFM ordered states were reported by Wu et al by a combination of neutron
scattering and $\mu$SR experimentsWu  _et al._ (2019). They found that for
$T<T_{N1}$, a second order phase transition yields an incommensurate SDW order
with a wave vector $\mathbf{q}=(0.44,~{}0,~{}0)$; while for $T<T_{N2}$, the
magnetic structure evolves into a coplanar commensurate AFM order with
$\mathbf{q}=(0.5,~{}0,~{}0)$. The schematic of these magnetic structures is
shown in Fig. 1(a). Furthermore, their $\mu$SR measurements on CeNiAs1-xPxO
manifested that the commensurate order exists only for $x\leq 0.1$, therefore
the QCP is probably between the incommensurate SDW and a paramagnetic phaseWu
_et al._ (2019), which is not expected in the general framework of local
quantum criticality. Additional microscopic local experiments are invited to
confirm the magnetic properties of CeNiAsO.

Herein, we employed 75As nuclear magnetic resonance (NMR) measurements on
field-aligned CeNiAsO powders, and both static and dynamic spin
susceptibilities are investigated. NMR spectra confirm two kinds of AFM
orderings below $T_{N1}=9.0(3)$ K and $T_{N2}=7.0(3)$ K, respectively. Both
NMR shift ($K$) and spin-lattice relaxation rate ($1/T_{1}$) manifest
anisotropic magnetic properties. The hyperfine interaction and influences of
the Kondo effect on $K$ and $1/T_{1}$ are also discussed.

Figure 1: (a) Local environment of As in CeNiAsO. The red arrows denote Ce
moments, which order in an incommensurate SDW with
$\mathbf{q}=(0.44,~{}0,~{}0)$ for $T_{N2}<T<T_{N1}$, and a commensurate AFM
with $\mathbf{q}=(0.5,~{}0,~{}0)$ for $T<T_{N2}$Wu  _et al._ (2019). In SDW
order, Ce moments might have a small $z$ component that is not shown here. The
blue arrows indicate the dipolar field at As sites calculated for specific
magnetic structures. (b) Simulated magnetization based on CEF theory shows c
as the hard axis, and $\mathbf{ab}$ as the easy plane. The inset depicts CEF
splitting. (c) Temperature dependence of magnetic susceptibility measured at
$B_{0}$=10 T, with field applied parallel and perpendicular to $\mathbf{c}$,
respectively. The insets display low-temperature $C/T$ and $d(T\chi_{c})/dT$,
from which the two magnetic transitions at $T_{N1}$ and $T_{N2}$ are clearly
seen. Note that the transition temperatures defined from $d(T\chi_{c})/dT$ are
relatively lower than those from $C/T$, because the former measurements were
made at a high field 10 T.

##  II<EMAIL_ADDRESS>Experimental details

A polycrystalline CeNiAsO sample of high purity was synthesized by solid-state
reaction as described previouslyLuo  _et al._ (2011). The sample quality was
verified by powder X-ray diffraction (XRD, PANalytical) with Cu K$\alpha$
radiation, and no impurity trace could be detected. 75As NMR measurements were
performed on field-aligned powders. The polycrystalline sample was thoroughly
ground into powders in a glove box, and mixed with Stycast 1266 with a weight
ratio $\approx 0.2$Young  _et al._ (2002). The mixture was placed in a strong
magnetic field of 14 T at 300 K in a physical property measurement system
(PPMS-14, Quantum Design) and held for 12 h before the Stycast was cured.
Because CeNiAsO is an easy-plane ($\mathbf{ab}$-plane) system (see below), the
field alignment was carried out on a rotating sample plate that rotates at a
slow speed ($\sim 5$ rpm). 75As (${}^{75}\gamma_{n}$=7.2919 MHz/T, $I$=3/2)
NMR spectra were recorded in a stepped frequency sweep spin-echo method at an
external field $B_{0}\sim 10$ T. The measurements were made for both
$\mathbf{B}_{0}\parallel\mathbf{c}$ and $\mathbf{B}_{0}\perp\mathbf{c}$. The
precise value of $B_{0}$ was calibrated by the 63Cu shift in a second coil
nearby. Spin-lattice relaxation rate ($1/T_{1}$) was measured in a standard
inversion recovery method on the central
($\frac{1}{2}\leftrightarrow-\frac{1}{2}$) transition, and $T_{1}$ was
extracted by fitting the recovery curve to

$M(t)=M(\infty)\\{1-2f[\frac{1}{10}\exp{(\frac{-t}{T_{1}})}+\frac{9}{10}\exp{(\frac{-6t}{T_{1}}})]\\},$
(1)

where $M(\infty)$, $f$ and $T_{1}$ are fitting parameters. Anisotropic
magnetic susceptibility of CeNiAsO was also measured on the aligned sample
with $B_{0}$=10 T in a PPMS equipped with a vibrating sample magnetometer
(VSM).

##  III<EMAIL_ADDRESS>Results and Discussion

###  III.1 3.1 CEF effect and magnetic anisotropy

Before presenting the NMR results, it is necessary to discuss briefly the
crystalline electric field (CEF) effect in CeNiAsO. The magnetic anisotropy of
a Ce-containing compound is usually described by the CEF effect whose
Hamiltonian in $C_{4v}$ point group is

$\hat{H}_{CEF}=B_{2}^{0}\hat{O}_{2}^{0}+B_{4}^{0}\hat{O}_{4}^{0}+B_{4}^{4}\hat{O}_{4}^{4},$
(2)

where $B_{l}^{m}$ ($l$=2,4; $m$=0,4) are CEF parameters, and $\hat{O}_{l}^{m}$
are Stevens operatorsHut . According to Wu’s inelastic neutron scattering
resultsWu  _et al._ (2019), $B_{2}^{0}\approx 26.7$ K, $B_{4}^{0}\approx-1$ K,
$B_{4}^{4}\approx 10.4$ K. It is found that the $j$=5/2 multiplet of Ce3+
splits into three Kramers doublets, with the first and second excited doublets
sitting at $\sim$226 and 790 K above the ground states, as shown in the inset
to Fig. 1(b). Simulated field dependent magnetizations at 300 K for field
parallel and perpendicular to $\mathbf{c}$-axis are provided in the mainframe
of Fig. 1(b). One clearly sees $M_{c}$ much smaller than $M_{ab}$, suggesting
$\mathbf{ab}$-plane as the magnetic easy-plane. This confirms the validity of
the way we aligned the powdered sample.

The temperature dependence of magnetic susceptibility measured under an
external field $B_{0}$=10 T is shown in Fig. 1(c). More technical detail about
the data analysis of magnetic susceptibility in aligned powders can be found
in Ref. Cheng  _et al._ (2019). Indeed, $\chi_{c}(T)$ is much smaller than
$\chi_{ab}(T)$ in the full window $2\leq T\leq 300$ K. The two
antiferromagnetic transitions are visible in the $d(T\chi_{c})/dT$ plot as
depicted in the inset to Fig. 1(c). Note that the transitions defined by such
are relatively lower than in our original reportLuo  _et al._ (2011), because
the large field 10 T likely decreases the transition temperatures of both AFM
orders. These susceptibility results will be compared with the NMR shifts in
the following subsections.

###  III.2 3.2 75As NMR spectra

The full 75As NMR spectra of aligned CeNiAsO under $B_{0}\sim 10$ T are shown
in Fig. 2, measured at 10 K (red) and 2 K (blue), respectively. For
$\mathbf{B}_{0}\parallel\mathbf{c}$ and $T=10$ K, three equally separated
peaks are resolved at 73.68, 65.96 and 81.46 MHz, corresponding to the central
($\frac{1}{2}\leftrightarrow-\frac{1}{2}$) and satellite
($\pm\frac{3}{2}\leftrightarrow\pm\frac{1}{2}$) transitions of the $I$=3/2
75As nucleus. The nuclear quadrupolar splitting frequency is extracted to be
$\nu_{Q}=$7.75 MHz at 10 K, and remains essentially unchanged down to 2 K,
deep inside the AFM ordered phase, manifesting that the principle axis of the
electric field gradient (EFG) does not change with AFM transitions. The value
of $\nu_{Q}$ is $\sim 22\%$ smaller than that in CeFeAsO (9.9 MHz)Ghoshray
_et al._ (2009), suggestive of a drastic change in the charge density
distribution around As and hence the character of Fe/Ni-As bonds in these two
compounds, which was confirmed previously by band structure calculationsXu
_et al._ (2008). Another salient feature of the spectrum at 2 K is that each
peak splits into two peaks, and the splittings are the same among central and
satellite transitions, consistent with an AFM ground state at low temperature.
We should point out that a small kink is seen at 72.08 MHz [denoted by a
asterisk in Fig. 2(a)], which should be attributed to a small amount of non-
aligned powders.

Figure 2: (a) The 75As frequency-sweep NMR spectra of CeNiAsO at 10 K (red)
and 2 K (blue) with the magnetic field $B_{0}\approx$ 10 T parallel with c.
The arrows denote all the central and satellite transitions of the $I$=3/2
75As nucleus. The asterisk indicates a small amount of non-aligned powder. (b)
ibid, but for field perpendicular to c.

Due to the symmetry, we assume $V_{zz}$ is along $\mathbf{c}$-axis, and
$V_{xx}=V_{yy}=-\frac{1}{2}V_{zz}$, where $V_{xx}$, $V_{yy}$ and $V_{zz}$
represent the EFG for the three principal axes. Therefore,
$\nu_{Q}^{\prime}=\nu_{Q}/2$ is expected for $\mathbf{B}_{0}\perp\mathbf{c}$.
Experimentally, we get $\nu_{Q}^{\prime}=3.9(1)$ MHz, as expected [Fig. 2(b)].
At 2 K, we also see two sets of peaks due to the AFM ordering. For both field
orientations, the central peak splits; it, therefore, is reasonable to infer
that the internal field at As sites has both components $\parallel\mathbf{c}$
and $\perp\mathbf{c}$. To show this more directly, we calculated the dipolar
fields at each As site exerted by Ce local moments and display them in Fig.
1(a) by blue arrows. In this calculation, the magnetic structure parameters
suggested by neutron scattering and $\mu$SR measurements were adoptedWu  _et
al._ (2019). At 2 K, the calculated dipolar field is ($-$0.0146, 0, 0.0252) T.
Since in our sample the powders are randomly oriented in the $\mathbf{ab}$
plane, we expect there is some distribution of internal field, and this
explains why the split peaks for $\mathbf{B}_{0}\perp\mathbf{c}$ are broader
and with longer tail than for $\mathbf{B}_{0}\parallel\mathbf{c}$.

Figure 3: 75As frequency-sweep NMR spectra of CeNiAsO at various temperatures
under $B_{0}\approx$ 10 T. (a-b) $\mathbf{B}_{0}\parallel\mathbf{c}$. (c-d)
$\mathbf{B}_{0}\perp\mathbf{c}$. The dashed lines indicate the Larmor
frequency. For clarity, the spectra for $T>T_{N1}$ shown in panels (a) and (c)
are amplified by a factor of $T$.

In order to get more detail across the magnetic transitions, we now focus on
the central peaks at different temperatures in Fig. 3. For better clarity, the
spectra are vertically offset, and the position of Larmor frequency $\nu_{0}=$
${}^{75}\gamma_{n}B_{0}$ is marked by a dashed line. Magnetic anisotropy can
be clearly seen by the opposite movement of peaks at lower temperature, cf
Fig. 3(a) and (c). For $\mathbf{B}_{0}\perp\mathbf{c}$, at 300 K, the
resonance peak is on the right-hand side of $\nu_{0}$, and it gradually drifts
to the left-hand side upon cooling, indicative that a sign change in NMR shift
probably occurs. See below.

Just below $T_{N1}$ and with $\mathbf{B}_{0}\parallel\mathbf{c}$, we see a
shoulder appear on each side of the central peak [cf 8.5 K in Fig. 3(b)]. As
$T$ decreases, the shoulders evolve into two broad peaks, and meanwhile the
height of original central peak in the paramagnetic state shrinks. Since NMR
is a microscopic local probe, such an evolution indicates phase segregation
between the emergent AFM phase and the paramagnetic phase. These patterns can
be fit to three Gaussian peaks, and one finds that the two AFM peaks are
broad, implying distribution of the internal field in this AFM order, which
might be understood by an incommensurate SDW order as proposed by neutron
scattering and $\mu$SR experimentsWu  _et al._ (2019). Near 7 K, the spectra
are very complicated, and are hard to be fitted to multi-peak functions. Below
6.8 K, two new well separate peaks can be resolved, and this should be
associated with the lower-temperature commensurate AFM order suggested by Wu
et alWu  _et al._ (2019). The window 6.8-7.3 K probably is a region where
these two AFM phases coexist. Such a phase coexistence is also observed for
$\mathbf{B}_{0}\perp\mathbf{c}$, in the similar temperature range [Fig. 3(d)].
It is interesting that for this field orientation, we do not see a clear trace
of phase segregation just below $T_{N1}$, the reason for which is still
unknown.

In spite of some unclear features, two conclusions can be drawn from the
temperature dependent 75As spectra. First, two different AFM orders appear at
low temperature, an incommensurate one between $T_{N_{1}}$ and $T_{N2}$, and a
lower-temperature phase that seems more likely a commensurate type. Second,
for both types of AFM orders, we see peak splittings in both
$\mathbf{B}_{0}\parallel\mathbf{c}$ and $\mathbf{B}_{0}\perp\mathbf{c}$. This
suggests that the internal fields at As sites have both in-plane and out-of-
plane components. We will get back to this issue later on.

###  III.3 3.3 75As NMR shifts and internal field

The NMR shifts ($K$) of 75As can be extracted from the central transition
lines. Taking account the correction from the second order quadropolar
effectRybicki  _et al._ (2013),

$\displaystyle\nu_{\parallel}$ $\displaystyle=^{75}\gamma_{n}B_{0}(1+K_{c}),$
(3a) $\displaystyle\nu_{\perp}$
$\displaystyle=^{75}\gamma_{n}B_{0}(1+K_{ab})+\frac{3\nu_{Q}^{2}}{16\cdot^{75}\gamma_{n}B_{0}},$
(3b)

and the derived $K_{c}$ and $K_{ab}$ are displayed in Fig. 4(a) as functions
of $T$. Both $K_{c}(T)$ and $K_{ab}(T)$ can be well fit to a Curie-Weiss
formula in the $T$ range 20-300 K, with the resultant Weiss temperatures
$\theta_{W}^{c}=-23(3)$ K and $\theta_{W}^{ab}=-11(2)$ K, confirming AFM-type
exchange couplings between Ce moments.

Figure 4: (a) 75As NMR shifts as functions of $T$, for field parallel and
perpendicular to $\mathbf{c}$. The open symbols represent the mean values in
the AFM phase. The solid lines are the Curie-Weiss fits to ${}^{75}K_{ab}(T)$
and ${}^{75}K_{c}(T)$ in the paramagnetic phase, leading to
$\theta_{W}^{ab}=-11(2)$ K and $\theta_{W}^{c}=-23(3)$ K, respectively. (b)
${}^{75}K$ vs. $\chi$ plots with $T$ as an implicit parameter. The linear fits
in the paramagnetic phase yield hyperfine coupling constants
$A_{\text{hf},ab}=-0.42(9)$ T/$\mu_{B}$, and $A_{\text{hf},c}=0.86(2)$
T/$\mu_{B}$.

In metals, the NMR shifts usually contain multiple contributions,

$K(T)=K^{o}+K^{s}(T),$ (4)

where the temperature independent $K^{o}$ arises from the orbital shift, and
$K^{s}(T)=A_{hf}\chi(T)$ is the spin (Knight) shift with $A_{hf}$ being the
hyperfine coupling constant. A $K-\chi$ plot is given in Fig. 4(b), from which
$A_{hf,c}=0.86(2)$ T/$\mu_{B}$ and $A_{hf,ab}=-0.42(9)$ T/$\mu_{B}$ are
obtained. We find that below $\sim 15$ K, the linear relation between $K(T)$
and $\chi(T)$ gradually collapses. In Kondo lattice compounds, this is usually
termed a Knight shift anomalyJiang  _et al._ (2014); Curro (2016). This
provides an estimate of the coherent Kondo scale $T^{*}\sim 15$ K. We should
mention that such a Knight shift anomaly has been observed in a large variety
of heavy-fermion materials, and multiple theoretical models have been proposed
that attribute the anomalous hyperfine coupling either to a temperature
dependence due to Kondo screeningKim  _et al._ (1995) or to different
occupations of CEF levelsOhama  _et al._ (1995). A more recent hypothesis is a
phenomenological two-fluid modelNakatsuji  _et al._ (2004); Yang and Pines
(2008, 2012); Shirer  _et al._ (2012) in which the hyperfine couplings to
localized $f$-electron (Fluid 1) spins and itinerant heavy-electron (Fluid 2)
spins are of different values. As the weight of the latter starts to increase
significantly below $T^{*}$, $K(T)$ deviates from its linear relation to
$\chi(T)$. It should also be noted that in CeNiAsO, the coherent temperature
$T^{*}$ is rather close to the single-ion Kondo scale $T_{K}$ estimated from
our previous entropy analysisLuo  _et al._ (2011).

Figure 5: Internal field at 75As sites as a function of $T$. (a),
$\mathbf{B}_{0}\parallel\mathbf{c}$. (b), $\mathbf{B}_{0}\perp\mathbf{c}$. The
hatched areas indicate phase coexistence.

Internal fields determined from the AFM-induced splitting (Fig. 3) are
summarized in Fig. 5. Two prominent features are identified. (i) For both
orientations, discontinuities are clearly visible in $B_{int}(T)$ around
$T_{N2}$, confirming a modification in magnetic order takes place. (ii) The
magnitude of $B_{int}$ is about $\sim$ 6-8 times that of the dipolar field
mentioned above (at 2 K). This manifests that the internal field at As sites
is dominated by the transferred hyperfine field. Unlike the long-range dipolar
interaction, the transferred hyperfine interaction is short-range and relies
on orbital hybridizationKitagawa  _et al._ (2008). The ratio
$A_{hf,c}/A_{hf,ab}\sim 2$ reminds us that such a hybridization is through the
As-$4p_{z}$ orbital. We therefore consider only the four nearest neighbor Ce
sites, and follow the symmetry analysis in Ref. Kitagawa  _et al._ (2008),

$\mathbf{B}_{\text{trh}}=\sum_{i=1}^{4}\mathbf{A}_{i}\cdot\mathbf{m}_{i},$ (5)

where $\mathbf{B}_{\text{trh}}$ is the transferred hyperfine field,
$\mathbf{m}_{i}$ is the moment at the $i$-th Ce site, $\mathbf{A}_{i}$ is the
hyperfine coupling tensor and takes the form as

$\mathbf{A}_{i}=\left[\begin{aligned}
\begin{array}[]{ccc}A_{aa}&A_{ab}&A_{ac}\\\ A_{ba}&A_{bb}&A_{bc}\\\
A_{ca}&A_{cb}&A_{cc}\\\ \end{array}\end{aligned}\right].$ (6)

In the paramagnetic state, the moment is uniform $\mathbf{m}_{i}=\mathbf{m}$,
and we have
$\mathbf{B}_{\text{trh}}=(4A_{aa}m_{x},~{}4A_{bb}m_{y},~{}4A_{cc}m_{z})$. The
diagonal components of $\mathbf{A}$ can be determined by comparing to the
experimental values of $A_{hf}$, which leads to $A_{aa}$=$A_{bb}=-0.11$
T/$\mu_{B}$, and $A_{cc}=0.22$ T/$\mu_{B}$.

For the lower-temperature AFM phase with wave vector
$\mathbf{q}=(0.5,~{}0,~{}0)$ Wu  _et al._ (2019), the moments
$\mathbf{m}_{1}$=$-\mathbf{m}_{2}$=$\mathbf{m}_{3}$=$-\mathbf{m}_{4}$=$\mathbf{m}$
Kitagawa  _et al._ (2008), and we get
$\mathbf{B}_{\text{trh}}=(4A_{ac}m_{z},~{}0,~{}4A_{ca}m_{x})$. It is
interesting to note that in order to have a non-vanishing in-plane hyperfine
field as suggested by our NMR measurements (cf Fig. 5), the $z$ component of
Ce moment should be non-zero. This is different from the coplanar AFM order
proposed by neutron and $\mu$SR experimentsWu  _et al._ (2019). However, we
should point out that our NMR measurements were made at external field of 10
T, while both neutron and $\mu$SR experiments were performed in the absence of
a magnetic field. It is possible that the $m_{z}$ component is field induced.
Either zero-field NMR or in-field neutron/muon experiment would be useful to
reconcile this bifurcation.

The situation for the incommensurate SDW phase at $T_{N2}<T<T_{N1}$ proposed
by Wu et alWu  _et al._ (2019) is too complicated for this analysis, and is
not discussed here.

###  III.4 3.4 Spin dynamics

We also investigated the spin dynamics of CeNiAsO by spin-lattice relaxation
rate measurements, and the results are summarized in Fig. 6. As is well known,
for conventional metals (like CuAbragam (1961)), the major mechanism to
transfer the spin temperature of a nuclear ensemble into equilibrium with the
lattice is via spin-flip scattering with conduction electronsSlichter (1990);
$1/T_{1}T$, therefore, is a constant proportional to $N^{2}(E_{F})$ [so-called
Korringa’s relation, where $N(E_{F})$ is the density of states at the Fermi
level]. In CeNiAsO, however, for both field orientations, $1/T_{1}$ is
essentially constants for $T$ well above $T_{N1}$, or in other words,
$1/T_{1}T\propto T^{-1}$. Such a Curie-Weiss-like behavior has been reported
in many other Ce compounds like CeIn3Kohori  _et al._ (1999), CePt2In7Sakai
_et al._ (2014), Ce2CoAl7Ge4Dioguardi  _et al._ (2017), etcBrüning  _et al._
(2008); Rybicki  _et al._ (2013); Dioguardi  _et al._ (2017), and can be
ascribed to relaxation dominated by spin fluctuations of Ce local moments.
Near 9 K, $1/T_{1}$ peaks for both field orientations, consistent with
critical slowing down at the magnetic transitions. The two transition
temperatures $T_{N1}$ and $T_{N2}$ are discernible in $1/T_{1}$, as shown in
the inset to Fig. 6(a-b). Below $T_{N2}$, $1/T_{1}$ decreases rapidly,
characteristic of long-range magnetic ordering.

Figure 6:  (a-b) $T$ dependence of $1/T_{1}$ at 10 T, for
$\mathbf{B}_{0}\parallel\mathbf{c}$ and $\mathbf{B}_{0}\perp\mathbf{c}$,
respectively. The insets show $1/T_{1}$ near the AFM transitions. (c)
Hyperfine-coupling normalized spin fluctuation rate
$R_{i}/A_{\text{hf,}i}^{2}$ as functions of $T$ for 75As. (d) Plots of
${}^{75}K/T^{1/2}$ vs. $1/T_{1}T$ in the paramagnetic phase for both field
orientations.

In general, the spin-lattice relaxation rate provides a measure of spin
fluctuations perpendicular to an applied $\mathbf{B}_{0}$, as described
byMoriya (1963)

$1/T_{1}T\propto\gamma_{n}^{2}\sum_{\mathbf{q}\perp\mathbf{B}_{0}}A_{\text{hf}}^{2}(\mathbf{q},\omega)\frac{\chi^{\prime\prime}(\mathbf{q},\omega)}{\omega},$
(7)

where $\chi^{\prime\prime}(\mathbf{q},\omega)$ is the imaginary part of the
dynamic susceptibility at wave vector $\mathbf{q}$ and frequency $\omega$.
Considering the magnetic anisotropy of the sample, $1/T_{1}T$ is decomposed
for further study:

$\displaystyle(1/T_{1}T)_{c}$ $\displaystyle=2R_{ab},$ (8a)
$\displaystyle(1/T_{1}T)_{ab}$ $\displaystyle=R_{ab}+R_{c},$ (8b)

where the spin fluctuation rate

$R_{i}\propto
A_{\text{hf},i}^{2}\sum_{\mathbf{q}_{\perp}}\frac{\chi^{\prime\prime}_{i}(\mathbf{q},\omega)}{\omega},$
(9)

with $i=ab,c$. Note that here the $\mathbf{q}$ and $\omega$ dependencies of
$A_{hf}$ are neglected. The temperature dependence of the hyperfine-coupling
normalized spin fluctuation rates $R_{i}/A_{\text{hf},i}^{2}$ are shown in
Fig. 6(c). $R_{i}/A_{\text{hf},ab}^{2}$ near $T_{N1}$ is about 5 times as
large as $R_{i}/A_{\text{hf},c}^{2}$. This suggests quasi-2D spin fluctuations
in CeNiAsO.

Finally, it is necessary to discuss the influence of the Kondo effect on spin
dynamics. Previous work on Kondo lattice systems suggested that the
$4f$-electron dominated $1/T_{1}T$ is proportional to the ratio of the static
susceptibility $\chi(T)$ and the dynamical relaxation rate $\Gamma(T)$ of the
$4f$ electronsMoriya and Ueda (1974); Kuramoto and Kitaoka (2000); MacLaughlin
(1989); Brüning  _et al._ (2008), where $\Gamma\propto T^{1/2}$ for $T>T^{*}$
Cox  _et al._ (1985). Since $K\propto\chi$ in this temperature range (Fig. 4),
then

$(1/T_{1}T)_{4f}\propto K(T)/T^{1/2}.$ (10)

This relation was found in many Ce-based Kondo compounds including
CeFePOBrüning  _et al._ (2008) that is isostructural to CeNiAsO. In Fig. 6(d),
we plot $K/T^{1/2}$ vs. $1/T_{1}T$ in the paramagnetic phase. For both field
directions, Eq. (10) is validated at high temperature, but is gradually
deviated for $T$ below 15 K. This provides additional evidence that a coherent
Kondo liquid gradually forms below $T^{*}\sim 15$ K in CeNiAsO.

###  III.5 3.5 Discussion and outlook

Relevant issues remain:

(1) It is still unknown whether the “phase segregation” observed at
$T_{N2}<T<T_{N1}$ is intrinsic or not. First, it is seen only in
$\mathbf{B}_{0}\parallel\mathbf{c}$, but not in
$\mathbf{B}_{0}\perp\mathbf{c}$; second, it disappears when entering the
lower-temperature ordered phase; and third, an XRD pattern (not shown here,
but equally good as in our original reportLuo  _et al._ (2011)) shows very
high sample quality without any noticeable impurity phase. Note that such a
phase segregation might not be seen with the resolution of neutron scattering
or $\mu$SR. If this phenomenon is intrinsic, considering the heat-capacity
results [see in Fig. 1(c)] we may infer a weakly first-order transition at
$T_{N1}$; it, therefore, is interesting to ask how this order is tuned to a
QCP under pressure.

(2) A previous studyLuo  _et al._ (2014) has manifested the realization of QCP
in CeNiAsO either by P-doping or pressure. $\mu$SR experiments on CeNiAs1-xPxO
suggested that the lower-temperature AFM phase is suppressed rapidly by
P-doping, and the QCP is between the incommensurate SDW and paramagnetic
phasesWu  _et al._ (2019). It is unknown whether the same situation also
appears when QCP is accessed by pressure which is a cleaner control parameter
than chemical doping. Moreover, an unconventional Kondo-breakdown type QCP was
suggested in CeNiAsOLuo  _et al._ (2014); however, the nature of quantum
fluctuations (e.g., reflected in scaling laws of Knight shift and spin-lattice
relaxation rateSi  _et al._ (2001) and whether critical fermionic degrees of
freedom should be involvedProchaska  _et al._ (2020); Kandala  _et al._
(2022), etc) near the QCP remains unclear, and microscopic evidence for a
Fermi-surface reconstructionKawasaki  _et al._ (2020) is also lacking.
Hopefully, 75As NMR under pressure will be helpful to clarify these open
questions.

All these require further investigations in the future.

##  IV<EMAIL_ADDRESS>Conclusions

By 75As NMR experiments, we investigated static and dynamic spin
susceptibilities of the AFM Kondo lattice material CeNiAsO. The NMR spectra
confirm that there are two kinds of AFM ordering that appear below
$T_{N1}=9.0(3)$ K and $T_{N2}=7.0(3)$ K. Quadrupolar splitting frequency
$\nu_{Q}\sim 7.75$ MHz remains essentially unchanged across the AFM
transitions. A Knight shift anomaly is observed below $T^{*}\sim 15$ K, which
gives a measure of the onset of coherent $c-f$ correlations. Quasi-2D
character of spin fluctuations is revealed by the highly anisotropic
hyperfine-coupling normalized fluctuation rates. The hyperfine field at As
sites transferred from Ce local mements contains both in-plane and out-of-
plane components, which is at odds with conclusions from neutron diffraction.
A more careful look into the magnetic structures is encouraged.

##  V Acknowledgments

We thank Shan Wu, Stuart E. Brown, Joe D. Thompson, Rui Zhou, and Long Ma for
insightful discussions and technical suggestions.

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