††thanks: This author contributed equally to this work††thanks: This author
contributed equally to this work

# Traversing the pyrochlore stability diagram; microwave-assisted synthesis
and discovery of mixed B-site Ln2InSbO7 family

Brenden R. Ortiz
<EMAIL_ADDRESS>Materials Department and California Nanosystems
Institute, University of California Santa Barbara, Santa Barbara, CA 93106,
USA  Paul M. Sarte
<EMAIL_ADDRESS>Materials Department and California Nanosystems Institute,
University of California Santa Barbara, Santa Barbara, CA 93106, USA  Ganesh
Pokharel  Materials Department and California Nanosystems Institute,
University of California Santa Barbara, Santa Barbara, CA 93106, USA  Michael
Garcia  Materials Department and California Nanosystems Institute, University
of California Santa Barbara, Santa Barbara, CA 93106, USA  Marcos Marmolejo
Materials Department and California Nanosystems Institute, University of
California Santa Barbara, Santa Barbara, CA 93106, USA  Stephen D. Wilson
<EMAIL_ADDRESS>Materials Department and California Nanosystems
Institute, University of California Santa Barbara, Santa Barbara, CA 93106,
USA

###### Abstract

The lanthanide pyrochlore oxides Ln2B2O7 are one of the most intensely studied
classes of materials within condensed matter physics, firmly centered as one
of the pillars of frustrated magnetism. The extensive chemical diversity of
the pyrochlores, coupled with their innate geometric frustration, enables
realization of a wide array of exotic and complex magnetic ground states.
Thus, the discovery of new pyrochlore compositions has been a persistent theme
that continues to drive the field in exciting directions. The recent focus on
the mixed $B$-site pyrochlores offers a unique route towards tuning both local
coordination chemistry and sterics, while maintaining a nominally pristine
magnetic sublattice. Here, we present a broad overview of the pyrochlore
stability field, integrating recent synthetic efforts in mixed B-site systems
with the historically established Ln2B2O7 families. In parallel, we present
the discovery and synthesis of the entire Ln2InSbO7 family (Ln: La, Pr, Nd,
Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu) located near the boundary of the
pyrochlore stability field using a rapid, hybrid mechanicochemical/microwave-
assisted synthesis technique. Magnetic characterization on the entire class of
compounds draws striking parallels to the stannate analogs, suggesting that
these compounds may host a breadth of exotic magnetic ground states.

††preprint: APS/123-QED

##  I Introduction

The cubic pyrochlore structure A2B2O7 (Fig. 1) is considered as one of the
canonical frustrated lattices in three dimensions Subramanian  _et al._
(1983); Gardner  _et al._ (2010); Greedan (2006); Rau and Gingras (2019);
Hallas  _et al._ (2018). In the case of the lanthanide pyrochlore oxides
(Ln2B2O7), the trivalent lanthanide Ln3+ ions form the hallmark frustrated
network of corner-sharing tetrahedra (Fig. 1). Even considering only the
lanthanide pyrochlore oxides, the number of possible combinations of Ln3+ and
B4+ is staggering Subramanian  _et al._ (1983); Gardner  _et al._ (2010);
Fuentes  _et al._ (2018); Lumpkin and Aughterson (2021). The flexibility to
decorate the frustrated lattice with lanthanide elements of varying single-ion
anisotropy, moment size, and radii – combined with the steric and chemical
flexibity offered by the B-site – have cemented the pyrochlore family as a
fruitful arena for the pursuit of the emergence of exotic and novel magnetic
ground states. Experimental and theoretical studies alike have proposed the
Ln2B2O7 pyrochlores as a means to realize a myriad of properties ranging from
unconventional long-range order Champion  _et al._ (2003); Savary  _et al._
(2012); Zhitomirsky  _et al._ (2012); Wong  _et al._ (2013); Champion and
Holdsworth (2004); Rau and Gingras (2019); Hallas  _et al._ (2018); Gardner
_et al._ (2010); Greedan (2006); Li  _et al._ (2016); Raju  _et al._ (1999);
Subramani  _et al._ (2021), unconventional spin-glass behavior Reimers  _et
al._ (1988); Clark  _et al._ (2014); Raju  _et al._ (1992); Greedan (2006);
Zhou  _et al._ (2008a); Singh and Lee (2012); Jana  _et al._ (2003); Shinaoka
_et al._ (2011); Zhou  _et al._ (2010); Taniguchi  _et al._ (2009); Reimers
_et al._ (1991a); Gaulin  _et al._ (1992); Gingras  _et al._ (1997),
topologically-nontrivial electronic states Guo and Franz (2009a, b); Pesin and
Balents (2010); Otsuka  _et al._ (2021); Hozoi  _et al._ (2014); Kargarian
_et al._ (2011); Yang and Kim (2010); Ezawa (2018); Shinaoka  _et al._ (2014),
spin ices Harris  _et al._ (1997); Bramwell  _et al._ (2001); Harris  _et al._
(1998); Ramirez  _et al._ (1999); Bramwell and Gingras (2001); Bramwell and
Harris (2020); den Hertog and Gingras (2000); Anderson (1956); Champion and
Holdsworth (2004); Ruff  _et al._ (2005); Yavors’kii  _et al._ (2008); Harris
_et al._ (1998); Melko and Gingras (2004), magnetic monopole quasiparticles
Sarte  _et al._ (2017); Morris  _et al._ (2009); Castelnovo  _et al._ (2008);
Bramwell  _et al._ (2009); Paulsen  _et al._ (2016), quantum spin liquids Gao
_et al._ (2019); Benton (2018); Wen  _et al._ (2017); Savary and Balents
(2017); Sibille  _et al._ (2015); Lee  _et al._ (2012); Balents (2010); Ross
_et al._ (2011a); Savary and Balents (2013, 2016); Zhou  _et al._ (2017),
cooperative paramagnetism Gardner  _et al._ (1999, 2001); Ueland  _et al._
(2010); Conlon and Chalker (2009); Keren  _et al._ (2004), and other spin-
liquid-like states Motome and Furukawa (2010); Liu  _et al._ (2019a).

Figure 1: Isometric view of the cation sublattice of the cubic $Fd\bar{3}m$
pyrochlore lattice (left), where the lanthanide sublattice consists of a 3D
network of corner-sharing tetrahedra. The pyrochlore structure is an ordered
superstructure variant of the defect fluorite structure, where the
8-coordinate A- and 6-coordinate B-site sites are structurally distinct
(right).

Considering the rich interplay between chemistry and physics present in the
pyrochlore materials, the discovery of new pyrochlore compositions has been a
constant theme over the last few decades Subramanian  _et al._ (1983); Gardner
_et al._ (2010); Rau and Gingras (2019); Hallas  _et al._ (2018). Given the
vast chemical space, the community naturally sought out empirical “rules” to
predict the relative stability of pyrochlore structures using simple chemical
metrics. These approaches are reminiscent of the Goldschmidt tolerance factor
for perovskites Goldschmidt (1926), and aim to provide simple guidelines and
chemical intuition. Within pyrochlores, the “radius-ratio rules” have been the
most successful, which constrain $R_{\rm{min}}$ $\leq$
$R{\rm{{}_{Ln^{3+}}}}$/$R{\rm{{}_{B^{4+}}}}$ $\leq$ $R_{\rm{max}}$ Subramanian
_et al._ (1983); Gardner  _et al._ (2010); Wiebe and Hallas (2015); Zhou and
Wiebe (2019). Despite the apparent simplicity and variations in the specifics
of $R_{\rm{min}}$ and $R_{\rm{max}}$, the radii-ratio rules have been applied
with great success Subramanian  _et al._ (1983); Gardner  _et al._ (2010).
Naturally, the diversity of experimentally accessible Ln2B2O7 pyrochlore
phases saw a rapid expansion in the 1960’s Wiebe and Hallas (2015),
particularly with rapid technological advances in extreme synthesis
conditions, particularly those at high pressure. Human ingenuity has continued
to realize metastable Ln2B2O7 phases considered outside the pyrochlore
stability field, including the germanates Hallas  _et al._ (2014, 2012); Zhou
_et al._ (2012); Li  _et al._ (2014); Hallas  _et al._ (2016a) and plumbates
Hallas  _et al._ (2015).

Beyond the search for the prototypical Ln2B2O7 phases, one can also consider
permutations that build upon the basic structure while maintaining the
hallmark frustrated Ln3+ sublattice. The first mention of mixed B-site
pyrochlore phases emerged with Blöte Blöte  _et al._ (1969), where the six
coordinate B4+ ion was replaced by the isoelectronic combination of
B${}_{i}^{3+}$B${}_{ii}^{5+}$. This is reminiscent of exploratory searches for
more complex diamond-like semiconductors where silicon can be split
iteratively into increasingly complex chemical compositions (e.g. Si, GaAs,
CuGaSe2) bound by charge balance. Unlike the diamond-like semiconductors,
where the permutations reduce the global symmetry of the unit cell (Si:
$Fd\bar{3}m$, GaAs: $F\bar{4}3m$), the B-site in
Ln2B${}_{\text{i}}^{3+}$B${}_{\text{ii}}^{5+}$O7 is nominally disordered,
yielding an average structure with the same global symmetry as the ternary
pyrochlore ($Fd\bar{3}m$). Despite the potential for a large number of
additional permutations on the pyrochlore structure, systematic studies of the
mixed B-site pyrochlores have been scarce so far Subramanian  _et al._ (1983).

Part of the historical hesitation behind the mixed B-site pyrochlores comes
from the negative connotation of “disorder” and its association with “dirty”
samples Martin  _et al._ (2017). Recently, however, there has been a surge of
renewed interest in the mixed B-site pyrochlores Gomez  _et al._ (2021); Sarte
_et al._ (2021); Nandi  _et al._ (2021); Rutherford (2021); Beare (2021);
Mauws  _et al._ (2021), driven by the emerging picture of the profound effects
that disorder can play on cooperative magnetic properties. Recent experimental
studies on Pr2Zr2O7 suggest the possibility that disorder may in fact provide
a key route in the realization of both exotic, and possibly novel magnetic
ground states Wen  _et al._ (2017); Martin  _et al._ (2017). The studies that
do exist for mixed B-site pyrochlores are fascinating in their own right, and
the most thorough investigations have indicated that the mixed Ga3+Sb5+
compositions are remarkably similar to the Ln2Ti2O7 titanates in both
structure and properties Gomez  _et al._ (2021); Sarte  _et al._ (2021); Blöte
_et al._ (1969); Nandi  _et al._ (2021). As such, we postulate that the
richness of the Ln2B2O7 phases may be mirrored in their mixed B-site
counterparts, but with additional chemical degrees of freedom.

In this work we aim to provide the first systematic study of a mixed B-site
family throughout the entire $f$-block in the context of the rich history of
the lanthanide Ln2B2O7 phases. We begin by providing an updated pyrochlore
stability field diagram, integrating recent synthetic efforts in the
metastable Ln2B2O7 phases and mixed B-site Ln2B${}_{i}^{3+}$B${}_{ii}^{5+}$O7
pyrochlores. Simultaneously, we present our synthesis and discovery of the
Ln2InSbO7 series of compounds. These compounds serve as a structural analog to
the Ln2Sn2O7 stannates, much as the Ln2GaSbO7 family corresponds to the
respective titanates. The Ln2InSbO7 compounds straddle the pyrochlore
stability window, and producing high-quality powders using classical synthesis
methods is difficult Strobel  _et al._ (2010). As such, we also present a new
synthesis technique – a hybrid mechanochemical-microwave method – for the
pyrochlore oxides which can produce phase-pure crystalline material in $<$30
min heat treatments. We discuss the magnetic properties of the entire
Ln2InSbO7 series down to 60 mK, analyzing the data in the context of other
Ln2B2O7 analogs and other mixed B-site pyrochlores. Our work demonstrates the
rich and diverse role the mixed B-site materials can play when deciphering
complex structure-property relationships in the lanthanide pyrochlores.

##  II Experimental Methods

###  II.1 Synthesis

Polycrystalline samples of the Ln2InSbO7 family (Ln: La, Pr, Nd, Sm, Eu, Gd,
Tb, Dy, Ho, Er, Tm, Yb, Lu) were synthesized from stoichiometric amounts of
the pre-dried lanthanide oxides: La2O3 99.999 % Alfa, Pr6O11 99.99 % Alfa,
Nd2O3 99.99 % Alfa, Sm2O3 99.9 % Alfa, Eu2O3 99.9 % Alfa, Gd2O3 99.99 % Alfa,
Tb4O7 99.9 % Alfa, Dy2O3 99.9 % Alfa, Ho2O3 99.99 % Alfa, Er2O3 99.9 % Alfa,
Er2O3 99.9 % Alfa, Tm2O3 99.9 % Alfa, Yb2O3 99.998 % Alfa, Lu2O3 99.999 %
Alfa, and the two metalloid oxides: In2O3 (99.9 % Alfa) and Sb2O5 (99.998 %
Alfa). Two synthetic routes were investigated.

The first corresponds to a method building on the classical heating profiles
previously reported for similar pyrochlores, while integrating
mechanicochemical methods. Stoichiometric mixtures of the pre-dried oxide and
the two metalloid oxides were combined into a tungsten carbide ball-mill vial
and milled for 60 min. The resulting powders were extracted, ground in an
agate mortar, and sieved through a 50 $\mu$m sieve. These powders were loaded
into 2 mL high-density alumina crucibles (CoorsTek) and annealed in a box
furnace at 1250°C for 48 h.

The second route involves abandoning classical heating profiles and instead
combines mechanicochemical methods with microwave-assisted synthesis.
Stoichiometric mixtures of the pre-dried oxide and the two metalloid oxides
were combined into tungsten carbide ball-mill vials and milled for an initial
60 min cycle. The resulting powder was extracted, ground in an agate mortar to
break up any agglomerates, and milled again for an additional 90 min. After
this second cycle, 10 mL of anhydrous ethanol was added to the vial, and the
sample was milled a third time for an additional 30 min. The resulting slurry
was dried in a 80°C oven, resulting in an extremely fine, homogenous powder.

Each precursor powder was loaded directly into a 2 mL alumina crucible
(CoorsTek), which was subsequently nested within a larger 10 mL alumina
crucible filled with 7 g of granular activated charcoal (DARCO 12-20 mesh,
Sigma-Aldrich). The crucibles were then placed within an assembly of low
density alumina foam measuring approximately 15$\times$15$\times$15 cm.
Finally, the total assembly was placed in a 2.45 GHz multimode microwave oven
with a maximum output of 1.2 kW (Panasonic model NN-SN651B). For samples
containing large lanthanide ions (La–Gd), the alumina crucible was exposed
directly to air. For smaller lanthanide ions (Tb–Lu), an alumina frit and
secondary 2 mL crucible were used as an effective “lid” to reduce heat loss
and increase the maximum temperature.

Phase purity was examined with powder x-ray diffraction (XRD) measurements at
room temperature on a Panalytical Empyrean diffractometer (Cu
K${}_{\alpha_{1,2}}$) in standard Bragg-Brentano ($\theta$-2$\theta$)
geometry. Rietveld refinements of the powder XRD patterns were performed using
TOPAS Academic v6 Coelho (2018). Structural models and visualization utilized
the VESTA software package Momma and Izumi (2011).

###  II.2 Magnetic characterization: dc & ac magnetic susceptibility

Measurements of the temperature- and field-dependent dc magnetization were
performed on a 7 T Quantum Design Magnetic Property Measurement System (MPMS3)
SQUID magnetometer in vibrating-sample magnetometry (VSM) mode. Powder of each
Ln2InSbO7 sample was placed in a polypropylene capsule and subsequently
mounted in a Quantum Design brass holder. The zero-field-cooled (ZFC) and
field-cooled (FC) DC magnetization was collected continuously in sweep mode
with a ramp rate of 2 K/min in the presence of an external DC field of 10000
Oe. The ZFC isothermal dc magnetization was collected continuously in sweep
mode with a ramp rate of 100 Oe/sec.

The temperature dependence of the ac magnetization was measured on a Quantum
Design 14 T Dynacool Physical Property Measurement Systems (PPMS) employing
the ac susceptibility option for the dilution refrigerator (ACDR). Phase-pure
powder of each member of the Ln2InSbO7 was cold pressed with a Carver press,
and a portion of the resulting pellet with approximate dimensions of 1x1x0.5
mm was adhered to a sapphire sample mounting post with a thin layer of GE
varnish. All ac measurements were collected under ZFC conditions in the
absence of an external dc magnetic field.

##  III Results & Discussion

###  III.1 Pyrochlore Stability Field

Part of the allure of the pyrochlore lattice is the chemical diversity offered
by the choice of A-site and B-site elements Subramanian  _et al._ (1983);
Gardner  _et al._ (2010); Lumpkin and Aughterson (2021); Talanov and Talanov
(2021); Trump  _et al._ (2018); Rau and Gingras (2019); Hallas  _et al._
(2018); Wiebe and Hallas (2015); Zhou and Wiebe (2019). As established
previously, the community leverages the empirically-derived “radius-ratio
rules” to predict and synthesize new pyrochlore materials. For trivalent
lanthanide oxide pyrochlores Ln2B2O7, the relationship was previously
established decades ago as 1.46 $\leq$
$R{\rm{{}_{Ln^{3+}}}}$/$R{\rm{{}_{B^{4+}}}}$ $\leq$ 1.80 Subramanian  _et al._
(1983). More recently, the rules are often quoted as 1.36 $\leq$
$R{\rm{{}_{Ln^{3+}}}}$/$R{\rm{{}_{B^{4+}}}}$ $\leq$ 1.71 Gardner  _et al._
(2010); Wiebe and Hallas (2015); Zhou and Wiebe (2019). The pyrochlore
stability rules have been applied with great success in discovery of stable
and metastable Ln2B2O7 phases and continue to play a fundamental role in
materials discovery.

While the applicability of the “radii-ratio rules” within the mixed B-site
systems is assumed, there has not been a exhaustive review of the B-site
systems in the context of the pyrochlore stability field. Recall that mixed
B-site alloys must additionally observe the charge neutrality conditions in
the permutation of Ln2B2O7 compositions to
Ln2B${}_{\text{i}}^{3+}$B${}_{\text{ii}}^{5+}$O7. As such, there are a
relatively finite number of combinations that can be expected to obey both the
radii-rules and charge-neutrality conditions. Figure 2 provides a graphical
review of known trivalent lanthanide pyrochlore-like oxide (gray) compositions
of the form Ln2B2O7 or Ln2B${}_{\text{i}}^{3+}$B${}_{\text{ii}}^{5+}$O7
Sleight (1969); Subramanian  _et al._ (1983); Klee and Weitz (1969); Michel
_et al._ (1974); Perez y Jorba (1962); Collongues  _et al._ (1961); Roth
(1956); Fomina and Pal’guev (1977); Subramanian  _et al._ (1983); Gundovin
_et al._ (1975); Klee and Weitz (1969); Besson  _et al._ (1966); Spiridinov
_et al._ (1968); Sibille  _et al._ (2016a); Shlyakhtina  _et al._ (2006);
Strobel  _et al._ (2010); Shlyakhtina  _et al._ (2014, 2018); Song  _et al._
(2019); Sato  _et al._ (1976); Shlyakhtina  _et al._ (2014, 2018); McQueen
_et al._ (2008); Zouari  _et al._ (2008); Subramanian  _et al._ (1983);
Kennedy  _et al._ (1997); Strobel  _et al._ (2010); Ortiz  _et al._ ; Clark
_et al._ (2013); Subramanian  _et al._ (1983); Greedan  _et al._ (1987);
Subramanian  _et al._ (1980); Tkachenko and Fedorov (2003); McCarthy (1971);
Ranganathan  _et al._ (1983); Shaplygin and Lazarev (1973); Sleight (1968);
Chien and Sleight (1978a); Subramanian  _et al._ (1983); Matsuhira  _et al._
(2011a); Klicpera  _et al._ (2020); Blacklock and White (1980); Yanagishima
and Maeno (2001); Lazarev and Shaplygin (1978); Ismunandar  _et al._ (2000);
Berndt  _et al._ (2015); Knop  _et al._ (1968); Montmory and Bertaut (1961);
Subramanian  _et al._ (1983); Taira  _et al._ (1999); Gaultois (2015); Kennedy
(1995); Bertaut  _et al._ (1959); Sleight (1968); Subramanian  _et al._
(1983); Strobel  _et al._ (2010); Kobayashi  _et al._ (2001); Gomez  _et al._
(2021); Blöte  _et al._ (1969); Whitaker and Greaves (2014); Bongers and Van
Meurs (1967); Casado  _et al._ (1985); Villars  _et al._ (2015); T. F. and
Copenhaver (1972); Subramanian  _et al._ (1983); Knop  _et al._ (1969, 1965);
Haipeng  _et al._ (2016); Xiaoge  _et al._ (2016); Li  _et al._ (2009);
Subramanian  _et al._ (1983); Kitayama and Katsura (1976); Soderholm  _et al._
(1982); Troyanchuk (1990); Yokokawa  _et al._ (1992); Shin-ike  _et al._
(1977); Shimakawa  _et al._ (1999); Subramanian  _et al._ (1988); Greedan  _et
al._ (1996); Shimazaki  _et al._ (2000); Shannon and Sleight (1968); Becker
and Felsche (1987); Hallas  _et al._ (2014, 2012); Zhou  _et al._ (2012); Li
_et al._ (2014); Hallas  _et al._ (2016a, 2015); Clark and Morley (1976).

In our search for lanthanide pyrochlore oxides, we have chosen to maintain the
nominal purity of both the Ln3+ and oxygen sublattices, although many other
variants on the pyrochlore lattice ($e.g.$ mixed anion oxy-fluoride Wells
(1987); Phraewphiphat  _et al._ (2018), oxy-nitrides Clark  _et al._ (2014),
alkali A-site Ross  _et al._ (2017), etc.) have been reported. For graphical
simplicity, compositions with a limited number ($<4$) of members reported
($e.g.$ Ln2AlSbO7 Che  _et al._ (2020)) have been omitted. White boxes mark
compositions with no reported synthetic data. Non-pyrochlore compositions are
marked in red, and include: 1) phase competition with other compounds, 2)
polymorph competition with fluorite-type or lower symmetry structures, or 3)
persistent (substantial) phase impurity and off-stoichiometry. Special
attention has been paid to pyrochlore phases that have been stabilized in the
pyrochlore structure through the application of hydrostatic pressure $>$1 MPa
($e.g.$ Ln2Ge2O7 Hallas  _et al._ (2014, 2012); Zhou  _et al._ (2012); Li  _et
al._ (2014); Hallas  _et al._ (2016a), Ln2Pb2O7 Hallas  _et al._ (2015)).
These metastable compositions are distinguished by a dark red coloration.

Figure 2: Stability-field map for Ln2B2O7 and
Ln2B${}_{\text{i}}^{3+}$B${}_{\text{ii}}^{5+}$O7 lanthanide pyrochlore oxides.
Known pyrochlore structures reported in the literature are superimposed with a
minimized stability field (black border). “Non-pyrochlore” null results (red)
and pyrochlores stabilized with high pressure (dark red) are distinguished
from “no data” (white).

Using a simple scoring algorithm, we have recalculated the limits of
$R_{\text{min}}$ $\leq$ $R{\rm{{}_{Ln^{3+}}}}$/$R{\rm{{}_{B^{4+}}}}$ $\leq$
$R_{\text{max}}$. Contrary to the often quoted range [$R_{\rm{min}}$,
$R_{\rm{max}}$] of [1.36Å, 1.71Å] Gardner  _et al._ (2010); Wiebe and Hallas
(2015); Zhou and Wiebe (2019), the best fit, denoted by the black boundary in
Fig. 2, was calculated to be [$R_{\rm{min}}$, $R_{\rm{max}}$] of [1.46Å,
1.82Å], in excellent agreement with the historical estimate provided by
Subramanian _et al._ Subramanian  _et al._ (1983). Our updated stability
diagram suggests that the mixed B-site pyrochlores largely abide by the
empirical radii rules discovered decades prior. This is interesting, as one
could reasonably expect that the bond strain and disorder induced by the mixed
B-site could negatively impact the energetics of the ordered pyrochlore
structure. The predictive accuracy of the model for all known compounds is
$\sim$94%. We note that the range [$R_{\rm{min}}$, $R_{\rm{max}}$] of [1.36Å,
1.71Å] Gardner  _et al._ (2010); Wiebe and Hallas (2015); Zhou and Wiebe
(2019) performs substantially worse, only capturing the known pyrochlore
phases with at 75% accuracy.

Upon closer inspection of Fig. 2, many compositions that border the edge of
the stability field were noted to have mixed results reported in the
literature – where coexistance and competition of the pyrochlore and defect
fluorite phases is contested Simeone  _et al._ (2017); Lumpkin and Aughterson
(2021). While we have tried to survey the results independently, the binary
demarcation of pyrochlore/non-pyrochlore is an oversimplification at best. The
literature is rife with such results that highlight the complexity of defect
formation Sibille  _et al._ (2017); Blanchard  _et al._ (2012) throughout the
stability field. Colloquially, our review of the literature serves as a
reminder that the stability field, although convenient, is an oversimplified
construct. One must respect that the defect thermodynamics and kinetics will
be unique for each composition. These defect mechanisms are crucial for many
key systems (e.g. Yb2Ti2O7 Chang  _et al._ (2012); Gaudet  _et al._ (2016a);
Ross  _et al._ (2012); Yasui  _et al._ (2003); Yaouanc  _et al._ (2011a); Ross
_et al._ (2011b); D’Ortenzio  _et al._ (2013); Chang  _et al._ (2014); Hallas
_et al._ (2018); Arpino  _et al._ (2017), Tb2Ti2O7 Ruminy  _et al._ (2016);
Gardner  _et al._ (1999, 2003); Gingras  _et al._ (2000); Chapuis  _et al._
(2010); Takatsu  _et al._ (2011); Yaouanc  _et al._ (2011b); Hamaguchi  _et
al._ (2004); Fennell  _et al._ (2014); Siddharthan  _et al._ (1999)) whose
ground state properties are intimately tied with the nuances of both defects
and disorder. It is important to note that such considerations must be taken
into account, particularly when moving into compositions with intentional
disorder on the B-site.

Figure 3: (a) Room temperature x-ray diffraction patterns for Ln2InSbO7
family. All compounds crystallize in cubic symmetry with pre-Gd compositions
(La…Gd) displaying characteristic (311) and (331) pyrochlore superlattice
reflections. (b,top) Linear dependence of the refined cubic lattice parameter
and cell volume with trivalent lanthanide Shannon radius. (b,bottom) Deviation
between predicted and observed intensity of (331) superlattice reflection as a
function of composition. The suspected crossover from the pyrochlore structure
for pre-Gd compositions (La…Gd) to a defect fluorite-like structure for post-
Gd compositions (Tb…Yb) is mirrored by the shift of the variable
$x$-coordinate of the 48$f$ oxygen.

###  III.2 Ln2InSbO7: Synthesis & Structure

While the effective radii of the combination of InSb is approximately 0.70 Å,
nearly identical to that of Sn (0.69 Å) Shannon (1976), the discrepancy is
large enough such that the InSb compositions straddle the limits of pyrochlore
stability field in Fig. 2. This is consistent with previous work by Strobel
Strobel  _et al._ (2010), where only Gd2InSbO7 was able to be synthesized in a
phase-pure form. In contrast to the case of the much smaller combination of
GaSb, approximating that of Ti, preliminary work by Strobel suggested that
lighter Ln3+ cations appeared to completely destabilize the cubic pyrochlore
structure, replaced by lower symmetry derivatives corresponding to numerous
unidentifiable reflections present in diffraction patterns Strobel  _et al._
(2010).

Empirical experimentation with Ln2InSbO7 compositions in the current study,
when combined with our previous reports on the Ln2GaSbO7 phases, suggests that
diffusion kinetics are sluggish in the mixed oxide compositions. Thus, the
successful synthesis of homogeneous powders free of both impurity phases and
unreacted reagents often require several cycles of classical “shake-and-bake”
processing, consisting of multiple heating cycles with intermittent grinding.
However, often these methods are insufficient to produce phase-pure powders.
Such is the case for Ln2InSbO7 Strobel  _et al._ (2010); Blöte  _et al._
(1969), and some select Ln2GaSbO7 compositions Gomez  _et al._ (2021); Blöte
_et al._ (1969), where impurities persist indefinitely. As an attempt to
circumvent the limitation of slow reaction kinetics, we incorporated
mechanochemical methods (see Methods § II.1), creating extremely fine,
homogenous, glassy precursor phases. As illustrated in Fig. 3, in the case of
La…Gd, a single annealing treatment (18 h) on the glassy precursor phase is
sufficient to produce phase-pure powders. The successful synthesis of phase-
pure Ln2InSbO7 cubic pyrochlore oxides suggests that prior limitations Strobel
_et al._ (2010) were a consequence of the classically established processing
methods commonly employed to synthesize pyrochlore oxides, and not limitations
of thermodynamics.

However, Ln2InSbO7 compositions with small lanthanide ions (Tb…Yb) did not
produce cubic pyrochlores, and instead exhibited diffraction patterns fraught
with numerous additional reflections, reminiscent of previous exploratory work
Strobel  _et al._ (2010). A combination of scanning electron microscopy (SEM)
and energy dispersive spectroscopy (EDS) confirmed the heated powders remained
chemically homogenous and single phase, suggesting a reduction in symmetry
from cubic to monoclinic or triclinic crystal systems. We noted an extreme
sensitivity of the additional peaks with anneal time and temperature,
suggestive of complex crystallization kinetics in the trans-Gd compounds.

To probe the crystallization process _ex situ_ , we designed a series of
quenching experiments to generate a crude time-temperature phase diagram. We
determined that the precursor, which is largely amorphous, first crystallizes
into a cubic phase within a very short time frame ($\sim$1 hr). As the anneal
proceeds further, the cubic phase gradually degrades and transforms into the
lower symmetry structure. Additional experiments to control atmospheric
conditions and test potential volatility issues failed to stymie the
decomposition of the cubic phase, suggesting that the symmetry reduction is a
thermodynamic effect. In this respect, although the emergence of the cubic
phase from the glassy precursor is _kinetically_ favored, the high symmetry
phase is ultimately _thermodynamically_ unstable for the smaller Ln3+ ions.

The synthetic observations summarized above are consistent with Fig. 2, where
Ln2InSbO7 compositions for early lanthanides are predicted to be cubic, while
destabilizing midway through the lanthanide series. However, the observation
that the cubic phase was kinetically favored provided a means to stabilize
pyrochlore-like phases for the heavier lanthanides. As described in Methods
§II.1, microwave-assisted heating Levin  _et al._ (2019); Michael P áMingos
and Baghurst (1991); Rao  _et al._ (1999); Mingos (1993); Whittaker and Mingos
(1994) of the mechanicochemically synthesized amorphous precursors provided a
direct experimental means of achieving the extreme heat/cooling rates required
to successfully quench in the cubic Ln2InSbO7 phases (Fig. 3(a)). It is worth
noting that trans-Gd compounds often benefit from small amounts of excess
Sb2O5 to compensate for increased volatility during the extreme ramp rates. We
note that the microwave-assisted mechanicochemical methods also work for
pyrochlore phases that are accessible through classical heating methods. Thus,
the combination of microwave-assisted heating and mechanochemical methods
enables the rapid synthesis of the pyrochlore oxides, on the order of minutes
_in lieu_ of days, allowing for wide experimental surveys of chemical spaces
in a short amount of time.

At first glance, it appears that the entire Ln2InSbO7 series can be stabilized
as cubic pyrochlores. As illustrated in Fig. 3(a), XRD confirmed that all
Ln2InSbO7 compositions crystallize in the cubic crystal system. No impurity
phases can be identified within the resolution of X-ray diffraction, and the
lattice parameter and cell volume trend linearly with increasing Ln3+ radii
(Fig. 3(b, top)), consistent with reports for other pyrochlore families
Subramanian  _et al._ (1983). For lighter lanthanides (La…Gd), Ln2InSbO7
compositions are best described with the pyrochlore $Fd\bar{3}m$ structure,
possessing the characteristic (311) and (331) “pyrochlore” superlattice
reflections Subramanian  _et al._ (1983); Gardner  _et al._ (2010); Maram  _et
al._ (2018); Ciomaga Hatnean  _et al._ (2016). However, it is worth noting
that due to the particular elements involved, these superlattice peaks in
Ln2InSbO7 are extremely weak. For the ideal pyrochlore structure with no
disorder present between the $A$\- and $B$-sites, the (311) and the (331)
superlattice reflections in La2InSbO7 are predicted to be 0.2% and 1.4% of the
main (222) reflection, respectively.

In the case of trans-Gd compositions (Tb…Lu), assignment of the unit cell is
more nuanced. Although these materials do maintain cubic symmetry (Fig. 3(a)),
the (311) and (331) superlattice reflections become vanishingly small, as
illustrated in Fig. 3(b, bottom). The black trace denotes the “fit fraction”
of the (331) reflection, where the “fit fraction” is defined by:

$F=100\%\times\left(1-\frac{I_{\text{hkl,Calc.}}-I_{\text{hkl,Obs.}}}{I_{\text{hkl,Calc.}}-I_{\text{hkl,Bkg.}}}\right)$
(1)

where $F$ was constructed such that perfect agreement with the pyrochlore
structure is defined as 100%, and complete suppression of the (331)
corresponds to 0%. The loss of the (331) peak as the Ln3+ radius shrinks is
mirrored by a gradual shift in the position of the $48f$ oxygen ion to the
higher symmetry point at (3/8, 1/8, 1/8). This shift towards a symmetric
oxygen coordination is likely associated with a gradual disordering of the
$A$\- and $B$-sublattices Lumpkin and Aughterson (2021); Popov  _et al._
(2016). However, we caution that oxygen is notoriously difficult to fully
characterize with standard X-ray diffraction. Incremental increases in
disorder are consistent with recent results in the pyrochlore zirconates,
where a combination of XANES, synchrotron, and neutron diffraction indicate
that the defect fluorite to pyrochlore transition across the series is gradual
Blanchard  _et al._ (2012). However, in the case of Ln2InSbO7, the situation
has an additional layer of complexity since the cubic phase is not the
thermodynamic ground state for trans-Gd compositions. Thus, we move to
conservatively classify the Ln2InSbO7 compositions from (Tb…Lu) as defect
fluorite phases. Unfortunately, as is the case of Ln2InSbO7, since the In3+,
Sb5+, and Ln3+ ions all share relatively similar electron densities, it is
difficult to directly quantify the degree of A- and B-site disordering.
However, loss of the (311) and (331) superlattice reflections is typically
sufficient to suggest suppression of pyrochlore-like domains.

Despite our tentative classification of the trans-Gd Ln2InSbO7 phases as
defect fluorite phases, we feel it appropriate to highlight recent results
that demonstrate how the transition from pyrochlore to defect fluorite is not
always a simple matter. Recent work on Tb2Hf2O7 Sibille  _et al._ (2017)
demonstrates that that the loss of the pyrochlore superlattice reflections
does not necessarily arise from cation disordering. In the case of the
hafanate, a combination of neutron powder diffraction and resonant X-ray
diffraction confirmed the presence of an ordered “pyrochlore” cation
sublattice. Instead, the anionic sublattice experiences progressive
disordering, marked by a high density of oxygen Frenkel defects. Sibille  _et
al._ (2017) Reflecting on our observations in the Ln2InSbO7 compounds, such an
effect could conceivably be convolved in our observation of shifts in the
48$f$ oxygen position, but ultimately lies outside our experimental confidence
at this time. Further experiments including neutron diffraction are underway.
However, as we will demonstrate in our magnetization measurements, many of the
trans-Gd compositions retain magnetic properties that are strikingly similar
to their pyrochlore stannate and titanate analogs.

Revisiting Fig. 2, it is clear that the structural trends identified in the
Ln2InSbO7 series are consistent with its location in the predicted pyrochlore
stability field. Despite having a nearly identical effective $B$-site radius
as the stannates (0.69 Å) Shannon (1976), the Ln2InSbO7 series (0.70 Å) does
not share the structural stability of the stannates. This striking drop in the
phase stability of the pyrochlore structure is mirrored by other large mixed
$B$-site pyrochlores such as Ln2ScNbO7 Jiang  _et al._ (2020); Zouari  _et
al._ (2008); McQueen  _et al._ (2008) (0.693 Å) and Ln2ScTaO7 Shlyakhtina  _et
al._ (2014, 2018); Sato  _et al._ (1976); Song  _et al._ (2019) (0.693 Å),
both of whom have effective Shannon radii Shannon (1976) closely comparable to
that of tetravalent tin. As our discussion shifts towards the magnetic
properties of the Ln2InSbO7 family of compounds, we will aim to make
comparisons between not only the stannates, but also with their heavier
analogs ($e.g.$ hafanates and zirconates), and other mixed B-site pyrochlores.

Figure 4: Combined temperature dependence of the ac and dc magnetization with
corresponding Curie-Weiss fits, and the isothermal dc magnetization of (a)
Nd2InSbO7, (b) Gd2InSbO7, and (c) Er2InSbO7. For temperatures $T<4$ K, the ac
susceptibility was measured at various frequencies $f_{ac}$ in the absence of
a dc field. For $T>1.8$ K, the dc susceptibility was measured in an external
field $H_{dc}$ of 100 Oe.

###  III.3 dc and ac magnetic susceptibility

Drawing analogies to the stannates, members of the Ln2InSbO7 family have been
organized into four categories based on their proposed magnetic ground state:
1) long-range antiferromagnetic order, 2) dynamic magnetic ground states, 3)
dipolar spin ices, and 4) nonmagnetic singlet states. As shown below, despite
mirroring a stability field similar to the hafanate counterparts Subramanian
_et al._ (1983); Gundovin  _et al._ (1975); Klee and Weitz (1969); Besson  _et
al._ (1966); Spiridinov  _et al._ (1968); Sibille  _et al._ (2016a);
Shlyakhtina  _et al._ (2006), the properties of the Ln2InSbO7 series are
remarkably consistent with the magnetic properties of the stannates, the non-
mixed $B$-site family with whom they share the closest structural properties
Subramanian  _et al._ (1983); Kennedy  _et al._ (1997). Remarkably, even in
the cases of trans-Gd compositions where the structure would be conservatively
classified as defect fluorite, the transport properties of the Ln2InSbO7
family remain reminiscent of the behavior exhibited by their pyrochlore
stannate counterparts.

####  III.3.1 Long-range antiferromagnetic order: Nd2InSbO7, Gd2InSbO7 &
Er2InSbO7

Nd2InSbO7: As illustrated in Fig. 4, a $\lambda$-type peak at TN = 0.37 K is
present in the ac susceptibility of Nd2InSbO7. Exhibiting no clear frequency
dependence, such a peak is consistent with a transition into a long-range
magnetic ordered state. As the isothermal magnetization suggests the presence
of $\langle 111\rangle$ Ising single-ion anisotropy Bramwell  _et al._ (1999),
combined with the small magnitude of the Curie-Weiss temperature, our data
supports the possibility that Nd2InSbO7 assumes non-coplanar all-in-all-out
magnetic order that has been reported for all other Nd-based pyrochlores Gomez
_et al._ (2021); Scheie  _et al._ (2021); Bertin  _et al._ (2015); Xu  _et
al._ (2015); Lhotel  _et al._ (2015); Anand  _et al._ (2017, 2015); Matsuhira
_et al._ (2013); Tomiyasu  _et al._ (2012); Mauws  _et al._ (2021); Yasui  _et
al._ (2001); Ku  _et al._ (2018).

While possessing many of the generic features of Nd-based pyrochlores,
Nd2InSbO7 exhibits a key distinguishing feature: a exceptionally low Curie-
Weiss temperature of +0.1 K. Recently, Gomez _et al._ Gomez  _et al._ (2021)
proposed that chemical pressure plays a crucial role in realization of moment
fragmentation Brooks-Bartlett  _et al._ (2014). The authors argue that any
contraction of the lattice would decrease the value of the Curie-Weiss
constant as a result of the enhancement of the antiferromagnetic Nd3+-O2--Nd3+
superexchange pathways, which would exclude moment fragmentation in favor of
long-range antiferromagnetic order Bertin  _et al._ (2015); Benton (2016).
Solely utilizing the lattice parameter, both TN and $\theta\rm{{}_{CW}}$ for
Nd2InSbO7 compare favorably to the general trends observed among all Nd-based
pyrochlores summarized by Fig. 8 in Gomez _et al._ Gomez  _et al._ (2021).

We suspect that Nd2InSbO7 may exhibit the same rich physics present in the
ScNb Scheie  _et al._ (2021); Mauws  _et al._ (2021), Hf Sibille  _et al._
(2018), Zr Benton (2016); Petit  _et al._ (2016a); Zoghlin  _et al._ (2021)
analogs due to its proximity to the crossover point in the sign of the Curie-
Weiss temperature. Recently, Scheie _et al._ Scheie  _et al._ (2021) suggested
that both Nd2ScNbO7 and Nd2Zr2O7 share a common, unusual fluctuating magnetic
ground state. This presents the exciting possibility that Nd2InSbO7 may
exhibit moment fragmentation. With the presence of controlled chemical
disorder, Nd2InSbO7 may provide additional insights into the physical origins
of the strong quantum fluctuations Benton (2016); Xu  _et al._ (2020); Kimura
_et al._ (2013) underlying both the nature of moment fragmentation and the
magnetic ground state present in the heavier Nd-based pyrochlores.

Gd2InSbO7: With a half-filled 4$f^{7}$ electronic configuration ($L$=0,
$S$=7/2) Jensen and Mackintosh (1991), Gd3+ is expected to possess minimal
spin anisotropy, particularly smaller than Ho3+, Dy3+, and Tb3+ Raju  _et al._
(1999); Cashion  _et al._ (1968); Glazkov  _et al._ (2005). Combined with
dominant antiferromagnetic exchange when placed on a pyrochlore lattice, Gd-
based pyrochlores are excellent candidates for the experimental realization of
a classical Heisenberg pyrochlore antiferromagnet and its predicted ground
state degeneracy Canals and Lacroix (1998, 2000); Moessner and Chalker (1998a,
b); Nawa  _et al._ (2018); Reimers (1992); Reimers  _et al._ (1991b); Villain
(1979).

As is the case for the other Gd2B2O7 pyrochlores (Ti Bonville  _et al._
(2003); Cashion  _et al._ (1968); Luo  _et al._ (2001), Sn Bonville  _et al._
(2003); Matsuhira  _et al._ (2002); Luo  _et al._ (2001), Sc/Nb Beare (2021),
Hf Durand  _et al._ (2008), Zr Durand  _et al._ (2008), Pb Hallas  _et al._
(2015)), the presence of minimal single-ion anisotropy results in a Curie-
Weiss-like magnetization response over a large temperature range. The presence
of dominant antiferromagnetic interactions with a $\theta\rm{{}_{CW}}$=$-$8.7
K, results in the deviation of the isothermal magnetization from the Brillouin
function in the low field limit before its saturation at $g_{J}J$=7
$\mu\rm{{}_{B}}$ Freitas and Gardner (2011).

In stark contrast with Gd2ScNbO7 Beare (2021), Gd2InSbO7 exhibits a frequency
independent $\lambda$-type anomaly at $T\rm{{}_{N}}$=0.78 K in the ac
susceptibility, corresponding to the onset of long-range order. In the case of
ScNb Beare (2021), a Mydosh parameter of 0.020(1) places Gd2ScNbO7 in the
conventional spin glass regime Mydosh (1993); Zhou  _et al._ (2008a). The
glassy nature of Gd2ScNbO7 is consistent with the bifurication of the ZFC/FC
dc magnetization, linear low-temperature specific heat, and $\mu$SR results.
Instead, Gd2InSbO7 has a sharp $\lambda$-type anomaly which is highly
reminiscent of Gd2Sn2O7 Wills  _et al._ (2006); Bonville  _et al._ (2003).
Unlike the titanate, which exhibits two closely spaced transitions Bonville
_et al._ (2003); Champion  _et al._ (2001); Stewart  _et al._ (2004), the
stannate assumes the $\mathbf{k}$ = (0,0,0) $\Gamma_{7}$ “Palmer-Chalker”
state Wills  _et al._ (2006); Stewart  _et al._ (2008). In fact, such a
collinear antiferromagnetic state is theoretically predicted for a Heisenberg
antiferromagnet with dipolar interactions Palmer and Chalker (2000), and
corresponds to the proposed state for Gd2Pt2O7 Hallas  _et al._ (2016b). Such
a magnetic structure is much simpler to determine experimentally when compared
to the proposed multi-$\mathbf{k}$ structure of the titanate that still
remains the subject of debate Stewart  _et al._ (2004); Paddison  _et al._
(2021); Champion  _et al._ (2001); Raju  _et al._ (1999); Petrenko  _et al._
(2004); Brammall  _et al._ (2011); Javanparast  _et al._ (2015).

The presumed long-range antiferromagnetic order of of Gd2InSbO7 is
particularly interesting in the context of the spin-glass ScNb. Beare (2021) A
generic mixed $B$-site Gd-based pyrochlores generally exhibits the three key
ingredients of a spin glass: 1) frustration, 2) disorder, and 3) competing
interactions. However, our conceptual intuition may be limited, particularly
in the face of a recent report of the effects of sample quality in Gd2Zr2O7 Xu
(2017); Durand  _et al._ (2008). The clear absence of glassy behavior in
Gd2InSbO7, despite substantial (intentional) chemical disorder, demonstrates
that other subtle perturbative terms such as beyond nearest neighbor exchange
terms Wills  _et al._ (2006); Cépas and Shastry (2004) may be particularly
influential in the selection of its magnetic ground state.

Er2InSbO7: As is the case for their Heisenberg Gd3+ counterparts, the
inclusion of dipolar interactions to the isotropic exchange Hamiltonian for XY
pyrochlores (e.g. Er3+) with dominant antiferromagnetic interactions is
predicted to lift any extensive classical degeneracy, and uniquely select the
Palmer-Chalker state Palmer and Chalker (2000); Guitteny  _et al._ (2013a);
Champion  _et al._ (2003). Experimentally, however, this is not the case for
the majority of XY pyrochlores Dun  _et al._ (2015); Poole  _et al._ (2007);
Lago  _et al._ (2014); Yaouanc  _et al._ (2013a); Cai  _et al._ (2016); Gaudet
_et al._ (2016a); Yasui  _et al._ (2003); Dun  _et al._ (2015); Hallas  _et
al._ (2016a). Exemplified by Er2Ti2O7, where long-range magnetic order is
realized _via_ a quantum order-by-disorder mechanism Champion  _et al._
(2003); Savary  _et al._ (2012); Zhitomirsky  _et al._ (2012); Wong  _et al._
(2013); Champion and Holdsworth (2004), XY pyrochlores present a versatile
experimental platform for the effects of multi-phase competition Hallas  _et
al._ (2018).

The presence of multi-phase competition has been used extensively to account
for key experimental signatures that are characteristic of the XY pyrochlores
Hallas  _et al._ (2018). These include: multiple heat capacity anomalies,
suppression of the ordering temperature, unconventional spin dynamics, defect
sensitivity, and pressure-dependent magnetic ground states. Recently, the
application of multi-phase competition has been extended Yan  _et al._ (2017);
Jaubert  _et al._ (2015); Guratinder  _et al._ (2019); Scheie  _et al._ (2020)
further to address the vastly different magnetic ground states observed among
each of the two families of XY pyrochlores.

In Er-based pyrochlores, members with a smaller lattice parameter tend to
select for long-range antiferromagnetic order in $\Gamma_{5}$ Poole  _et al._
(2007); Dun  _et al._ (2015), while their larger counterparts assume the
Palmer-Chalker state at significantly lower Néel temperatures Hallas  _et al._
(2017); Petit  _et al._ (2017). The effects of multi-phase competition becomes
particularly apparent when comparing the titanate that orders at 1.23 K with
$\psi_{2}$ ($\Gamma_{5}$) Champion  _et al._ (2003); Poole  _et al._ (2007);
Rau  _et al._ (2016), while its stannate counterpart begins to order at 108 mK
in the Palmer-Chalker state Petit  _et al._ (2017); Yahne  _et al._ (2021);
Guitteny  _et al._ (2013a). The significant suppression of the second order
magnetic transition, combined with the multiscale dynamics, is consistent with
the placement of Er2Sn2O7 in close proximity to the $\Gamma_{7}$/$\Gamma_{5}$
boundary Guitteny  _et al._ (2013a); Yan  _et al._ (2017); Hallas  _et al._
(2018). The titanate, however, exhibits little sample dependence and a robust
ground state as it is located deep within $\psi_{2}$ of $\Gamma_{5}$ Yan  _et
al._ (2017); Gaudet  _et al._ (2016b).

Figure 5: Temperature dependence of the out-of-phase component
$\chi^{\prime\prime}$ of the ac susceptibility of Er2InSbO7 (left) with
corresponding Arrhenius analysis (right) for frequencies below 1 kHz.

As illustrated in Fig. 4(c), both the temperature-dependent and isothermal
magnetization mirrors the titanate Champion  _et al._ (2003); Cao  _et al._
(2009); Bonville  _et al._ (2013); Petrenko  _et al._ (2011) and the stannate
Sarte  _et al._ (2011); Matsuhira  _et al._ (2002); Bondah-Jagalu and Bramwell
(2001); Al Ghamdi  _et al._ (2014). Above 20 K, the magnetic susceptibility
follows the Curie-Weiss law for dominant antiferromagnetic interactions
($\theta\rm{{}_{CW}}$=$-$11.5 K) between large Er3+ moments
($\mu\rm{{}_{eff}}$=9.7 $\mu\rm{{}_{B}}$). The isothermal magnetization does
not follow a Brillouin curve, saturating at $\sim$4.2 $\mu\rm{{}_{B}}$
$<<g_{J}J$. The large moment suggests a large dipolar term, and when combined
with the XY single-ion anisotropy of Er3+, a Palmer-Chalker state
theoretically should be realized Palmer and Chalker (2000); Guitteny  _et al._
(2013a); Champion  _et al._ (2003, 2003); Poole  _et al._ (2007). In the case
of Er-based pyrochlores, B-site disorder does not preclude the assumption of
long-range magnetic order. Samples produced along the Er2Sn2-xTixO7 solid
solution retained the $\psi_{2}$ ($\Gamma_{5}$) ground state of the titanate
until a quantum crossover at $x=1.7$ to $\Gamma_{7}$ of the stannate Shirai
_et al._ (2017).

In Er2InSbO7, the transition to a long-range ordered state is consistent with
the distinct peak at $T_{N}$ = 0.56 K present in the ac susceptibility.
However, unlike the Nd and Gd analogs, the prominent peak exhibits a clear
frequency dependence at low frequencies. In Er2Sn2O7, the frequency-dependence
was attributed to spins located at magnetic domain boundaries Tardif  _et al._
(2015); Lhotel  _et al._ (2011). As illustrated in Fig. 5, the frequency
dependence does not follow Arrhenius-type behavior. Instead, our data exhibits
extreme non-linearity for frequencies above 1 kHz. We note that this behavior
may still be consistent with the stannate, as all published ac data is
collected over a narrow low-frequency regime from 5-200 Hz Guitteny  _et al._
(2013a); Petit  _et al._ (2017). We hypothesize that the non-Arrhenius
behavior in Er2InSbO7 may be a reflection of the slow dynamics that have
previously identified in Er2Sn2O7 Petit  _et al._ (2017); Guitteny  _et al._
(2013a). Ultimately, it would be interesting to determine if both the stannate
and titanate analogs exhibit non-linearity in the high frequency limit.

While Er2InSbO7 does exhibit clear linear behavior for lower frequencies, its
activation energy of 12.5 K is over an order of magntiude larger than that of
its stannate analog (0.9 K) Guitteny  _et al._ (2013a); Petit  _et al._
(2017). Such an increase is consistent with the effects of disorder, as has
been proposed in $A$-site disordered pyrochlores Snyder  _et al._ (2004a,
2001a); Ehlers  _et al._ (2006), although it has been noted that $B$-disorder
may have the opposite effect Ke  _et al._ (2007). Regardless, the distinct
similarities between Er2InSbO7 and its stannate analogs suggest the existence
of multi-scale dynamics, where slow dynamics may persist and coexist with
static long-range magnetic order below TN.

The Néel temperature of Er2InSbO7 is unique among those values reported for
all other Er3+-based pyrochlores. Naïvely, with a lattice constant comparable
to the stannate, it is expected that the resulting weaker exchange tensor
would depress the Néel temperature of Er2InSbO7 towards the 0 K limit Hallas
_et al._ (2018). Instead, its large TN=0.56 K is comparable to that of the
much smaller titanate Champion  _et al._ (2003); Cao  _et al._ (2009);
Bonville  _et al._ (2013); Petrenko  _et al._ (2011). Such a large Néel
temperature suggests that Er2InSbO7 is located far from a phase boundary in
the the classical ground-state phase diagram Guitteny  _et al._ (2013a); Yan
_et al._ (2017); Hallas  _et al._ (2018), although it important to note that
without neutron diffraction, the identity of $\Gamma$ and its respective
$\psi$ both remain for now an open question.

Figure 6: Combined temperature dependence of the ac and dc magnetization with
corresponding Curie-Weiss fits, and the isothermal dc magnetization of (a)
Yb2InSbO7, (b) Tb2InSbO7, and (c) Sm2InSbO7. For temperatures $T<4$ K, the ac
susceptibility was measured at various frequencies $f_{ac}$ in the absence of
a dc field. For $T>1.8$ K, the dc susceptibility was measured in an external
field $H_{dc}$ of 100 Oe.

####  III.3.2 Dynamic Magnetic Ground States: Yb2InSbO7, Tb2InSbO7 & Sm2InSbO7

Yb2InSbO7: Corresponding to the second, and arguably most prominent family of
the XY pyrochlores Hallas  _et al._ (2018), Yb-based pyrochlores have been
subject to intense interest since the proposal of quantum spin ice behavior in
Yb2Ti2O7 Ross  _et al._ (2011a); Gingras and McClarty (2014); Applegate  _et
al._ (2012); Petit (2020); Scheie  _et al._ (2020); Pan  _et al._ (2016). The
subsequent explosion of experimental and theoretical studies have propelled
research into Yb-based magnetism in various frustrated geometries, including
the triangular Baenitz  _et al._ (2018); Liu  _et al._ (2018); Ranjith  _et
al._ (2019a); Bordelon  _et al._ (2019); Ranjith  _et al._ (2019b); Schmidt
_et al._ (2021); Li  _et al._ (2015); Balents (2010); Paddison  _et al._
(2017) and Shastry-Sutherland lattices Sriram Shastry and Sutherland (1981);
Kim and Aronson (2013); Bernhard  _et al._ (2011); Kikuchi  _et al._ (2009);
Liu  _et al._ (2014a); Dublenych (2012). Fueled by the enhanced quantum
fluctuations that are a result of the $J{\rm{{}_{eff}}}=1/2$ degrees of
freedom originating from the thermally well-isolated Kramers doubet single-ion
ground state Hodges  _et al._ (2001); Hallas  _et al._ (2016a); Gaudet  _et
al._ (2015); Malkin  _et al._ (2004); Hallas  _et al._ (2018); Yan  _et al._
(2017); Gardner  _et al._ (2010), Yb-based magnetism has remained a fruitful
arena for the search of quantum spin liquid states Hallas  _et al._ (2018);
Hermele  _et al._ (2004); Banerjee  _et al._ (2008); Savary and Balents
(2012); Benton  _et al._ (2012); Lee  _et al._ (2012); Savary and Balents
(2013); Gingras and McClarty (2014).

As is the case for their Er3+ counterparts, multi-phase competition in Yb-
based pyrochlores is particularly influential on the magnetic properties of
the selected ground state Scheie  _et al._ (2020); Jaubert  _et al._ (2015).
In addition to extremely depressed transition temperatures
($T\rm{{}_{N}},T\rm{{}_{C}}\ll 1$ K), Yb-based pyrochlores have demonstrated
particular sensitivity to sample preparation and defect formation Ross  _et
al._ (2012); Dun  _et al._ (2014); Hallas  _et al._ (2016c, 2018); Arpino  _et
al._ (2017); Yaouanc  _et al._ (2011a); D’Ortenzio  _et al._ (2013); Lhotel
_et al._ (2014); Yaouanc  _et al._ (2013b); Gaudet  _et al._ (2016a); Bowman
_et al._ (2019); Sala  _et al._ (2014); Mostaed  _et al._ (2017). The case of
Yb2Ti2O7 is perhaps the most infamous. Minute changes in the Yb stoichiometry
have been demonstrated to broaden, depress, or completely suppress any
signature of long-range static magnetic order in heat capacity and
magnetization. Chang  _et al._ (2012); Gaudet  _et al._ (2016a); Ross  _et
al._ (2012); Yasui  _et al._ (2003); Yaouanc  _et al._ (2011a); Ross  _et al._
(2011b); D’Ortenzio  _et al._ (2013); Chang  _et al._ (2014); Hallas  _et al._
(2018); Arpino  _et al._ (2017). Interestingly enough, the application of a
moderate amount of pressure Kermarrec  _et al._ (2017) is sufficient to
recover the $\textbf{k}=0$ splayed ferromagnetic state. Altogether, these
results are consistent with the placement of the titanate near the
ferromagnetic $\Gamma_{9}$/antiferromagnetic $\Gamma_{5}$ phase boundary Yan
_et al._ (2017); Jaubert  _et al._ (2015); Ross  _et al._ (2011a).

Given the extreme fragility of the magnetic ground state in Yb2Ti2O7, one
would expect full suppression of the long-range order in the mixed B-site
Yb2GaSbO7. Indeed, Yb2GaSbO7 does not exhibit the characteristic
$\lambda$-type anomaly in its heat capacity Sarte  _et al._ (2021); Blöte  _et
al._ (1969) indicative of static long-range magnetic order in Yb3+-based
pyrochlores Dun  _et al._ (2013, 2014); Hallas  _et al._ (2016c); Lago  _et
al._ (2014); Yaouanc  _et al._ (2013a); Chang  _et al._ (2012); Gaudet  _et
al._ (2016a); Ross  _et al._ (2012); Yasui  _et al._ (2003); Champion  _et
al._ (2001); Cai  _et al._ (2016); Hodges  _et al._ (2002). At a cursory
glance this seems consistent with the deleterious effect of disorder, though
more thorough investigations demonstrate that only a minority of spins freeze,
with the majority remaining dynamic to base temperatures as confirmed by both
inelastic neutron scattering Sarte  _et al._ (2021) and $\mu$SR spectroscopy
Hodges  _et al._ (2011).

In fact, strong dynamical correlations have been shown to be ubiquitous among
the Yb3+-based pyrochlores Hallas  _et al._ (2016c, 2018), developing
coherence at a temperature T∗ $>$ TN,TC centered about a broad feature present
in the heat capacity Hodges  _et al._ (2002); Yaouanc  _et al._ (2013a);
Hallas  _et al._ (2016a); Cai  _et al._ (2016); Hallas  _et al._ (2016c).
These unconventional dynamics correspond to gapless $\mathbf{Q}$=0 excitations
Hallas  _et al._ (2016c); Gaudet  _et al._ (2016a); Dun  _et al._ (2013);
Hallas  _et al._ (2017); Petit  _et al._ (2017) and are remarkably robust
against disorder and doping effects, regardless of the presence (or
suppression) of long-range magnetic order Hallas  _et al._ (2018, 2016c);
Sarte  _et al._ (2011). Similar effects have been particularly well-documented
in Tb3+-based pyrochlores Rau and Gingras (2019); Kermarrec  _et al._ (2015);
Taniguchi  _et al._ (2013); Kadowaki  _et al._ (2018); Gardner  _et al._
(2003); Kanada  _et al._ (1999); Anand  _et al._ (2018); Gaulin  _et al._
(2015).

In the case of Yb2GaSbO7, the application of a modest 1 T field shifts
spectral weight from these low energy fluctuations into magnetic Bragg peaks
corresponding to $\Gamma_{9}$ long-range magnetic order Sarte  _et al._
(2021). Although corresponding to the zero field magnetic structure of the
titanate, the appearance of such static magnetic order despite the presence of
maximal $B$-site disorder at first is not expected, particularly when
considering stuffing of the order of 1-2% in the Yb2Ti2O7 suppresses long-
range ordering Hallas  _et al._ (2018); Ross  _et al._ (2012); Yaouanc  _et
al._ (2011a); Ross  _et al._ (2011b); D’Ortenzio  _et al._ (2013); Arpino  _et
al._ (2017). While mechanical pressure can be used to recover the magnetic
order in Yb2Ti2O7, the energy scale is quite disparate from the application of
a 1 T field in Yb2GaSbO7. This is evidence that antisite disorder (stuffing)
and $B$-site disorder operate independently – and should be experimentally
discernible.

As summarized in Fig. 6(a), the low temperature magnetic properties of
Yb2InSbO7 are reminiscent of its GaSb analog Sarte  _et al._ (2021) with
comparable Curie-Weiss parameters (3.5$\mu_{\text{eff}}$/Yb3+and
$\theta\rm{{}_{CW}}$=$-$1.4 K). In the low temperature limit, a frequency-
independent peak at 0.17 K dominates, reminiscent of both Yb2GaSbO7 and the
relatively sharp features present in pristine samples of both the titanate
Arpino  _et al._ (2017); Yasui  _et al._ (2003); Gaudet  _et al._ (2016a);
Scheie  _et al._ (2017); Yaouanc  _et al._ (2016); Chang  _et al._ (2012,
2014) and stannate Dun  _et al._ (2013); Yaouanc  _et al._ (2013a); Lago  _et
al._ (2014). The energy scale is comparable to the stannate
(T$\rm{{}_{C}}\sim$0.11–0.15 K Dun  _et al._ (2013); Yaouanc  _et al._
(2013a); Lago  _et al._ (2014)) which is proximal in terms of both average
$B$-site radius Shannon (1976) and cell volume. The comparison between InSb:Sn
is analogous to GaSb:Ti, where the smaller GaSb (0.35 K Sarte  _et al._
(2021)) mimics the respective titanate (0.26 K Yasui  _et al._ (2003); Gaudet
_et al._ (2016a); Ross  _et al._ (2012)).

Exploiting the analogy to its GaSb analog, it is more probable that the sharp
peak in both $\chi^{\prime}$ and $\chi^{\prime\prime}$ corresponds to a
freezing of a minority of spins instead of long-range ferromagnetic order. The
striking similarities between Yb2InSbO7 and Yb2GaSbO7 are particularly
noteworthy since Yb2InSbO7 is outside the pyrochlore stability field (Fig. 2)
and may be intermediate between the pyrochlore and defect fluorite structures.
The analogous behavior of InSb and GaSb may suggest that the cation sublattice
(up to a certain correlation length) remains intact, despite a highly
disordered anionic sublattice, as is the case for Tb2Hf2O7 Sibille  _et al._
(2017).

Although it remains to be seen whether Yb2InSbO7 will exhibit properties
consistent with Yb2GaSbO7 or its stannate analogs, the interaction between
long-range magnetic order and the mixed B-site pyrochlores is interesting.
Given that 1-2% of Yb disorder suppresses long-range magnetic order Hallas
_et al._ (2018); Ross  _et al._ (2012); Yaouanc  _et al._ (2011a); Ross  _et
al._ (2011b); D’Ortenzio  _et al._ (2013); Arpino  _et al._ (2017), the
observation that such suppression can be reversed by a modest field in
Yb2GaSbO7 Sarte  _et al._ (2021), with its much larger structural and steric
changes, suggests that $B$-site disorder may present a unique (and very
distinct) chemical means of reliably suppressing long-range ferromagnetic
order in Yb-based pyrochlores, while preserving the Yb3+ sublattice and its
underlying persistent dynamics Hallas  _et al._ (2016c, 2018).

Despite possessing a wealth of rich and complex set of phenomena that rivals
the Yb-based pyrochlores, theoretical efforts to address their Tb-based
counterparts have remained an enduring challenge Rau and Gingras (2019);
Gardner  _et al._ (2010). In contrast to the other lanthanide ions presented
so far, the non-Kramers doublet of the $J$=6 free-ion ground state manifold of
Tb3+ is separated from its first excited doublet by only $\sim$1.4 meV Princep
_et al._ (2015); Zhang  _et al._ (2014); Mirebeau  _et al._ (2007); Gingras
_et al._ (2000); Gardner  _et al._ (2010, 1999). Corresponding to a gap
roughly an order of magnitude smaller than Yb3+, the application of the
pseudospin-1/2 model is not necessarily straightforward, necessitating the
incorporation of more subtle effects, possibly even on equal footing with
$\hat{\mathcal{H}}\rm{{}_{CEF}}$ Molavian  _et al._ (2007, 2009); Hallas  _et
al._ (2020). Such subtle effects include virtual crystal field transitions,
secondary corrections that introduce anistropic exchange and may potentially
stabilize a quantum spin ice ground state in the cooperative paramagnet
Tb2Ti2O7 Molavian  _et al._ (2007); Liu  _et al._ (2019b); Molavian  _et al._
(2009); Kao  _et al._ (2003); Molavian and Gingras (2009); Gingras and
McClarty (2014); Gingras  _et al._ (2000).

Tb2InSbO7: As is the case for its Yb3+ counterpart, multi-phase competition is
particularly influential on the magnetic ground state of Tb2Ti2O7 Rau and
Gingras (2019); Takatsu  _et al._ (2016); Guitteny  _et al._ (2013b); Petit
_et al._ (2012); Wakita  _et al._ (2016). Highly reminiscent of Yb2Ti2O7, the
magnetic ground state of Tb2Ti2O7 is extremely sample dependent Ruminy  _et
al._ (2016); Gardner  _et al._ (1999, 2003); Gingras  _et al._ (2000); Chapuis
_et al._ (2010); Takatsu  _et al._ (2011); Yaouanc  _et al._ (2011b);
Hamaguchi  _et al._ (2004); Fennell  _et al._ (2014); Siddharthan  _et al._
(1999) with deviations in stoichometry on the level of 1-2% yielding
quadrupolar order Taniguchi  _et al._ (2013); Takatsu  _et al._ (2016);
Kadowaki  _et al._ (2015), consistent with the placement of the titanate in
close proximity to the boundary between quantum spin ice and quadrupolar order
Rau and Gingras (2019). Contrast such behavior to that exhibited by the
stannate which assumes “soft” (or dynamic) spin ice ordering corresponding to
Tb3+ moments oriented 13.3° relative to the local $\langle 111\rangle$
Mirebeau  _et al._ (2005); Bert  _et al._ (2006); Dahlberg  _et al._ (2011);
Chapuis  _et al._ (2007); De Reotier  _et al._ (2006); Mirebeau  _et al._
(2005); Rule  _et al._ (2007, 2009).

As summarized in Fig. 6(b), the isothermal magnetization of Tb2InSbO7 is
consistent with Ising single-ion anisotropy characteristic of Tb3+-based
pyrochlores Rau and Gingras (2019); Gingras and McClarty (2014); Gingras  _et
al._ (2000); Zhang  _et al._ (2014); Hallas  _et al._ (2014); Sibille  _et
al._ (2017). The high temperature dc magnetization is qualitatively similar to
both its stannate Dahlberg  _et al._ (2011); Mirebeau  _et al._ (2005);
Bondah-Jagalu and Bramwell (2001); Matsuhira  _et al._ (2002) and titanate
Gingras  _et al._ (2000); Luo  _et al._ (2001); Gardner  _et al._ (1999)
analogs with comparable Curie-Weiss parameters of $\mu\rm{{}_{eff}}$ = 10
$\mu\rm{{}_{B}}$ and $\theta\rm{{}_{CW}}$ = $-11.9$ K. However, the sharp
feature indicative of long-range order in the stannate Dahlberg  _et al._
(2011) is absent, replaced by a broad, frequency-independent feature centered
about $\sim$1.5 K in $\chi^{\prime}$. These properties are highly reminiscent
of Tb2SnTiO7 Gaulin  _et al._ (2015). In the titanate Luo  _et al._ (2001);
Hamaguchi  _et al._ (2004); Gardner  _et al._ (2003); Lhotel  _et al._ (2012),
stannate Dahlberg  _et al._ (2011); Mirebeau  _et al._ (2005), and Tb2GaSbO7
Blöte  _et al._ (1969), the broad feature has been attributed to the buildup
of short-range magnetic correlations.

More complexity is evident in the out-of-plane $\chi^{\prime\prime}$ ac
magnetization signal (Fig. 7). We first observe an upturn in the magnetization
around 2 K, normally associated with an increase in ferromagnetic interactions
and mirrored by the broad feature in $\chi^{\prime}$. A shoulder peak in
$\chi^{\prime\prime}$ can be seen near 1.2 K, particularly in the low-
frequency regime. This feature corresponds with the peak in the AC
magnetization shown in Figure 6. However, unlike the pristine Tb2Sn2O7, our
data lacks the characteristic sharpness of the ferromagnetic ordering peak in
the stannate, and it is difficult to extract a frequency dependence due to
convolution with the rising signal. The general shape of $\chi^{\prime\prime}$
agrees remarkably well with x=0.1 Ti-doped Tb2Sn2-xTixO7.Dahlberg  _et al._
(2011) We postulate that the InSb disorder plays a similar role to the Ti-
doping, destroying the long-range ferromagnetic order. An additional feature
emerges around 400 mK, a similar energy scale to a 300 mK feature observed in
the pristine Tb2Sn2O7. However, while prior reports observed an anomalously
high Mydosh parameter (0.34),Dahlberg  _et al._ (2011) we estimate that the
400 mK feature exhibits $\Delta T_{f}/T_{f}\Delta log(f)=0.04$, an order of
magnitude lower and more akin to results observed in the doped Tb2Sn2-xTixO7
samples.

Presuming that Tb2InSbO7 sits on the boundary between the defect fluorite and
pyrochlore stability fields, the observation that Tb2InSbO7 exhibits behavior
extremely reminiscent to lightly doped Tb2Sn2-xTixO7 is somewhat perplexing.
However, if you consider the case of the corresponding halfnate Tb2Hf2O7,
where despite the presence of substantial anionic disorder (oxygen Frenkel
defects), the cation sublattice remains ordered in the pyrochlore motif
Shlyakhtina  _et al._ (2006); Sibille  _et al._ (2016a, 2017); Matsuhira  _et
al._ (2002); Subramanian  _et al._ (1983), the similarities of Tb2InSbO7 to
its Tb2Sn2-xTixO7 analog can be somewhat reconciled. The question as to why
$\chi^{\prime\prime}$ retains such complex structure in Tb2InSbO7 that is so
characteristic of both the titanate and stannate, despite the known
sensitivity of Tb-based pyrochlores to chemical disorder Ruminy  _et al._
(2016); Gardner  _et al._ (1999, 2003); Gingras  _et al._ (2000); Chapuis  _et
al._ (2010); Takatsu  _et al._ (2011); Yaouanc  _et al._ (2011b); Hamaguchi
_et al._ (2004); Fennell  _et al._ (2014); Siddharthan  _et al._ (1999), may
assist in addressing the physics that underlies the distinct sample dependence
in the selection of the magnetic ground state for Tb-based pyrochlores Rau and
Gingras (2019).

Figure 7: The out-of-phase ac susceptibility ($\chi^{\prime\prime}$) in
Tb2InSbO7 exhibits a complex spectra with multiple peaks and strong frequency-
dependence (left). A subsequent Arrhenius fit to the frequency dependence of
the 400 mK feature yields activation energy $E_{a}/R=10.7$ K and Mydosh
parameter $\delta$T${}_{\text{F}}=0.04$ (right).  Figure 8: Combined
temperature dependence of the ac and dc magnetization with corresponding
Curie-Weiss fits, and the isothermal dc magnetization of (a) Ho2InSbO7 and (b)
Dy2InSbO7. For temperatures $T<4$ K, the ac susceptibility was measured at
various frequencies $f_{ac}$ ranging from 10 Hz to 9984 Hz in the absence of a
dc field. Here we show data collected for For $T>1.8$ K, the dc susceptibility
was measured in an external field $H_{dc}$ of 100 Oe.

Despite exhibiting a frequency independent peak at 0.62 K that compares
favorably to both its titanate (TN = 350 mK Mauws  _et al._ (2018)) and
stannate (TN = 440 mK Peçanha-Antonio  _et al._ (2019)) analogs, the peak
observed in the case of Sm2InSbO7 is significantly broader. Although a natural
suspicion is that such broadness is a reflection of poor crystallinity, this
is not the case for Sm2InSbO7 which exhibits clear pyrochlore superstructure
reflections and crystallinity comparable to all other pre-Gd InSb compounds.
In fact, such behavior is highly reminiscent of its zirconate analog, which
despite exhibiting excellent crystallinity with no detectable Sm/Zr antisite
defects, does not exhibit a sharp $\lambda$-type anomaly in the heat capacity
down to 0.1 K Xu (2017). Instead, Sm2Zr2O7 has a broad peak centered at 500 mK
corresponding to the full $Rln2$ of the ground state doublet, consistent with
a buildup of short range magnetic correlations Xu (2017); Singh  _et al._
(2008).

Such a comparison to its zirconate analog is further strengthened by clear
similarities in both the temperature and field dependence of the dc
magnetization. As is the case for all Sm3+-based pyrochlores, the temperature
dependence of the dc magnetization in Fig. 6(c) does not exhibit a simple
Curie-Weiss-like behavior due to crystal field effects Singh  _et al._ (2008);
Malkin  _et al._ (2010); Peçanha-Antonio  _et al._ (2019). Restriction of the
Curie-Weiss analysis to a narrow temperature range yields a Curie-Weiss
temperature $\theta\rm{{}_{CW}}$ of $-1.0$ K and an effective paramagnetic
moment $\mu\rm{{}_{eff}}$ of 0.5 $\mu\rm{{}_{B}}$/Sm3+, comparing favorably
with those values recently reported for the zirconate Xu (2017)
($\theta\rm{{}_{CW}}$=$-1.27$ K and $\mu\rm{{}_{eff}}$=0.39
$\mu\rm{{}_{B}}$/Sm3+). As is the case for the zirconate, the isothermal
magnetization does not saturate up to $\mu_{o}H{\rm{{}_{ext}}}\sim$ 7 T,
instead the magnetization is linear, consistent with a pseudospin-1/2 model
assuming Ising anisotropy with a $g$ factor significantly less than 1 Xu
(2017); Singh  _et al._ (2008). Such single-ion anisotropy is consistent with
the $|\pm 3/2\rangle$ dipolar-octupolar doublet previously reported for the
stannate Peçanha-Antonio  _et al._ (2019), titanate Mauws  _et al._ (2018);
Peçanha-Antonio  _et al._ (2019), and zirconate Xu (2017).

Possessing a moment over a magnitude smaller relative to the dipolar spin ice
candidates Ho3+ and Dy3+, the corresponding reduction in the dipolar term that
is characteristic of all Sm3+-based pyrochlores Mauws  _et al._ (2018);
Peçanha-Antonio  _et al._ (2019); Singh  _et al._ (2008); Bondah-Jagalu and
Bramwell (2001); Donnerer  _et al._ (2016), suggesting that exchange will be
particularly influential in the magnetic ground state of Sm2InSbO7. When
combined with the slightly larger lattice parameter, there is a possibility
that quantum fluctuations may be strong enough to prohibit the assumption of
the long-range antiferromagnetic all-in-all-out magnetic order that usually
accompanies antiferromagnetically coupled Ising moments on the pyrochlore
lattice Gomez  _et al._ (2021); den Hertog and Gingras (2000); Guruciaga  _et
al._ (2014); Ikeda and Kawamura (2008); Bramwell and Harris (1998); Opherden
_et al._ (2017); Sibille  _et al._ (2015). As is the case for the zirconate Xu
(2017), strong dynamics may persist in Sm2InSbO7 down to the lowest measurable
temperatures, placing Sm2InSbO7 as a potential quantum spin ice candidate.

####  III.3.3 Dipolar spin ices: Ho2InSbO7 & Dy2InSbO7

The discovery of the the dipolar (“classical”) spin ice state in both Ho2Ti2O7
Harris  _et al._ (1997); Bramwell  _et al._ (2001); Harris  _et al._ (1998)
and Dy2Ti2O7 Ramirez  _et al._ (1999) spurred a flurry of interest in
lanthanide pyrochlore oxides that has spanned the past two decades Gardner
_et al._ (2010); Bramwell and Harris (2020); Giblin  _et al._ (2018); Hallas
_et al._ (2018); Rau and Gingras (2019); Diep (2013). When decorated with
large Ho3+ or Dy3+ moments that exhibit local $\langle 111\rangle$ Ising
single-ion anisotropy amd net effective ferromagnetic exchange,Rosenkranz  _et
al._ (2000); Fukazawa  _et al._ (2002); Blöte  _et al._ (1969); Flood (1974);
Rosenkranz  _et al._ (2000); Jana  _et al._ (2002) Ho2Ti2O7 and Dy2Ti2O7
exhibit the macroscopically degenerate “two-in-two-out” motif fundamental to
the dipolar spin ice Bramwell and Gingras (2001); Bramwell and Harris (2020);
den Hertog and Gingras (2000); Anderson (1956); Champion and Holdsworth
(2004); Ruff  _et al._ (2005); Yavors’kii  _et al._ (2008); Harris  _et al._
(1998); Melko and Gingras (2004).

As illustrated in Fig. 8, both Ho2InSbO7 and Dy2InSbO7 exhibit quantitative
behavior highly reminiscent of both their titanate Matsuhira  _et al._ (2000);
Wiebe and Hallas (2015); Bramwell  _et al._ (1999); Harris  _et al._ (1997);
Ramirez  _et al._ (1999); Siddharthan  _et al._ (1999); Petrenko  _et al._
(2011); Fennell  _et al._ (2004); Cashion  _et al._ (1968) and stannate
Matsuhira  _et al._ (2002); Bondah-Jagalu and Bramwell (2001); Matsuhira  _et
al._ (2000); Wiebe and Hallas (2015); Bramwell  _et al._ (1999); Kadowaki  _et
al._ (2002) counterparts. The isothermal magnetization saturates to half of
the theoretical value, consistent with a pseudospin-1/2 model assuming Ising
anisotropy Bramwell  _et al._ (1999). Temperature-dependent susceptibility
measurements confirm the clear absence of long-range magnetic order. Above 3
K, both systems exhibit Curie-Weiss-like behavior with parameters similar to
those reported for the titanate and stannate. Frequency-dependent maxima at
0.80 K and 1.12 K appear in the ac susceptibility of Ho2InSbO7 and Dy2InSbO7,
respectively, comparable to those temperatures reported for their stannate
Matsuhira  _et al._ (2000, 2000) and titanate Matsuhira  _et al._ (2001,
2000); Snyder  _et al._ (2003, 2001b); Ehlers  _et al._ (2002) counterparts.
Such behavior is also consistent with other Ho- and Dy-based spin ices, where
frequency dependence is indicative of local spins freezing into the spin ice
state Zhou  _et al._ (2008b); Gardner  _et al._ (2011); Ehlers  _et al._
(2004, 2002); Snyder  _et al._ (2003); Hallas  _et al._ (2012); Zhou  _et al._
(2012); Snyder  _et al._ (2002, 2001b); Matsuhira  _et al._ (2001, 2000). An
Arrhenius analysis of the frequency-dependent ac susceptibility (Fig. 9)
yielded activation energies $E_{a}/R$ of 16.5 K and 7.6 K for Ho2InSbO7 and
Dy2InSbO7, respectively, that are comparable to those values reported
Matsuhira  _et al._ (2011b); Snyder  _et al._ (2004b); Yaraskavitch  _et al._
(2012); Snyder  _et al._ (2004b); Quilliam  _et al._ (2011); Matsuhira  _et
al._ (2001); Rutherford (2021) for the corresponding stannate and titanate
systems.

Figure 9: Arrhenius fit of the frequency dependence of the freezing transition
in Dy2InSbO7 (left) and Ho2InSbO7 (right), yielding activation energies of
$E_{a}/R=7.6$ K and 16.5 K, respectively, comparable to their stannate and
titanate analogs. Figure 10: Combined temperature dependence of the ac and dc
magnetization, and the isothermal dc magnetization of (a) Eu2InSbO7, (b)
Tm2InSbO7, and (c) Pr2InSbO7. For temperatures $T<4$ K, the ac susceptibility
was measured at various frequencies $f_{ac}$ in the absence of a dc field. For
$T>1.8$ K, the dc susceptibility was measured in an external field $H_{dc}$ of
100 Oe.

The realization of a dipolar spin ice for two trans-Gd members of the
Ln2InSbO7 family is not completely unexpected. Early studies on “stuffed” spin
ice Ho2(Ti2-xHox)O7-x/2 ($0\leq x\leq 2/3$) demonstrated that the spin ice
state is remarkably robust against significant amounts of disorder Lau  _et
al._ (2006a); Aldus  _et al._ (2013); Lau  _et al._ (2006b); Zhou  _et al._
(2007); Ehlers  _et al._ (2008). The resistance to local disorder appears to
persist up to the defect fluorite limit, where both $A$ and $B$-sites become
indistinguishable. The presence of antisite disorder naturally accounts for
both the negative Weiss temperature and reduced activation energy barrier Ke
_et al._ (2007); Lau  _et al._ (2006a). However there are noteworthy
differences between the Ln2InSbO7 systems and the “stuffed” spin ices. In
particular, the magnitude of the normalized values of both $\chi^{\prime}$ and
$\chi^{\prime\prime}$ are much larger than their stuffed spin ice counterparts
Lau  _et al._ (2006a). The freezing transitions of the InSb pyrochlores are
also much sharper with minimal temperature suppression, reminiscent of their
pure stannate Matsuhira  _et al._ (2000, 2000) and titanate Matsuhira  _et
al._ (2001, 2000); Snyder  _et al._ (2003, 2001b); Ehlers  _et al._ (2002)
analogs.

Recently it was shown that in many Dy-based pyrochlores, anti-site disorder
between the A- and B-sites may destroy the spin ice state. This effect was
observed for the “stuffed” spin ice
${\text{Dy}}_{2}({\text{Dy}}_{x}{\text{Ti}}_{2-x}){\text{O}}_{7-x/2}$ Ueland
_et al._ (2008) and Dy2Zr2O7, the latter of which hosts a disordered spin
liquid-like magnetic ground state Ramon  _et al._ (2019). Recall that
Dy2InSbO7 is initially considered to be within the defect fluorite regime of
the stability diagram, naively characterized by gradual disordering of the A-
and B-sites. In this sense, the prototypical spin-ice behavior seen in Figure
8 is surprising. Potentially both Ho2InSbO7 and Dy2InSbO7 may exhibit the same
anionic sublattice disorder at Tb2Hf2O7 Sibille  _et al._ (2017), which would
preserve the Dy3+ and Ho3+ cation sublattice and the requisite spin-ice state.
Consistent with this thesis, the spin state has been shown to remarkably
robust against disorder isolated to the non-magnetic $B$-site sublattice.
Examples include the doped systems Ho2(Ti2-xHox)O7-x/2 ($x\lesssim$ 0.3) Lau
_et al._ (2006a), Dy2Sn2-xSbxO7+x/2 Ke  _et al._ (2007), and Dy2Ti2-xFexO7 Liu
_et al._ (2014b). Similar effects are observed in the mixed B-site analogs
Dy2ScNbO7 Ke  _et al._ (2007); Rutherford (2021), Dy2GaSbO7 Nandi  _et al._
(2021), and Dy2FeSbO7 Nandi  _et al._ (2017) as well.

As is the case for all trans-Gd members, additional measurements, ($e.g.$
neutron diffraction and PDF, heat capacity, $etc.$) will be necessary to
distinguish between which particular scenario: “stuffed” ($A^{3+}_{B^{4+}}$
anti-site) or $B$-site ($B_{i}B_{ii}$) disordered spin ice applies to both the
Ho3+ and Dy3+ members.

####  III.3.4 Nonmagnetic Singlet Ground States: Tm2InSbO7, Eu2InSbO7 &
Pr2InSbO7

Eu2InSbO7: Both Eu- and Tm-based pyrochlores host trivial non-magnetic spin
singlet ground states Chien and Sleight (1978b); Bondah-Jagalu and Bramwell
(2001); Pal  _et al._ (2018); Dasgupta  _et al._ (2007); Zinkin  _et al._
(1996). In the case of Eu3+, the combination of the 4$f^{6}$ electronic
configuration, strong spin-orbit coupling, and $D_{3d}$ point group symmetry
yields a non-magnetic ${}^{7}F_{0}$ ground state Pal  _et al._ (2018);
Dasgupta  _et al._ (2007); Chattopadhyay  _et al._ (2004). With a gap of over
100 K to its first excited magnetic state ${}^{7}F_{1}$, the Van Vleck term
dominates the low temperature magnetism of Eu3+ pyrochlores Bondah-Jagalu and
Bramwell (2001); Strobel  _et al._ (2010); Sheetal and Yadav (2022). The
dominance of the Van Vleck term for Eu2InSbO7 is consistent with the
temperature independence of the magnetic susceptibility over a large
temperature range below 100 K (Fig. 10(a)), in stark contrast to the behavior
predicted by the Curie-Weiss law. As was the case for the titanate Pal  _et
al._ (2018); Dasgupta  _et al._ (2007) and stannate Bondah-Jagalu and Bramwell
(2001) analogs, the isothermal magnetization does not saturate but continues
linearly up to 7 T, consistent with antiferromagnetic exchange Pal  _et al._
(2018). The increase in the magnetization in the low temperature limit can be
attributed to the presence of small amounts of magnetic impurities that
eventually freeze.

Tm2InSbO7: In the case of Tm3+, the ground state is a non-Kramers doublet that
is weakly split due to crystal field effects – resulting in a non-magnetic
singlet state at sufficiently low temepratures. Bondah-Jagalu and Bramwell
(2001); Blöte  _et al._ (1969); Zinkin  _et al._ (1996); Taira  _et al._
(2002); Lea  _et al._ (1962). As is the case for both the stannate Bondah-
Jagalu and Bramwell (2001) and titanate Zinkin  _et al._ (1996) analogs, the
isothermal magnetization of Tm2InSbO7 does not saturate, with values over an
order of magnitude smaller than its predicted saturation magnetization Jensen
and Mackintosh (1991). The temperature dependence of the magnetic
susceptibility illustrated in Fig. 10(b) does not exhibit Curie-Weiss-like
behavior, instead it is reminiscent of the behavior predicted for a
singlet–triplet level scheme Zinkin  _et al._ (1996); Greedan (1992). The
slight upturn and resulting non-linearity at the low temperature limit is
attributed to the presence of small amounts of magnetic impurities. The
frequency dependence of the ac susceptibility suggests that these impurities
freeze in the low temperature limit.

Pr2InSbO7: In contrast to their Tm3+ and Eu3+ counterparts, Pr-based
pyrochlores can host a plethora of exotic magnetic ground states including the
possibility of the experimental realization of quantum spin ice in the
stannate Zhou  _et al._ (2008b); Sarte  _et al._ (2017), hafanate Sibille  _et
al._ (2016b, 2018); Anand  _et al._ (2016), and zirconate Kimura  _et al._
(2013); Martin  _et al._ (2017); Wen  _et al._ (2017); Petit  _et al._
(2016b). Pr2Zr2O7 in particular has garnered significant attention as
theoretical and experimental treatments have identified $J_{zz}>$ 0 that is
much larger than both $|J_{\pm}|$ and $|J_{\pm\pm}|$, placing Pr2Zr2O7 near
the QSI limit Kimura  _et al._ (2013); Petit  _et al._ (2016b); Martin  _et
al._ (2017). In these systems, the crystalline electric field acting on the
$J=4$ manifold yields a non-Kramers doublet that is well-thermally isolated
($\Delta>$ 10 meV), and thus can be projected onto an effective $J$=1/2
description Kimura  _et al._ (2013); Zhou  _et al._ (2008b); Savary and
Balents (2017); Bonville  _et al._ (2016). Thus, Pr3+-based pyrochlores can
provide an experimental platform to investigate the complex, and often subtle
interplay that result from geometric frustration, strongly correlated
electrons, and orbital degrees of freedom Onoda and Tanaka (2011).

However, the zirconate is also fraught with substantial chemical and
structural disorder Wen  _et al._ (2017); Martin  _et al._ (2017); Trump  _et
al._ (2018). Though the physical origin of the disorder is still a subject of
debate, ($e.g.$ lattice strain Martin  _et al._ (2017), Pr3+ off-centering
Koohpayeh  _et al._ (2014)) the effect is to induce a temperature-independent
continuum of scattering Wen  _et al._ (2017). The profound influence of
structural disorder in the zirconate is intuitive, considering that Pr3+ is a
non-Kramers ion. It was shown that any perturbation away from ideal $D_{3d}$
site symmetry can be mapped onto the introduction of transverse fields onto
the pseudospin manifold Wen  _et al._ (2017); Martin  _et al._ (2017).
Although such a strong random transverse theoretical treatment was shown to be
consistent with experimental observations, the role of exchange disorder plays
remains an open question Rau and Gingras (2019). Theoretical attempts to
address such a question have only focused on both extremes. In the case of
strong exchange and weak exchange disorder, a disordered variant of a QSI is
expected, while in the case of the exchange being significantly weaker than
disorder, one would expect a frozen quadrupolar gas Savary and Balents (2017);
Benton (2018). Instead, the intermediate regime, where the strength of the
exchange and exchange disorder are comparable, has remained largely unexplored
and represents an open question of great interest Rau and Gingras (2019).

At a cursory level, Pr2InSbO7 presents an opportunity to address this
intermediate regime. Its significantly smaller lattice parameter relative to
the zirconate should result in an increase in strength of the Pr3+-O2--Pr3+
superexchange pathways. Instead, as illustrated in Fig. 10(a), both dc and ac
magnetization results suggest a trivial non-magnetic singlet state resembling
that of Tm2InSbO7. The absolute magnitude of the isothermal magnetization is
suppressed, and combined with the lack of saturation is inconsistent with a
pseudospin-1/2 model assuming Ising anisotropy. Further, many quantum spin ice
candidates Zhou  _et al._ (2008b) exhibit a weakly frequency-dependent feature
at base temperatures – but this feature is replaced with a Curie-like tail in
Pr2InSbO7 that may be attributed to the freezing of small amounts of
impurities.

A natural conclusion would be that the splitting of the ground state doublet
into a non-magnetic singlet state was a result of maximal $B$-site disorder,
in strong contrast with the minimal levels of disorder in the zirconate Wen
_et al._ (2017); Martin  _et al._ (2017); Koohpayeh  _et al._ (2014). However,
other mixed $B$-site Pr-based pyrochlores such as Pr2Sn2-xTixO7 Sarte  _et
al._ (2017) and Pr2ScNbO7 pri  have been identified as spin liquid-like
candidates. It is possible that the various magnetic ground states stem from
the deviation between the Shannon radii Shannon (1976) of In3+ and Sb5+
($\delta$=0.2), which is substantially larger than that of Pr2Sn2-xTixO7
($\delta$=0.085) or Pr2ScNbO7 ($\delta$=0.105). The role of the deviation
$\delta$ in the selection of the magnetic ground state is a promising avenue
for future investigation, with direct experimental routes in the form of the
plethora of mixed $B$-site combinations currently synthetically available,
each with differing values of $\delta$.

##  IV Conclusion

With a wide array of possible compositions and the hallmark frustrated
lattice, mixed-site pyrochlores allow access to exotic properties ranging from
moment fragmentation, quantum spin ices, spin liquids, magnetic monopoles, and
many others Gardner  _et al._ (2010); Wiebe and Hallas (2015); Hallas  _et
al._ (2018); Rau and Gingras (2019). In this work we analyzed the known
stability field for the lanthanide oxide pyrochlores Ln2B2O7. We merged recent
synthetic results for the mixed B-site compositions with the previously
established families, providing a broad review of the existing Ln-B-O
compounds. We find the slightly modified “radius-ratio rule” of $1.43$ $\leq$
$R{\rm{{}_{Ln^{3+}}}}$/$R{\rm{{}_{B^{4+}}}}$ $\leq$ $1.82$ to perform across
the entire family with a predictive accuracy of $\sim$94%.

In tandem with our review of pyrochlore compositions, we presented the phase-
pure, rapid synthesis of the entire Ln2InSbO7 mixed B-site family. Our
synthetic improvements enable synthesis of the entire family, improving upon
prior reports wherein only the Gd and Nd variants were produced with
sufficient purity Strobel  _et al._ (2010). Furthermore, we discuss how
microwave-heating methods can stabilize the cubic phase throughout the entire
family, despite small Ln3+ (Lu…Tb) showing the proclivity to crystallize in
low symmetry (monoclinic/triclinic) forms. We have provided empirical evidence
that the InSb family may transition towards the defect fluorite phase for
small Ln3+ (Lu…Tb) while compounds with Ln3+ (La…Gd) are consistent with the
pyrochlore $Fd\bar{3}m$ structure.

This work additionally presents a broad survey of the magnetic properties
throughout the entire Ln2InSbO7 mixed B-site family. We provide analysis and
comparisons to the relevant Ln2B2O7 analogs, examining how the magnetic
properties of the InSb compounds compare to the parent structures. We find
that Nd2InSbO7, Gd2InSbO7, and Er2InSbO7 exhibit properties consistent with
the antiferromagnetic ordering typical of their analogous stannates.
Yb2InSbO7, Tb2InSbO7, and Sm2InSbO7 show properties consistent with a dynamic
ground state. Ho2InSbO7 and Dy2InSbO7 exhibit behavior consistent with the
canonical dipolar spin ices, with energetics consistent with their stannate
equivalents. Finally, Eu2InSbO7, Tm2InSbO7, and Pr2InSbO7 host non-magnetic
singlet ground states. Such behavior is not unexpected for Eu2InSbO7 and
Tm2InSbO7 due to their local crystal field configurations. However, in the
case of Pr3+, the non-magnetic ground state appears to arise from a splitting
of the non-Kramers ground state doublet due to the large Shannon radii
contrast between In3+(VI) and Sb5+(VI).

Our survey of the existing lanthanide pyrochlore oxides combined with the
magnetic characterization of the Ln2InSbO7 family provides a fresh perspective
on the properties of mixed B-site pyrochlores. The chemical diversity of the
mixed B-site systems have become a highlight of recent pyrochlore research, as
the community reexamines the influence of disorder and local sterics on the
realization of complex quantum phenomenon in the pyrochlore phases. Striking
similarities between mixed B-site systems and the analogous mono B-site
pyrochlores continues to emerge, forcing us to reevaluate the presumed
deleterious role of disorder. Instead, chemical disorder becomes a tool –
enabling controlled local chemical changes while maintaining a pristine Ln3+
magnetic sublattice.

##  V Acknowledgments

The authors acknowledge fruitful conversations with A. A. Aczel, J. A. M.
Paddison, C. R. Wiebe, B. A. Frandsen, and A. Krajewska. B.R.O. and P.M.S.
acknowledge financial support from the University of California, Santa
Barbara, through the Elings Fellowship. This work was supported by DOE, Office
of Science, Basic Energy Sciences under Award DE-SC0017752 (S.D.W., P.M.S.,
B.R.O.) This work used facilities supported via the UC Santa Barbara NSF
Quantum Foundry funded via the Q-AMASE-i program under award DMR-1906325. The
research made use of the shared experimental facilities of the NSF Materials
Research Science and Engineering Center at UC Santa Barbara (DMR- 1720256).
The UC Santa Barbara MRSEC is a member of the Materials Research Facilities
Network MRL .

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