]Current address: Laboratory for Multiscale materials eXperiments, Paul
Scherrer Institute, 5232 Villigen PSI, Switzerland

# Impact of disorder in Nd-based pyrochlore magnets

Mélanie Léger  Institut Néel, CNRS and Université Grenoble Alpes, 38000
Grenoble, France  Laboratoire Léon Brillouin, CEA, CNRS, Université Paris-
Saclay, CE-Saclay, F-91191 Gif-sur-Yvette, France  Florianne Vayer  Institut
de Chimie Moléculaire et des Matériaux d’Orsay (UMR CNRS 8182), Université
Paris-Saclay, F-91405 Orsay, France  Monica Ciomaga Hatnean  [  Department of
Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom  Françoise
Damay  Laboratoire Léon Brillouin, CEA, CNRS, Université Paris-Saclay, CE-
Saclay, F-91191 Gif-sur-Yvette, France  Claudia Decorse  Institut de Chimie
Moléculaire et des Matériaux d’Orsay (UMR CNRS 8182), Université Paris-Saclay,
F-91405 Orsay, France  David Berardan  Institut de Chimie Moléculaire et des
Matériaux d’Orsay (UMR CNRS 8182), Université Paris-Saclay, F-91405 Orsay,
France  Björn Fåk  Institut Laue Langevin, 6, rue Jules Horowitz, BP 156
F-38042 Grenoble, France  Jean-Marc Zanotti  Laboratoire Léon Brillouin, CEA,
CNRS, Université Paris-Saclay, CE-Saclay, F-91191 Gif-sur-Yvette, France
Quentin Berrod  Univ. Grenoble Alpes, CNRS, CEA, Grenoble INP, IRIG, SyMMES,
38000 Grenoble, France  Jacques Ollivier  Institut Laue Langevin, 6, rue Jules
Horowitz, BP 156 F-38042 Grenoble, France  Jan P. Embs  Laboratory for Neutron
Scattering and Imaging, Paul Scherrer Institut, 5232 Villigen PSI, Switzerland
Tom Fennell  Laboratory for Neutron Scattering and Imaging, Paul Scherrer
Institut, 5232 Villigen PSI, Switzerland  Denis Sheptyakov  Laboratory for
Neutron Scattering and Imaging, Paul Scherrer Institut, 5232 Villigen PSI,
Switzerland  Sylvain Petit<EMAIL_ADDRESS>Laboratoire Léon Brillouin,
CEA, CNRS, Université Paris-Saclay, CE-Saclay, F-91191 Gif-sur-Yvette, France
Elsa Lhotel<EMAIL_ADDRESS>Institut Néel, CNRS and Université
Grenoble Alpes, 38000 Grenoble, France

###### Abstract

We study the stability of the antiferromagnetic all-in–all-out state observed
in dipolar-octupolar pyrochlores that have neodymium as the magnetic species.
Different types of disorder are considered, either affecting the immediate
environment of the Nd3+ ion, or substituting it with a non-magnetic ion.
Starting from the well studied Nd2Zr2O7 compound, Ti substitution on the Zr
site and dilution on the Nd magnetic site with La substitution are
investigated. The recently discovered entropy stabilized compound NdMox, which
exhibits a high degree of disorder on the non magnetic site is also studied.
Using a range of experimental techniques, especially very low-temperature
magnetization and neutron scattering, we show that the all-in–all-out state is
very robust and withstands substitutional disorder up to large rates. From
these measurements, we estimate the Hamiltonian parameters and discuss their
evolution in the framework of the phase diagram of dipolar-octupolar
pyrochlore magnets.

##  I Introduction

Geometrically frustrated magnetism is a forefront research topic within
condensed matter physics, as testified by the wealth of exotic phenomena
discovered over the past years [1, 2]. To explore this physics, pyrochlore
oxide compounds ($R_{2}T_{2}$O7, $R$ is a rare-earth and $T$ is usually a
transition metal), in which both $R$ and $T$ sites form a corner sharing
tetraheda lattice are a rich playground. Depending on the rare-earth element,
different spin anisotropy (Ising, easy-plane, Heisenberg) and interactions
(antiferromagnetic, ferromagnetic, long-range, anisotropic) come into play.
This results in amazing physics, ranging from delicate balance between ferro-
and antiferromagnetism, order by disorder mechanisms to Coulomb and quantum
spin liquid phases [3, 4].

Ho2Ti2O7 and Dy2Ti2O7 are for instance experimental realizations of the
celebrated “spin ice” state [5]. They correspond to the case of ferromagnetic
interactions between rare earth magnetic moments, constrained to lie along the
local $\langle 111\rangle$ axes (linking the center of each tetrahedron to its
vertices) by a crystal field scheme featuring a strong Ising anisotropy.

In contrast to spin ice, Nd-based magnets, Nd2Zr2O7 [6, 7, 8], Nd2Sn2O7 [9]
and Nd2Hf2O7 [10, 11] are Ising antiferromagnets, hence, at first glance, more
conventional materials. The Nd3+ magnetic moments indeed exhibit a strong
local $\langle 111\rangle$ Ising anisotropy with an effective value of about
2.4-2.5 $\mu_{\rm B}$. Magnetization measurements together with powder neutron
diffraction experiments further reveal that these compounds undergo an
antiferromagnetic transition towards an all-in–all-out magnetic configuration,
where the four spins point inwards to or outwards from the center of a
tetrahedron. This situation corresponds in principle to the canonical un-
frustrated case. Surprisingly, experiments demonstrate a much more complex
case: in Nd2Zr2O7 and Nd2Hf2O7 the measured ordered magnetic moment is only
about 1 $\mu_{B}$, thus less than half of the full moment. Furthermore, the
Curie-Weiss temperature obtained from the magnetic susceptibility is positive,
indicating ferromagnetic interactions that should stabilize a spin ice state,
contrasting with the all-in–all-out ground state. In addition, inelastic
neutron scattering measurements reveal distinctive excitations, encompassing a
flat mode at finite energy, above which dispersive modes appear [12, 13, 14].
Both modes have remarkable structure factors, the flat mode being
characterized by pinch points, suggesting dynamic spin ice correlations, while
the dispersive mode is characterized by a half moon pattern stemming out of
the pinch point positions.

To solve this conundrum, Benton [15] showed that it is necessary to consider
the dipolar-octupolar nature [16] of the Nd3+ magnetic moment in the
pyrochlore lattice symmetry. At low temperature, the relevant degree of
freedom is an effective spin $1/2$ whose $z$ component can be interpreted as
the magnetic moment, while its $x$ and $y$ components represent combinations
of octupolar moments. The observed ordered phase corresponds to an ordering of
this pseudo-spin, which adopts an all-in–all-out structure, each spin pointing
however in an inclined “direction” in the $({\bf x},{\bf z})$ local plane. It
is therefore physically both a magnetic and octupolar structure, the ordered
dipole moment being reduced compared to the ground state doublet value because
of the octupolar contribution. It was further shown that this peculiar
ordering competes with a ferromagnetic Coulomb phase, which gives rise to spin
ice correlations above the ordering temperature [17, 18], so that the system
may be close to a $U(1)$ spin liquid state [16, 15, 19, 20].

The aim of this work is to study the stability of this unconventional all-
in–all-out state when disorder is introduced into the Nd2Zr2O7 compound.
Different kinds of disorder are considered: on the one hand, substitution on
the magnetic site, replacing neodymium by lanthanum, and on the other,
substitution on the zirconium (non-magnetic) site. Two cases are envisaged:
substitution by titanium, and another original route, which consists in an
entropy stabilized mixing of five different cations at the zirconium site. All
those different substitutions contribute to modifying the environment and
exchanges interactions between Nd3+ ions.

We show that the all-in–all-out state is very robust and withstands disorder
and / or dilution up to very large levels. This paper is organized as follows.
The first section focuses on experimental techniques and the sample synthesis.
The next sections describe the crystalline structures, crystal field schemes
and magnetic ground states. We then investigate the low energy excitations,
propose effective exchange parameters and briefly present the in field
evolution of the structure. We finally discuss the influence of the different
types of disorder, by comparing our results with the properties of previously
reported Nd-based pyrochlore compounds.

##  II Experimental techniques

###  II.1 Sample synthesis

Three types of samples were synthesized for this study. The Nd2(Zr1-xTix)2O7
samples are the same as studied in Ref. 21, and were also grown as single
crystals. The (Nd1-yLay)2Zr2O7 samples were synthesized in polycrystalline
form using solid state reactions between starting powders Nd2O3, La2O3 and
ZrO2 in a stoichiometric mixture [22]. The powders were mixed and pressed in
pellets to facilitate the chemical reaction. To eventually obtain
(Nd1-yLay)2Zr2O7 compounds, several heat treatments were necessary. Finally,
three samples were obtained: Nd1.8La0.2Zr2O7 (10 %-substituted sample),
Nd1.6La0.4Zr2O7 (20 %-substituted sample) and Nd1.2La0.8Zr2O7 (40
%-substituted sample). They were then characterized by X-ray diffraction
measurements to check the purity of the phase.

The NdMox (Nd2(TiZrHfScNb)2O7) samples were synthesized, as described in Ref.
23.

###  II.2 Bulk magnetic measurements

The “high” temperature magnetization measurements were performed on a Quantum
Design MPMS VSM-SQUID magnetometer from 2 to 300 K. The powder samples were
packed in cellophane with an approximate spherical shape. The data were
corrected from demagnetizing effects with the demagnetization factor for a
sphere.

Magnetization and AC susceptibility measurements were performed down to 80 mK
using a superconducting quantum interference device (SQUID) magnetometer
equipped with a 3He-4He dilution refrigerator developed at the Institut Néel-
CNRS Grenoble [24]. The powder samples were placed in small rectangular
pouches made from thin copper sheets and covered with Apiezon grease to favor
proper thermalization. The NdMox polycrystalline sample was pressed into a
parallelepiped shape.

The specific heat of the NdMox sample was measured with a Quantum Design PPMS
equipped with a 3He system. To estimate the lattice contribution, a YMox
(Y${}_{2}[$(TiZrHfSn)0.205Ge${}_{0.18}]_{2}$O7) sample was also measured.

###  II.3 Powder neutron diffraction

For the Nd2(Zr1-xTix)2O7 and (Nd1-yLay)2Zr2O7 samples (except the 40 %
substituted-sample), powder neutron diffraction experiments were carried out
at G4.1 (LLB-Orphée, France) operated with $\lambda=2.43$ Å and equipped with
a 3He-4He dilution refrigerator. A dedicated vanadium sample holder was used,
loaded with 4He pressure of about 10 bars to ensure thermalization.

Neutron powder diffraction data for the NdMox and Nd1.2La0.8Zr2O7 samples were
acquired with the high resolution neutron powder diffractometer HRPT [25] at
Paul Scherrer Institute. The data for NdMox were collected at 300 K with
neutron wavelengths $\lambda=1.494$ and 1.1545 Å in the high intensity mode of
the instrument and for the Nd1.2La0.8Zr2O7 with $\lambda=1.044$ and 2.450 Å in
the medium resolution mode of the instrument. The Nd1.2La0.8Zr2O7 sample has
been found to contain 4% wt. of a Nd(OH)3 impurity, which was included into
the refinements.

###  II.4 Inelastic neutron scattering

For the (Nd1-yLay)2Zr2O7 samples with $y=10$ and 20 %, inelastic neutron
scattering measurements were performed on the time-of-flight spectrometer
PANTHER (ILL, France) [26] to determine the crystal electric field (CEF)
excitations [27]. The spectrometer was operated with an incident energy
$E_{i}=50$ meV, yielding a good compromise in terms of energy resolution and
flux. The samples were placed in an orange cryostat capable of cooling down to
1.5 K, and data were taken up to 200 K with 50 K steps. The scattering from
the empty sample holder was measured and subtracted from the raw data.

Similar experiments were performed on the triple axis spectrometers 2T (LLB-
Orphée, France) for Nd2(Zr1-xTix)2O7 samples, and at EIGER (PSI, Switzerland)
[28] for the NdMox and Nd1.2La0.8Zr2O7 samples. We used a constant final wave
vector $k_{f}=2.662$ Å-1, yielding an energy resolution at the elastic line of
the order of 1 meV. Several $Q$-scans were measured with $1<Q<6$ Å-1 and the
spectra were then assembled to produce intensity maps as a function of $Q$ and
energy transfer $\omega$. A pyrolytic graphite (PG) filter was used to
eliminate spurious signals due to harmonics in the incident beam being
scattered from the elastic incoherent response of the sample and sample
holder. Depending on the strength of such scattering, the length of the filter
may not be sufficient to completely eliminate this effect. At 2T we also
measured the aluminum can sample holder under the same conditions to eliminate
background and spurious scattering as far as possible by subtraction. When
measuring Nd1.2La0.8Zr2O7 at EIGER we used a long filter ($\sim 70$ mm), which
also minimizes the spurious scattering. For the measurements of NdMox at EIGER
we used the normal short filter ($\sim 36$ mm) and time constraints prevented
the measurement of the empty holder. We have therefore fitted the spurious
peaks at 17.5 meV ($2k_{i}=3k_{f}$) and 43.5 meV ($k_{i}=2k_{f}$) using
Gaussian profiles and subtracted these from the data. This treatment works
well for the peak at 17.5 meV but is less effective for the peak at 43.5 meV.

To determine the low energy magnetic response, typically below 1 meV, we used
the time of flight spectrometers SHARP (LLB CRG @ILL) [29] with an incident
wavelength of 5.1 Å for the (Nd1-yLay)2Zr2O7 samples with $y=10$ and 20 %, and
FOCUS (PSI, Switzerland) with incident wavelengths of 5 and 5.75 Å, for the
NdMox sample. The samples were placed in a Cu sample holder and inserted into
a dilution refrigerator, reaching temperatures below 100 mK. The
(Nd1-yLay)2Zr2O7 spectra present a Gaussian profile at the elastic position,
due to incoherent and Bragg scattering. To extract the low temperature
inelastic magnetic signal, data measured at 5 K were used as a reference.
Nevertheless, because a low energy quasielastic signal is still visible at
this temperature, amplified by the detailed balance factor, these data cannot
be directly used for subtraction. The spectrum measured at 5 K was thus
reduced to a series of energy scans, for each $Q$ position. Each scan was then
fitted in a small energy interval
$-\omega_{1}<\omega<\omega_{1},~{}\omega_{1}\approx$ 0.4 meV to a function
consisting in a Lorentzian function of width $\gamma$ on top a Gaussian
profile of width $\sigma$:

$I_{\rm
model}=I_{o}e^{-4\ln{2}\times(\omega/\sigma)^{2}}+(1+1/(e^{\omega/T}-1))\frac{A}{T}\frac{\omega}{(\omega^{2}+\gamma^{2})}$

The Gaussian and Lorentzian contributions represent the elastic and
quasielastic responses respectively. We then considered the quantity:

$S(Q,\omega)=I_{\rm
exp}(Q,\omega,T)-I_{o}e^{-4\ln{2}\times(\omega/\sigma)^{2}}$

The low energy excitations of the Nd2(Zr1-xTix)2O7 samples were probed on
single crystals on the IN5 (ILL) time-of-flight spectrometer. Data for the
$x=2.5$ % single crystal sample [30] are reported in Ref. 18. The $x=10$ %
sample was measured in the $(100)-(010)$ scattering plane with incident
wavelengths of 6, 8 and 10 Å [31].

##  III Structural properties

Sample | Ti/La occupancy | $x$(O2) | Lattice parameter | $T_{\rm CW}$ | $\mu_{\rm eff}$ | $T_{\rm N}$ | $m_{\rm ord}$ | $\epsilon_{0}$
---|---|---|---|---|---|---|---|---
[%] | [Å] | [mK] | [$\mu_{\rm B}$] | [mK] | [$\mu_{\rm B}$] | [meV]
Nd2Zr2O7 | - | 0.3366 | 10.69 | 175 | 2.45 | 370 | $1.15\pm 0.05$ | 0.075
Nd2Zr1.95Ti0.05O7 | 2.8 | 0.3360 | 10.67 | 145 | 2.46 | 420 | $1.36\pm 0.04$ | 0.07
Nd2Zr1.8Ti0.2O7 | 11.2 | 0.3359 | 10.64 | 180 | 2.49 | 430 | $1.33\pm 0.03$ | 0.075
Nd1.8La0.2Zr2O7 | 9.6 | 0.3346 | 10.69 | 275 | 2.61 | 330 | $1.27\pm 0.06$ | 0.05
Nd1.6La0.4Zr2O7 | 20.5 | 0.3344 | 10.70 | 220 | 2.75 | 240 | $1.25\pm 0.06$ | 0.025
Nd1.2La0.8Zr2O7 | 40 | 0.3330 | 10.73 | 170 | 2.61 | - | - | -
NdMox | - | 0.330 | 10.56 | 30 | 2.60 | 550 | $1.22\pm 0.09$ | 0.11
Table 1:  Structural (at room temperature) and magnetic parameters for the
investigated polycrystalline samples. The values for the pure compound have
been updated compared to Ref. [6, 7] from new measurements on D1B and D2B
(ILL). For all compounds, space group is $Fd\bar{3}m$ (#227), the atoms are:
Nd(La) - $16d$ $(1/2,1/2,1/2)$ position, Zr(Ti,$M$) - $16c$ $(0,0,0)$
position, O1 - $8b$ $(3/8,3/8,3/8)$ position, and O2 - $48f$ $(x,1/8,1/8)$
position. The Curie-Weiss temperatures and effective moments have been
obtained from Curie-Weiss fits between 4 and 50 K for all samples. Note that
the obtained Curie-Weiss temperature is affected by the fitting range, as well
as the demagnetization effects, which may explain the differences with some
values reported in the literature. For Nd1.2La0.8Zr2O7, no ordering
temperature could be observed down to 75 mK. For Nd2Zr2O7, Nd2Zr1.95Ti0.05O7
and Nd2Zr1.8Ti0.2O7, the gap values $\epsilon_{0}$ were obtained for single
crystal samples. Figure 1: (a) X-ray and (b) neutron diffractograms (taken at
HRPT) measured on NdMox (red points) at room temperature together with the
refinement with a pyrochlore structure (black line). The difference between
measurements and calculations is shown in blue. Green bars show the Bragg
positions.

As shown below, all the compounds discussed in this article were found to
crystallize in the $Fd{\bar{3}}m$ space group. Rietveld refinement using
FullProf [32] was performed to refine the lattice parameter as well as the
position of the $48f$ oxygen atom, which is the single positional parameter in
the pyrochlore structure. For the substituted samples, the lanthanum occupancy
could not be determined with a good accuracy as neodymium and lanthanum
elements have very similar scattering lengths.

The Nd2(Zr1-xTix)2O7 samples structural properties are discussed in Ref. [18].
(Nd1-yLay)2Zr2O7 structural properties at ambient temperature were determined
from X-ray diffraction for $y=10$ and 20 % and neutron scattering for $y=40$
%. The percentages of lanthanum occupancy are consistent with the synthesis
expectations. The NdMox structure was refined from X-ray and neutron
diffraction measurements (See Figure 1). Accounting for a possible anti-site
disorder, which could occur between the Nd3+ and the Sc3+ ions, did not
improve the refinement.

The obtained structural properties are summarized in Table 1. They show that
the lattice parameter and the oxygen position are little affected by
substitution in Nd2Zr2O7 based compounds. The lattice parameter seems to
slightly decrease with the introduction of Ti, and to increase with the
introduction of La, which is consistent with Ref. 33. In the case of NdMox,
the lattice parameter is significantly reduced compared to the pure Nd2Zr2O7
compound.

##  IV Crystal electric field

The inelastic neutron scattering spectra recorded at about 1.5 K for all
samples are shown in Figure 2. For the Ti and La-substituted compounds, the
spectra are characterized by two lines, centered around 23-25 and 34-38 meV,
the latter peak being broader. In (Nd1-yLay)2Zr2O7with $y=10$ and 20 %, the
instrument resolution allows to distinguish a shoulder on the higher energy
mode that can be due to the presence of two very close modes. The energy
widths are almost identical (considering the instruments resolution), whatever
the doping (see also Table 2). For the high entropy compound, the situation is
very similar, with excitations centered at about the same energies. Their half
width at half maximum, however, is much larger, about 6-8 meV. Phonon
excitations, characterized by their larger intensity at high $Q$, are also
visible in all samples around 15 meV.

Sample | $E_{1}$ | $\Delta_{1}$ | $E_{2}$ | $E_{3}$ | $\Delta_{2,3}$
---|---|---|---|---|---
Nd2Zr2O7 [8] | 23.4 | - | 34.4 | 35.7 | -
Nd2Zr1.95Ti0.05O7* | 23.2 | 3.9 | 35.9 | 5.9
Nd2Zr1.8Ti0.2O7* | 22.8 | 4.8 | 36.3 | 5.7
Nd1.8La0.2Zr2O7† | 23.8 | 1.9 | 34.5 | 36.5 | 1.5
Nd1.6La0.4Zr2O7† | 24.2 | 1.9 | 35.0 | 38.4 | 2.4
Nd1.2La0.8Zr2O7* | 25.4 | 2 | 37 | 6
NdMox * | 21 | 6 | 36 | 8
Table 2:  Energy and half-width at half maximum of the CEF levels measured on
the different Nd-based pyrochlore compounds, in meV. Compounds marked with a *
were measured on triple axis spectrometers, where the energy resolution is
expected to be about 3.5 and 5 meV at 23 and 36 meV respectively. Compounds
marked with a † were measured on PANTHER with a resolution of about 2 meV
[26]. Figure 2:  Inelastic neutron scattering spectra at $T=1.5-1.6$ K in
Nd2(Zr1-xTix)2O7 with $x=2.5$ and 10 % measured on 2T, (Nd1-yLay)2Zr2O7 with
$y=10$ and 20 % measured on PANTHER and Nd2(TiZrHfScNb)2O7 and
(Nd1-yLay)2Zr2O7 with y=40 % measured on EIGER.

These results should be discussed in the perspective offered by the reference
compound Nd2Zr2O7. Its crystal field scheme has been studied in references [7,
6, 8], and has levels at about 23, 34, 35 and 104 meV. All the studied
compounds thus have very similar schemes compared to the “pure” compound (the
104 meV level could not be observed within the energy range of our
measurements). While this result could have been expected for the La-
substitution, it is more surprising for the other compounds, namely
Nd2(Zr1-xTix)2O7 and NdMox, for which the local environment of the Nd3+ ion is
directly perturbed by the substitutions and the disorder on the $B$ site.

In order to estimate how the small variations in the CEF level energies
reflect in the crystal field coefficients, we have analyzed the data using the
multiplet description of the CEF levels previously used in Refs. 6, 8. In this
case, the crystal field Hamiltonian is written [34]:

$\displaystyle V_{\rm CEF}$ $\displaystyle=$
$\displaystyle\sum_{k=0,2,4,6}B_{k,0}C_{k,0}$ (2)
$\displaystyle+\sum_{k=2,4,6}\sum_{m=1}^{k}B_{k,m}\left(C_{k,-m}+(-1)^{k}C_{k,m}\right)$

The $C_{k,m}$ operators, usually called Wybourne operators, are defined as
$C_{k,m}=\sqrt{\frac{4\pi}{2k+1}}Y_{k,m}$, where $Y_{k,m}$ are the spherical
harmonic functions. The Wybourne coefficients $B_{k,m}$ are considered as
empirical coefficients. The $B_{k,m}$ parameters which are not zero in the
$D_{3d}$ symmetry were computed using the energy level values detailed in
Table 2. The obtained values are close to the ones obtained in the pure
compound and differ only by about 5 % (See Appendix A).

In the Nd2(Zr1-xTix)2O7 and (Nd1-yLay)2Zr2O7 samples, the levels are not
significantly broadened, so one can assume that the Nd3+ crystal field states
are homogeneous within the sample. In the case of NdMox, disorder induces a
larger distribution that could correspond to local variations up to about 35 %
of the $B_{k,m}$ values (See Appendix A). This may affect locally the
properties of the Nd3+ magnetic ion, and in particular the value of the
magnetic moment. Nevertheless, it is worth noting that the dipolar-octupolar
character of the ground doublet, which confers to Nd2Zr2O7 its peculiar
properties, remains present in the obtained wavefunctions. This is especially
interesting since it makes possible the study of the disorder impact on the
collective properties of the Nd substituted compounds, without considering as
a first approximation the local variations in the Nd3+ single ion properties.

##  V Magnetic ground state

For all samples, magnetization measurements performed between 2 and 300 K show
a continuous behavior, excluding the possible existence of magnetic
transitions due to magnetic impurities. Estimations of the Curie-Weiss
temperature $T_{\rm CW}$ and the effective moment $\mu_{\rm eff}$ for all
samples were performed by fitting the inverse susceptibility $\chi^{-1}$
(obtained from magnetization data, assuming that $M$ is linear in $H$) as a
function of temperature using a Curie-Weiss law $\chi=C/(T-T_{\rm CW})$, where
$C=\mu_{0}{\cal N}_{\rm A}\mu_{\rm eff}^{2}/(3k_{\rm B}$) as shown in Figure 3
($\mu_{0}$ is the vacuum permittivity, ${\cal N}_{\rm A}$ the Avogadro
constant, and $k_{\rm B}$ the Boltzmann constant). A Van Vleck contribution
was also included in the fit and gives for all samples values between 0.0035
and 0.0039 emu/mol Nd, in agreement with the literature [6, 8]. The obtained
values are summarized in Table 1. All the Curie-Weiss temperatures are
positive, similarly to the pure compound, which suggests effective
ferromagnetic interactions. Although the determination of $T_{\rm CW}$ is only
approximate (the fit is very sensitive due to the small values, and $T_{\rm
CW}$ is affected by demagnetization corrections), the following tendencies are
observed: in the Ti substituted samples, the Curie-Weiss temperatures are in
the same range or smaller than in the pure sample. Surprisingly, in the La-
substituted samples, where the magnetic Nd3+ ions are diluted, Curie-Weiss
temperatures are larger than in the pure sample, reaching the pure sample
value only for the 40 % substituted sample. In NdMox, $T_{\rm CW}$ is strongly
reduced and is close to zero.

It is also interesting to discuss the obtained effective moments. They are all
larger than in the pure compound, reaching 2.75 $\mu_{\rm B}$ in the
(Nd1-yLay)2Zr2O7 compound with $y=20$ %. In magnetization as a function of
field measurements, the magnetization is almost saturated at 100 mK and 8 T
for all samples (see Appendix B). The magnetization reaches values
corresponding to half of the moment value, which is the expected result for
polycrystalline pyrochlore compounds with local $\langle 111\rangle$
anisotropy [35]. A finite slope is nevertheless observed at high field. While
effective moment values between 2.5 and 2.6 $\mu_{\rm B}$ are consistent with
the crystal electric field parameters deduced from neutron scattering, the
high value of 2.75 $\mu_{\rm B}$ obtained for the 20 % La-substituted compound
does not seem compatible with the INS results. It may be due to an
overestimation of the La substitution rate in the sample, which would
naturally induce an overestimated average Nd magnetic moment.

Figure 3: Inverse approximate susceptibility $H_{i}/M$ vs temperature $T$
between 4 and 50 K with an applied field $H\approx 1000$ Oe, where $H_{i}$ is
the internal field obtained after demagnetization corrections. Lines
correspond to fits using a Curie-Weiss law with a Van Vleck component. The
resulting Curie-Weiss temperatures and effective moments are summarized in
Table 1.

The existence of a magnetic transition below 1 K was probed through AC
susceptibility and magnetization measurements. An antiferromagnetic transition
is observed in all samples except the most diluted (Nd1-yLay)2Zr2O7 sample.
This transition manifests as a maximum in $\chi^{\prime}$ and
$\chi^{\prime\prime}$, whose position does not depend on frequency (see Figure
4), as well as a peak in magnetization. For NdMox, a peak in the specific heat
(see Appendix B) confirms the existence of a transition (Note that it was not
possible to cool down to low enough temperature to reach the transition
temperature of the other samples with our specific heat set-up).

The Néel temperatures of the La-substituted compounds are smaller than in the
pure powder Nd2Zr2O7 (330 and 240 mK for $y=10$ and 20 % respectively), which
may be expected due to the dilution of the magnetic Nd3+ ions. The transition
is even suppressed in the measured range ($T>75$ mK) for the $y=40$ % sample
for which a paramagnetic behavior is observed down to the lowest temperature.
Conversely, in other substituted samples, the Néel temperature $T_{\rm N}$ is
larger (See Table 1), which indicates that disorder has reinforced the
antiferromagnetic ordering. It is worth noting that the ratio $T_{\rm
N}/T_{\rm CW}$ is much larger in NdMox and to a lesser extent in
Nd2(Zr1-xTix)2O7 than in Nd2Zr2O7 and (Nd1-yLay)2Zr2O7 samples, pointing to
different magnetic couplings as discussed below. In all samples, the
susceptibility peaks are wider than in the pure sample, suggesting a small
distribution of the transition temperatures. In addition, in the
(Nd1-yLay)2Zr2O7 samples, and especially in the $y=20$ % one (see Figure
4(b)), the real part of the susceptibility keeps increasing at low
temperature, which may be due to the persistence of disordered Nd magnetic
moments even below the Néel temperature.

Figure 4: AC susceptibility, in-phase $\chi^{\prime}$ (blue, left) and out-of-
phase $\chi^{\prime\prime}$ (red, right), as a function of temperature
measured with an applied field $H_{\rm ac}=1.8$ Oe and $f=0.57$ Hz for
(Nd1-yLay)2Zr2O7 with (a) $y=10$ %, (b) $y=20$ % and for (c) NdMox.

The magnetic structure associated with this magnetic ordering was determined
from neutron diffraction measurements using a Fullprof refinement [32].
Examples of diffractograms measured on (Nd1-yLay)2Zr2O7 and NdMox are shown in
Figure 5(a) and 6(a), where it can be seen that the magnetic structure has a
${\bf k}={\bf 0}$ propagation vector. (220) and (311) peaks have a strong
magnetic intensity at low temperature, while the (002) peak has not, which is
characteristic of the all-in–all-out structure [36] observed in the pure
Nd2Zr2O7 compound [7]. The temperature dependence of the ordered magnetic
moment $m_{\rm ord}$ could also be determined (See Figures 5(b) and 6(b)). At
low temperature, the obtained ordered moment is always larger than in the pure
compound (See Table 1) and reaches about 1.35 $\mu_{\rm B}$ for the Ti-
substituted samples. It is nevertheless far from the full moment of the
Nd3+ion. These results thus suggest that the ground state of all the
substituted compounds is similar to the pure compound, i.e. an
antiferromagnetic all-in–all-out state in which the ordered moment has a
dipolar-octupolar character [15].

Figure 5:  (a) Neutron diffractogram measured on G4.1 for (Nd1-yLay)2Zr2O7
with $y=10$ % at 45 mK (red dots) and 950 mK (blue squares). The red line is a
refinement to an all-in–all-out magnetic structure. (b) Ordered magnetic
moment as a function of temperature determined from the diffractogram. The
line is a guide to the eye. Figure 6:  (a) Neutron diffractogram measured for
NdMox by integrating the inelastic data measured on FOCUS between -0.1 and 0.1
meV at 70 mK (red dots) and 750 mK (blue squares). The red line is a
refinement to an all-in–all-out magnetic structure. (b) Ordered magnetic
moment as a function of temperature determined from the diffractogram. The
line is a guide to the eye.

##  VI Low energy excitations

A key feature of this dipolar-octupolar ordered state is the existence of low
energy excitations with peculiar characteristics: flat mode with pinch points
from which stem dispersive excitations. We have thus performed measurements in
order to check whether these excitations are affected by disorder and
substitutions. In the case of 2.5 % titanium substituted single crystals, we
have shown that the excitations remain almost intact [18]. In the $x=10$ %
sample, these excitations are still present at almost the same energy (the
energy of the flat mode is about 70 $\mu$eV), but features are broadened,
especially the pinch points (see Figure 7).

Figure 7:  Low energy excitations measured on Nd2(Zr1-xTix)2O7 with $x=10$ %
at 50 mK on IN5. (a) Spectra along $(2k0)$ showing the flat mode and the
dispersive branches measured with $\lambda=6$ Å. (b) Energy cut at $E=70\pm
0.05\ \mu$eV showing the broadened pinch points measured with $\lambda=8$ Å.
(c) $Q$-cut measured with $\lambda=10$ Å showing the persistence of a gap in
the excitations.

The lack of single crystals for (Nd1-yLay)2Zr2O7 and NdMox prevents the
observation of pinch points in the excitations of these samples. Nevertheless,
inelastic measurements show in all measured samples the persistence of a flat
mode as can be seen in Figures 8 and 9. The energy $\epsilon_{0}$ of the flat
mode is decreased to about 50 and 25 $\mu$eV in (Nd1-yLay)2Zr2O7 with $y=10$
and 20 % respectively, while it is increased in the NdMox compound
($\epsilon_{0}\approx 100\ \mu$eV). Dispersive excitations are hardly seen in
the NdMox sample but are present in the (Nd1-yLay)2Zr2O7 and qualitatively
similar to the ones of the pure Nd2Zr2O7 compound; still, features are
broadened, so that the gap looks almost suppressed in (Nd1-yLay)2Zr2O7 with
the present resolution (70 $\mu$eV).

Figure 8:  Low energy excitations of (Nd1-yLay)2Zr2O7 measured at 50 mK on
SHARP (a) for $y=10$ % and (c) $y=20$ %.The plotted data were obtained
following the procedure explained in Section II. (b)(d) Calculations performed
with the parameters detailed in Table 3, and including a 60 $\mu$eV broadening
(full width at half maximum) for each spin wave mode in order to mimic the
experimental resolution. Figure 9:  Energy scan in NdMox measured on FOCUS at
70 mK (red squares) and 1 K (blue dots), in which the signal has been
integrated over the $Q$-range $[0.5;1.6]\ \AA^{-1}$.

These results confirm the conclusion reached in the previous section: the
magnetic state in the substituted compounds is of the same nature as in the
parent compound Nd2Zr2O7. Disorder and / or dilution seems to renormalize the
parameters (ordering temperature, ordered magnetic moment, gap) and to induce
some distribution but without affecting strongly the ground state itself.

##  VII Determination of the exchange parameters

To go further in the understanding of the role of disorder, it is necessary to
quantitatively estimate the microscopic parameters which govern the magnetic
state, namely the magnetic couplings.

CEF measurements show that in spite of the loss of symmetry induced by local
disorder, the magnetic moment of the Nd3+ ions is well described by a dipolar-
octupolar wavefunction. As a result, the XYZ Hamiltonian, which was shown to
be relevant for this kind of magnetic ions [16] can be used to investigate the
magnetic couplings in the system. This Hamiltonian is written:

${\cal H}=\sum_{\langle i,j\rangle}\left[{\sf
J}_{x}\tau^{x}_{i}\tau^{x}_{j}+{\sf J}_{y}\tau^{y}_{i}\tau^{y}_{j}+{\sf
J}_{z}\tau^{z}_{i}\tau^{z}_{j}+{\sf
J}_{xz}(\tau^{x}_{i}\tau^{z}_{j}+\tau^{z}_{i}\tau^{x}_{j})\right]$ (3)

where $\tau^{x,y,z}_{i}$ are the components of the pseudo-spin and ${\sf
J}_{a}$ the coupling parameters ($a=x,y,z,xz$). $\tau^{z}$ is oriented along
the local $\langle 111\rangle$ axes of the tetrahedra, and corresponds to the
magnetic component probed by magnetization or neutron scattering (at low $Q$).
The dipolar-octupolar character of the pseudo-spin implies that $\tau^{x}$ and
$\tau^{z}$ components transform like a pseudo-vector (or a dipole) but that
the $\tau^{y}$ component transforms as an octupole.

The ${\sf J}_{xz}$ term can be removed from Equation 3 by rotating the $x$ and
$z$ axes in the $({\bf x},{\bf z})$ plane by an angle $\theta$ [16, 15]:

$\displaystyle{\cal H}_{\rm XYZ}=$ $\displaystyle\sum_{\langle
i,j\rangle}\left[\tilde{\sf
J}_{x}\tilde{\tau}^{\tilde{x}}_{i}\tilde{\tau}^{\tilde{x}}_{j}+\tilde{\sf
J}_{y}\tilde{\tau}^{\tilde{y}}_{i}\tilde{\tau}^{\tilde{y}}_{j}+\tilde{\sf
J}_{z}\tilde{\tau}^{\tilde{z}}_{i}\tilde{\tau}^{\tilde{z}}_{j}\right]$ (4)
$\displaystyle{\rm with}\quad\tan(2\theta)=\frac{2{\sf J}_{xz}}{{\sf
J}_{x}-{\sf J}_{z}}$

where the $\tilde{\sf J}_{a}$ ($a=x,y,z$) are the exchange parameters in the
rotated frame $({\bf\tilde{x}},{\bf\tilde{z}})$ (See Appendix C).

When an all-in–all-out state is stabilized as in the compounds studied here,
the $\theta$ angle is simply related to the ordered magnetic moment $m_{\rm
ord}$ along $\bf z$ and the saturated magnetic moment $m_{\rm sat}$ [15] by
the equation:

$\cos(\theta)=\frac{m_{\rm ord}}{m_{\rm sat}}$ (5)

Accounting for zero-point quantum fluctuations slightly affects this equation,
leading to [15, 14]:

$\cos(\theta)\approx\frac{m_{\rm ord}}{0.9m_{\rm sat}}$ (6)

The coupling parameters can be obtained from the measured magnetic scattering
function $S(Q,\omega)$ using spin wave theory, as previously done for Nd2Zr2O7
and Nd2Zr1.95Ti0.05O7 [18, 14]. In the absence of single crystals, the
dispersing branches in $S(Q,\omega)$ are broadened by the powder average,
making an accurate determination of the parameters difficult. Nevertheless, in
the specific case studied here, distinctive features are present in the
spectra, which enable an estimation of the parameters, even for powder
samples: the gap to the flat mode $\epsilon_{0}$, the energy $\epsilon_{1}$ of
the dispersion at the Brillouin zone boundary (${\bf Q}=(110),\ (112),\
(221)$) and the top of the dispersion at the Brillouin zone center
$\epsilon_{2}$ (${\bf Q}=(220),\ (113)$). They are given by the following
relations [15, 14]:

$\displaystyle\epsilon_{0}$ $\displaystyle=\sqrt{(3|\tilde{\sf
J}_{z}|-\tilde{\sf J}_{x})(3|\tilde{\sf J}_{z}|-\tilde{\sf J}_{y})}$ (7)
$\displaystyle\epsilon_{1}$ $\displaystyle=\sqrt{(3|\tilde{\sf
J}_{z}|+\tilde{\sf J}_{x})(3|\tilde{\sf J}_{z}|+\tilde{\sf J}_{y})}$ (8)
$\displaystyle\epsilon_{2}$ $\displaystyle=3\sqrt{(|\tilde{\sf
J}_{z}|+\tilde{\sf J}_{x})(|\tilde{\sf J}_{z}|+\tilde{\sf J}_{y})}$ (9)

$\epsilon_{0}$ can directly be observed in powder spectra but it is not as
straightforward for $\epsilon_{1}$ and $\epsilon_{2}$. In Nd2Zr2O7 and
Nd2Hf2O7, $\tilde{\sf J}_{y}$ is small compared to the other parameters, we
have thus set it to zero to better constrain the fit. By analyzing energy cuts
at several $Q$, we could then estimate $\epsilon_{2}$ and deduce the
parameters $\tilde{\sf J}_{x}$ and $\tilde{\sf J}_{z}$ of Nd1.8La0.2Zr2O7 and
Nd1.6La0.4Zr2O7, which are reported in Table 3 (note that the sign of
$\tilde{\sf J}_{z}$ is fixed by the all-in–all-out nature of the ground
state). The calculated powder average spectra corresponding to these
parameters are shown in Figures 8(b) and (d).

Sample | $\tilde{{\sf J}}_{x}$ | $\tilde{{\sf J}}_{y}$ | $\tilde{{\sf J}}_{z}$ | $\tilde{{\sf J}}_{x}/\tilde{{\sf J}}_{z}$ |  $\theta$ (rad)
---|---|---|---|---|---
Single crystals
Nd2Zr2O7 [18] | 1.18 | -0.03 | -0.53 | -2.20 | 1.23
Nd2Zr2O7 [14] | 1.06 | 0.16 | -0.53 | -1.97 | 0.98
Nd2Zr1.95Ti0.05O7 [18] | 0.97 | 0.21 | -0.53 | -1.83 | 1.08
Nd2Zr1.8Ti0.2O7 | 1.26 | -0.05 | -0.57 | -2.23 | 1.09
Nd2Hf2O7 [37] | 1.23 | 0.09 | -0.66 | -1.86 | 1.26
Polycrystalline samples
Nd1.8La0.2Zr2O7 | 1.19 | 0 | -0.48 | -2.50 | 0.99
Nd1.6La0.4Zr2O7 | 0.935 | 0 | -0.34 | -2.76 | 1.01
NdMox | 0.25 | 0 | -0.47 | -0.53 | 1.02
Nd2ScNbO7 | -0.07 | 0 | -0.37 | 0.19 | 1.08
Nd2GaSbO7 | -0.44 | 0 | -0.91 | 0.48 | 0.73
Table 3:  Estimated exchange couplings (in K) in Nd3+ pyrochlore magnets.
${\tilde{\sf J}}_{y}$ was fixed to zero in all samples, except Nd2Zr2O7,
Nd2(Zr1-xTix)2O7 and Nd2Hf2O7. $\theta$ was obtained according to Equation 6.
J parameters were determined using the dispersion inferred from inelastic
neutron scattering data only in Nd2Zr2O7, Nd2(Zr1-xTix)2O7, and
(Nd1-yLay)2Zr2O7 compounds (see Appendix D for Nd2Zr1.8Ti0.2O7). For other
ones (in italics), they were deduced from $\epsilon_{0}$, $\theta$ and $T_{\rm
CW}$. Nd2GaSbO7 and Nd2ScNbO7 parameters are deduced from values given in Ref.
38 and 39 respectively, following the procedure detailed in the text.

In systems where low energy inelastic neutron scattering measurements are not
of high enough quality so that only $\epsilon_{0}$ can be determined, the
Curie-Weiss temperature can also be used to estimate the coupling parameters.
Indeed, it is given by the relation [15]:

$T_{\rm CW}=\frac{{\sf J}_{z}}{2}=\frac{\tilde{\sf
J}_{z}\cos^{2}(\theta)+\tilde{\sf J}_{x}\sin^{2}(\theta)}{2}$ (10)

where the coupling parameters are in K. Nevertheless, as previously mentioned,
the determination of the Curie-Weiss temperatures $T_{\rm CW}$ from
susceptibility are far from accurate given their low values, so that the
resulting errors in the coupling parameters may be important. After
determining $\theta$ from Equation 6 and assuming $\tilde{\sf J}_{y}=0$,
$\tilde{\sf J}_{x}$ and $\tilde{\sf J}_{z}$ can thus be determined using
Equations 7 and 10. We used this method to estimate the couplings in NdMox. To
get a broader view, we also applied it to other Nd pyrochlore systems where
$m_{ord}$, $T_{\rm CW}$ and $\epsilon_{0}$ are known from the literature,
namely Nd2ScNbO7 and Nd2GaSbO7. It is however worth noting that when the
exchange parameters could be determined independently, the computed $T_{\rm
CW}$ is far from the experimental one, except in the study by Xu et al. [14].
In addition to the experimental uncertainty in $T_{\rm CW}$, the determination
of $\theta$ may also be at the origin of these discrepancies; the ordered
moment can indeed be underestimated due to thermalization issues. The coupling
parameters obtained by this method may thus be considered cautiously.

##  VIII Field induced behavior

Figure 10: $M/H$ versus temperature measured at several applied magnetic
fields $H$ in (Nd1-yLay)2Zr2O7 with (a) $y=10$ %, (b) $y=20$ % and in (c)
NdMox. All curves were measured using a ZFC-FC procedure.

The behavior of the substituted samples when a magnetic field is applied is
qualitatively similar to the Nd2Zr2O7 sample (except of course for
Nd1.2La0.8Zr2O7 which does not order down to 75 mK), as detailed in Ref. 21
for the Nd2(Zr1-xTix)2O7 compounds. The lack of single crystals for
(Nd1-yLay)2Zr2O7 and NdMox samples prevents from a detailed study under
applied magnetic field. Several key features can nevertheless be observed. As
shown in Figure 10, a freezing is observed below $T_{\rm N}$ that manifests as
a splitting in Zero Field Cooled - Field Cooled (ZFC-FC) magnetization
measurements as a function of temperature. This effect has already been
observed in several pyrochlore compounds, including Nd-based ones, and is
probably due to the trapping of magnetic moments at the domain-walls. This
irreversibility is suppressed when increasing the magnetic field, below 500 Oe
in Nd1.6La0.4Zr2O7, above 500 Oe in Nd1.8La0.2Zr2O7 and above 2000 Oe in
NdMox. This hierarchy is consistent with the different ordering temperatures
observed in the three compounds.

In both (Nd1-yLay)2Zr2O7 compounds, the magnetization at low field keeps
increasing when decreasing the temperature, as previously observed in the ac
susceptibility. This effect is more important in the $y=20$ % sample and may
be due to the existence of Nd spins which remain paramagnetic down to the
lowest temperature due to the dilution.

The field induced behavior was shown to be very similar to the pure compound
in the Nd2(Zr1-xTix)2O7 samples [21]. In (Nd1-yLay)2Zr2O7 and NdMox samples,
the field induced transitions could not be probed due to the polycrystalline
character of the samples. Nevertheless, in NdMox a clear inflection point
could be observed in the isothermal magnetization curve at low temperature
(See Figure 11), which may feature the field induced transition observed along
$[111]$ in Nd2Zr2O7 [7, 13, 14]. The transition field is however one order of
magnitude larger than in Nd2Zr2O7. It reaches 0.225 T at low temperature,
progressively decreases when increasing the temperature, reaching zero at the
Néel temperature.

Figure 11: $M$ versus $H$ at several temperatures in NdMox. The sample was
first “saturated” in -1 T and the field was ramped up to 1 T.

##  IX Discussion

The present study about the effects of disorder on Nd-based pyrochlores
highlights the robustness of the all-in–all-out ordered state with a dipolar-
otcupolar character stabilized in these compounds. Disorder introduced on the
transition metal site is expected to disturb the local symmetry of the Nd3+
ions, and thus to affect the single ion properties. Nevertheless, even if the
$D_{3d}$ symmetry is suppressed on a significant proportion of sites in the
most substituted Nd2(Zr1-xTix)2O7 sample or in NdMox, the crystal electric
field properties are not strongly modified. Disorder manifests as a broadening
of the crystal field excitations but the nature and the position of the levels
remain similar. As a consequence the data can be reproduced by crystal field
wavefunctions close to the ones obtained in pure samples. In particular, the
obtained Nd3+ ground doublet retains the dipolar-octupolar character which
enables the exotic properties reported in the pure compound. The main effect
seems to be on the value of the ground doublet moment, which increases
slightly with disorder.

###  IX.1 Substitution on the non-magnetic site

Upon substituting titanium on the zirconium site, the Curie-Weiss temperature
is only slightly modified, hence a small change of the exchange parameters,
which is not that surprising since structural parameters are very similar in
the substituted and the pure samples (See Table 1). Nevertheless, the Néel
temperature increases with the substitution rate, as well as the all-in–all-
out ordered moment along the $\langle 111\rangle$ directions. This indicates
that the ordering is reinforced and slightly pushed along the local $\langle
111\rangle$ directions, which means a smaller $\theta$ angle. In terms of
interactions, this would imply that the absolute value of the ${\tilde{\sf
J}}_{z}$ parameter is increased compared to ${\tilde{\sf J}}_{x}$ in these
substituted polycrystalline samples.

The exchange parameters determined from inelastic neutron scattering data
analysis on single crystals (see Table 3) do not however provide clear trends.
Strong variations are obtained even in the reported parameters for pure
Nd2Zr2O7 samples, which may be due to distinct fit methods but also result
from different crystal growth protocols. Indeed, these samples additionaly
have different $T_{\rm N}$ and ordered moments. As shown in Ref. 21, it is
also worth mentioning that single crystals have smaller ordering temperature
and ordered moment along ${\bf z}$ than polycrystalline samples. In spite of
the robustness of global properties with respect to substitutions, the details
of the microscopic parameters are thus sensitive to subtle variations in the
composition.

In NdMox, where disorder is much stronger, the Curie-Weiss temperature is
close to zero, while the ordering temperature and the flat mode energy
significantly increase. This corresponds to a much smaller ${\tilde{\sf
J}}_{x}$ value than in the pure compound, while ${\tilde{\sf J}}_{z}$ is
preserved. The $\theta$ value is however little affected, since the ordered
moment along $\langle 111\rangle$ remains similar. This stability of $\theta$
is surprising since it cannot be simply related to a microscopic parameter of
the system (See Equation 4). The main difference between NdMox and other
measured compounds arises from the significant decrease of the lattice
parameter (see Table 1). It is comparable to the one obtained in Nd2Sn2O7 [9],
and Nd2ScNbO7 [39, 40]. Nd2ScNbO7 has similar $T_{\rm CW}$ and ordered moment
as NdMox but a smaller Néel temperature (370 mK). Nd2Sn2O7 has a much larger
Néel temperature (910 mK) and z ordered moment (1.7 $\mu_{\rm B}$). Nd2GaSbO7
[38], which has the smallest reported lattice parameter (10.34 Å) in the
family has properties similar to Nd2Sn2O7. In all these compounds
characterized by a reduced lattice parameter, $|{\tilde{\sf
J}}_{x}/{\tilde{\sf J}}_{z}|$ is significantly decreased. In particular,
${\tilde{\sf J}}_{x}$ strongly decreases and possibly changes sign in
Nd2GaSbO7 and Nd2ScNbO7 (See Table 3). As illustrated in the phase diagram of
Figure 12, the competition between the high temperature Coulomb phase and the
all-in–all-out state is then much reduced, in favor of the all-in–all-out
state. Starting from Nd2Zr2O7 characterized by $|{\tilde{\sf
J}}_{x}/{\tilde{\sf J}}_{z}|\approx 2$, such samples are then located deeper
in the $\tilde{z}$ all-in–all-out phase. Note that this also results in a
higher transition temperature. Further studies are needed to enlighten the
relation between these interaction parameters and microscopic properties.

Figure 12: Sketch of the classical phase diagram for dipolar-octupolar
pyrochlores assuming $\tilde{\sf J}_{z}<0$. The $\tilde{x}$ all-in–all-out
phase is stabilized for $\tilde{\sf J}_{x}<\tilde{\sf J}_{z}<0$, while the
$\tilde{x}$ spin ice is favored for $\tilde{\sf J}_{x}>3|\tilde{\sf J}_{z}|$
(these two phases have their moments oriented along the $\tilde{x}$
direction). In quantum models, this $\tilde{x}$ spin ice phase is a $U(1)$
spin liquid phase, and the phase boundary is slightly shifted [19, 20]. In
between arises the $\tilde{z}$ all-in–all-out (with the moments oriented along
the $\tilde{z}$ direction) observed in pure and substituted Nd2Zr2O7. The
position of the different compounds listed in Table 3 are indicated by colored
rectangles.

###  IX.2 Substitution on the magnetic site

Even more interesting is the case of non-magnetic substitution on the magnetic
site. When diluting up to 20 % the magnetic ions with non-magnetic ions, our
measurements show only quantitative changes in the behavior but the physics
remain the same: (i) the system reaches an ordered all-in–all-out ground state
along the ${\tilde{z}}$ direction; (ii) the apparent reduced ordered moment
along ${\bf z}$ remains in the same range as the pure sample (see Table 1);
(iii) the excitations show the same characteristics, a gapped flat mode
associated to dispersive excitations (see Figure 8). The transition as well as
Curie-Weiss temperatures nevertheless decrease gradually with dilution.
$T_{\rm N}$ is below 75 mK at 40 % dilution, despite a positive $T_{\rm CW}$
of 170 mK and a substitution rate quite far from the percolation threshold (61
% of dilution) [41]. The inelastic spectra for these dilute systems with
$y=10$ and 20 % can be reproduced with average interaction parameters (See
Figure 8 and Table 3). As expected, these parameters decrease with dilution,
but interestingly the ratio $|{\tilde{\sf J}}_{x}/{\tilde{\sf J}_{z}}|$
increases. This is seen experimentally through the decrease of the spin ice
flat mode towards the elastic line. In Figure 12, this corresponds to moving
towards the boundary of the $U(1)$ spin liquid phase [15]. These results
suggest that by weakening the all-in–all-out ordering, magnetic dilution
favors the unconventional Coulomb phase observed in these compounds. To
confirm this hypothesis, it would be interesting to measure the magnetic
diffuse scattering in the 40 % sample, and see whether or not it shows the
spin ice like pattern down to very low temperature.

Figure 13: Calculated inelastic neutron scattering spectra along $(hh0)$ at
zero temperature and for different substitution rates. The top left figure
shows the spin wave calculations, to be compared with the 0 % calculation.

To go further in the understanding of magnetic dilution, we have performed
numerical simulations of the spin dynamics in a dilute system (see Appendix
E), assuming exchange parameters close to the pure system ($\tilde{\sf
J}_{x}=1,~{}\tilde{\sf J}_{z}=-0.5$ K). As shown in Figure 13, when the
dilution rate increases, the intensity of the dispersive mode decreases, while
its width increases. The flat mode persists even at high substitution rates,
but its position in energy decreases, approaching the elastic line. It
condenses for a substitution rate of about 50 %. Supernumerary flat levels
below the gap to the main flat mode are clearly present in the systems with
low dilution levels. We believe they reflect the presence of sites that no
longer have six, but less neighbors due to substitution. These calculations
are consistent with the dynamics observed in (Nd1-yLay)2Zr2O7 samples, i.e.
the persistence of the flat and dispersive modes, the broadening of the modes
and the decrease of the gap. This suggests that the microscopic exchange
parameters may not be strongly affected by the dilution on the Nd site. The
apparent increase of $|{\tilde{\sf J}}_{x}/{\tilde{\sf J}_{z}}|$ determined
based upon the average model is thus a consequence of the dilution and of the
random fields it creates, which apparently tend to favor the disordered
Coulomb phase with respect to the all-in–all-out ordered state.

##  X Conclusions

Our study of substitutions in Nd2Zr2O7 samples, on both the magnetic and non
magnetic sites, demonstrate the remarkable stability of the dipolar-octupolar
properties. The ground state remains this unusual all-in–all-out
antiferromagnetic state, characterized by a partially ordered magnetic moment
along the local $\langle 111\rangle$ directions. The excitations, which
exhibit both a flat mode and a dispersive (the so-called dynamic fragmentation
[15]), also persist in the samples where ordering could be observed.

In all studied samples here and in the literature, substitution on the non-
magnetic site, or replacement of zirconium by another element, tend to
decrease the lattice parameter. Concomitantly, our study tends to show that
the ratio $|{\tilde{\sf J}}_{x}/{\tilde{\sf J}}_{z}|$ decreases, which means
that the system moves away from the $U(1)$ spin liquid state [15]. While no
clear picture emerges up to now to explain the value of these couplings, this
suggests that in order to reach this unconventional spin liquid state in Nd-
based pyrochlore compounds, one should try to expand the lattice parameter.
The dilution on the magnetic site may be an option, since (Nd1-yLay)2Zr2O7
systems show an increase of $|{\tilde{\sf J}}_{x}/{\tilde{\sf J}}_{z}|$. The
presence of specific diffuse scattering in the more diluted sample, that would
indicate the persistence of the Coulomb phase, nevertheless remains to be
investigated.

##  XI Acknowledgments

We gratefully acknowledge E. Suard and C. Colin for additional diffraction
experiments at D2B and D1B at ILL respectively, C. Paulsen for the use of his
magnetometers and N. Dragoe.

This work is based on experiments performed at the Swiss spallation neutron
source SINQ, Paul Scherrer Institute, Villigen, Switzerland.

We acknowledge financial support from the Agence Nationale de la Recherche
under Grant No. ANR-19-CE30-0040 and ANR 19-CE30-0030-01, as well as from the
2FDN. The work at the University of Warwick was supported by EPSRC through
grant EP/T005963/1.

##  Appendix A CEF

The $B_{k,m}$ values determined from the CEF energy levels of the different
samples are summarized in Table 4. The corresponding wavefunctions for the
pure Nd2Zr2O7 sample are detailed in Table 5.

Sample | $B_{2,0}$ | $B_{4,0}$ | $B_{4,3}$ | $B_{6,0}$ | $B_{6,3}$ | $B_{6,6}$
---|---|---|---|---|---|---
Nd2Zr2O7 [8] | 49.2 | 408.9 | 121.6 | 148.1 | -98.0 | 139.1
Nd2Zr1.95Ti0.05O7 | 54.1 | 408.9 | 121.6 | 140.7 | -107.8 | 139.1
Nd2Zr1.8Ti0.2O7 | 52.9 | 408.9 | 121.6 | 137.0 | -112.7 | 139.1
Nd1.8La0.2Zr2O7 | 51.7 | 429.3 | 124.6 | 144.4 | -98.0 | 135.6
Nd1.6La0.4Zr2O7 | 54.1 | 449.8 | 133.8 | 148.1 | -102.9 | 132.1
Nd1.2La0.8Zr2O7 | 61.5 | 460.0 | 121.6 | 129.6 | -85.7 | 139.0
NdMox | 56.6 | 408.9 | 139.8 | 137.0 | -112.7 | 128.7
Table 4:  Wybourne coefficients in meV obtained for the different samples.
Figure 14: Energy of the crystalline field levels obtained by varying the
$B_{k,m}$ coefficients in an interval of $\pm$ 5 (red), 10 (yellow), 20
(green), 30 (cyan) and 50 (blue) % around the nominal values of Ref. 8. The
vertical and horizontal lines depict the variation range observed in the
various samples.

These parameters have been estimated from the average position of the energy
levels only. To estimate the uncertainty on these parameters, the energy
levels have been calculated in a systematic way, by varying the $B_{k,m}$
coefficients in an interval of 5, 10, 20, 30 and 50 % around the values
determined in Ref. 8 for Nd2Zr2O7 (See line 1 of Table 4).

Figure 14 presents the different CEF energy levels obtained, showing
$E_{2,3,4}$ as a function of $E_{1}$, in the form of a scatter graph. On those
graphs are also shown the energy range determined by INS experiments. These
calculations thus allow us to estimate the uncertainty range on the $B_{k,m}$
values still compatible with measurements. They also indicate which local
variation of the $B_{k,m}$ parameters could induce the broadening observed in
certain experiments.

$m_{J}$ | $E_{0}$ | $E^{\prime}_{0}$ | $E_{1}$ | $E^{\prime}_{1}$ | $E_{2}$ | $E^{\prime}_{2}$ | $E_{3}$ | $E^{\prime}_{3}$ | $E_{4}$ | $E^{\prime}_{4}$
---|---|---|---|---|---|---|---|---|---|---
0 | 23 meV | 34 meV | 35 meV | 104 meV
$J=9/2$ |  |  |  |  |  |  |  |  |  |
$-9/2$ | 0.894 |  |  |  | -0.346 | 0.223 |  |  |  |
$-7/2$ |  |  |  | 0.148 |  |  | -0.510 |  | 0.833 |
$-5/2$ |  |  | 0.735 |  |  |  |  | -0.486 |  | 0.442
$-3/2$ | 0.261 | 0.336 |  |  | 0.842 | -0.294 |  |  |  |
$-1/2$ |  |  |  | 0.652 |  |  | 0.681 |  | 0.306 |
$+1/2$ |  |  | 0.652 |  |  |  |  | 0.681 |  | -0.306
$+3/2$ | -0.336 | 0.261 |  |  | -0.294 | 0.842 |  |  |  |
$+5/2$ |  |  |  | 0.735 |  |  | -0.486 |  | -0.442 |
$+7/2$ |  |  | -0.148 |  |  |  |  | 0.510 |  | 0.833
$+9/2$ |  | 0.894 |  |  | -0.223 | -0.346 |  |  |  |
$J=11/2$ |  |  |  |  |  |  |  |  |  |
$-11/2$ |  |  |  |  |  |  |  | -0.082 |  | 0.075
$-9/2$ | 0.109 |  |  |  |  | 0.145 |  |  |  |
$-7/2$ |  |  |  | -0.055 |  |  | 0.147 |  |  |
$-5/2$ |  |  | 0.053 |  |  |  |  | -0.079 |  | -0.065
$-3/2$ |  |  |  |  |  | -0.098 |  |  |  |
$-1/2$ |  |  |  |  |  |  |  |  | 0.059 |
$+1/2$ |  |  |  |  |  |  |  |  |  | 0.059
$+3/2$ |  |  |  |  | -0.098 |  |  |  |  |
$+5/2$ |  |  |  | 0.053 |  |  | -0.079 |  | 0.065 |
$+7/2$ |  |  | 0.055 |  |  |  |  | -0.147 |  |
$+9/2$ |  | 0.109 |  |  | -0.145 |  |  |  |  |
$+11/2$ |  |  |  |  |  |  | -0.082 |  | -0.075 |
Table 5:  Coefficients of the Nd3+ wavefunction expansion in Nd2Zr2O7.

##  Appendix B Complementary macroscopic measurements

The magnetization curves measured at low temperature (typically around 100 mK)
on all the studied samples are shown in Figure 15, showing a finite slope at
high field.

Figure 15: Magnetization $M$ vs magnetic field $H$ measured in all samples up
to 8 T at about 100 mK.

The temperature dependence of the NdMox specific heat is shown on Figure 16,
showing the magnetic transition at 550 mK.

Figure 16: Magnetic contribution to the specific heat $C$ vs temperature $T$
measured in NdMox. The YMox specific heat was subtracted from the raw data
after a correction due to the different molecular weigths [42].

##  Appendix C XYZ Hamiltonian

To model the low energy properties of Nd3+ pyrochlore magnets, we have
introduced the XYZ Hamiltonian in Equations 3 and 4. The ${\sf J}$ and
$\tilde{{\sf J}}$ parameters in these equations are connected by the following
relations:

$\begin{cases}\tilde{{\sf J}}_{x}=&{\sf J}_{x}\cos^{2}\theta+{\sf
J}_{z}\sin^{2}\theta+{\sf J}_{xz}\sin 2\theta\\\ \tilde{{\sf J}}_{z}=&{\sf
J}_{x}\sin^{2}\theta+{\sf J}_{z}\cos^{2}\theta-{\sf J}_{xz}\sin 2\theta\\\
\tilde{{\sf J}}_{y}=&{\sf J}_{y}\end{cases}$

or

$\begin{cases}\tilde{{\sf J}}_{x}=&\frac{1}{2}\left({\sf J}_{x}+{\sf
J}_{z}+\sqrt{({\sf J}_{x}-{\sf J}_{z})^{2}+4{\sf J}_{xz}^{2}}\right)\\\
\tilde{{\sf J}}_{z}=&\frac{1}{2}\left({\sf J}_{x}+{\sf J}_{z}-\sqrt{({\sf
J}_{x}-{\sf J}_{z})^{2}+4{\sf J}_{xz}^{2}}\right)\\\ \tilde{{\sf J}}_{y}=&{\sf
J}_{y}\end{cases}$

and conversely:

$\begin{cases}{\sf J}_{xz}=&(\tilde{{\sf J}}_{x}-\tilde{{\sf
J}}_{z})\sin\theta\cos\theta\\\ {\sf J}_{x}=&\tilde{{\sf
J}}_{x}\cos^{2}\theta+\tilde{{\sf J}}_{z}\sin^{2}\theta\\\ {\sf
J}_{z}=&\tilde{{\sf J}}_{z}\cos^{2}\theta+\tilde{{\sf J}}_{x}\sin^{2}\theta\\\
\end{cases}$

##  Appendix D Interaction parameters for Nd2Zr1.8Ti0.2O7

Inelastic neutron scattering on IN5 with the field applied along $[001]$
allowed us to determine the dispersion in the Nd2Zr1.8Ti0.2O7 sample, and thus
to access the parameters of the XYZ Hamiltonian using the Spinwave software
[43, 44]. The obtained parameters are detailed in Table 3, and the
measurements together with the calculations are shown in Figure 17.

Figure 17: (a) Inelastic neutron scattering spectra measured in
Nd2Zr1.8Ti0.2O7 on IN5 with the field applied along $[001]$, and (b)
calculated dispersion with the parameters of Table 3.

##  Appendix E Numerical simulations

To better understand the remarkable stability of the all-in–all-out ground
state in Nd pyrochlores, numerical simulations of spin dynamics have been
carried out. To this end, a finite size system containing $N\times N\times N$
unit cells of the pyrochlore lattice was considered, among which $n$ sites
were chosen at random, to simulate the substitution by lanthanum. These sites
are therefore empty. As a first step, the mean-field equation based on
Equation 3 was solved until convergence. As a function of $n$, the pseudo-spin
ordered magnetic moment decreases gradually. It does not reach exactly zero at
the percolation threshold ($n/(16N^{3})\approx 0.61$), as expected due to the
finite size of the system [45].

Secondly, spin dynamics were calculated by solving the classical equations of
motion. This includes integrating the differential equations governing the
classical temporal evolution of a pseudo-spin at a given site. At the end of
this stage, the correlations $\langle S_{i}S_{j}(t)\rangle$ are computed,
which are Fourier transformed to obtain spectra as a function of energy
transfer $\omega$. As Nd3+ is a dipolar-octupolar Kramers ion, the actual
magnetic moment is related to the pseudo-spin as $S_{i}=\eta_{i}\tau^{z}_{i}$,
where $\eta_{i}$ denotes the local $\langle 111\rangle$ direction. In
addition, to benchmark the simulations, the $n=0$ calculations are compared to
the spin-wave spectrum. Exchange parameters close to that of the pure sample
$\tilde{\sf J}_{x}=1,~{}\tilde{\sf J}_{z}=-0.5$ K were used systematically.
Results are shown in Figure 13.

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