# Frustration, strain and phase co-existence in the mixed valent hexagonal
iridate Ba3NaIr2O9

Charu Garg  Department of Physics, Indian Institute of Science Education and
Research, Dr. Homi Bhabha Road, Pune 411008, India  Antonio Cervellino  Swiss
Light Source, Paul Scherrer Institute, CH-5232 Villigen, Switzerland  Sunil
Nair  Department of Physics, Indian Institute of Science Education and
Research, Dr. Homi Bhabha Road, Pune 411008, India  Centre for Energy Science,
Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune
411008, India

(September 3, 2024)

###### Abstract

Using detailed synchrotron diffraction, magnetization, thermodynamic and
transport measurements, we investigate the relationship between the mixed
valence of Ir, lattice strain and the resultant structural and magnetic ground
states in the geometrically frustrated triple perovskite iridate Ba3NaIr2O9.
We observe a complex interplay between lattice strain and structural phase co-
existence, which is in sharp contrast to what is typically observed in this
family of compounds. The low temperature magnetic ground state is
characterized by the absence of long range order, and points towards the
condensation of a cluster glass state from an extended regime of short range
magnetic correlations.

##  I Introduction

Geometrically frustrated magnets- where triangular lattice antiferromagnets
(TLAFs) are considered to be an archetype- remain at the forefront of
contemporary condensed matter Witczak-Krempa  _et al._ (2014); Ramirez (1994);
Moessner and Ramirez (2006). Of particular interest in the recent years have
been a number of Ruthenium and Iridium based perovskite variants which
stabilize in an inherently frustrated environment. In general, the
stabilization of a particular structural ground state depends on the tolerance
limit of the corresponding symmetry which in turn is related to the relative
ionic radii of the constituent elements. For instance, in the perovskite ABO3,
introducing a bigger element at the $A$ and $B$ sites can progressively tune
the lattice from a high symmetry hexagonal to a lower symmetry orthorhombic,
or even a monoclinic one Johnsson and Lemmens (2007). The same is true for the
double (A2BB${}^{{}^{\prime}}$O6) and triple layered perovskites
(A3BB${}^{{}^{\prime}}_{2}O_{9}$) as well, where it has been shown that $B$
site cations with higher atomic radii stabilizes in a lower symmetry Zhao  _et
al._ (2009); Nag  _et al._ (2018); Vasala and Karppinen (2015).

A relatively recent addition to this family of geometrically frustrated
magnets are the Barium based triple perovskite iridates of the form Ba3MIr2O9
(M=alkali metal, alkaline earth metal, 3$d$ transition metal or lanthanides).
The choice of the M-site cation strongly determines the crystallographic
symmetry, which in turn inordinately influences the magnetic ground states.
For example, in M=Zn2+, a close realization of the elusive J=0 state is
observed whereas for M= Mg2+, Sr2+ and Ca2+, deviation from the non magnetic
state in the form of antiferromagnetic exchange interactions, ferromagnetic
and weak dimer like features are observed respectively Nag  _et al._ (2018).
Another addition to this family is the newly reported Ba3CoIr2O9, where Co2+,
being a magnetic ion strongly influences exchange paths leading to weak
ferromagnetism at low temperature and the highest magneto-structural
transition temperature reported in the triple perovskite iridates Garg  _et
al._ (2020). On the other hand, Ba3BiIr2O9 has been reported to exhibit a
giant magneto-elastic transition accompanied by the opening of a spin gap
Miiller  _et al._ (2012). The structure-property relationships in these
systems are clearly driven by a complex interplay between the relative
strengths of competing spin-orbit coupling (SOC), electronic correlation (U),
and a hybridization interaction controlled by the Ir-O-Ir bond angle. Thus
small perturbations, such as changes in lattice parameters caused by
variations of different M ions, can tip the balance between the competing
energies and ground states.

In all the reported 6H hexagonal triple perovskite iridates, the ionic radii
of the $M$ site cation lies in the range of 0.605$\AA$-0.947$\AA$, beyond
which the internal pressure forces the lattice to stabilize in a lower
symmetry. For example, the Ba3CaIr2O9 system has been reported to stabilize in
$C2/c$ monoclinic symmetry Zhao  _et al._ (2009) which is in line with the
expected structural ground state based on the tolerance limit. Interestingly,
an exception appears to be the Ba3NaIr2O9 system, which - in spite of the
similar ionic radii of Na (1.02$\AA$) and Ca (1.00$\AA$) - has been reported
to stabilize in the high symmetry hexagonal structure at room temperatures. In
this report, we discuss this relatively un-investigated Na based triple
perovskite iridate, where iridium is forced to be in the unconventionally high
charge state of 5.5. We investigate polycrystalline specimens of this system
using a combination of high resolution synchrotron diffraction, magnetization,
resistivity and specific heat measurements. We observe that the lattice
appears to accommodate strain as the temperature is reduced, which in turn
precludes the stabilization of a lower symmetry structural phase. This is in
contrast to what is typically observed in this class of materials. On the
other hand, a very gradual and incomplete transformation to a low symmetry
orthorhombic phase is observed, and the high symmetry hexagonal phase survives
till the lowest measured temperatures. Measurements of the magnetization and
specific heat point towards the existence of a extended cooperative
paramagnetic regime characterized by short range magnetic correlations, which
condenses into a cluster glass like state at low temperatures.

##  II Experimental Details

Polycrystalline specimens of Ba3NaIr2O9 were synthesized by using the standard
solid state reaction route. Stoichiometric amounts of high purity BaCO3, Na2O3
and IrO2 were thoroughly ground and then sintered at 11000C under oxygen
atmosphere to maintain the high oxidation state of Iridium. The phase purity
was confirmed by x-ray diffraction using a Bruker D8 Advance diffractometer
with a Cu K$\alpha$ radiation. High resolution synchrotron x-ray diffraction
data was collected using the Materials Science (MS) X04SA beam line
(wavelength 0.56526$\lambda$) at the Swiss Light Source (SLS, PSI
Switzerland). The finely ground powder was loaded in a glass capillary of
diameter 0.3mm and was spun during the data acquisition at various
temperatures between 5 K and 300 K. The structure was analyzed by Rietveld
refinement using the FULLPROF suite Rodriguez-Carvajal (2001); Rietveld
(1969). The structures shown in the manuscript are drawn using Vesta Momma and
Izumi (2011). The homogeneity and stoichiometry of the compound were also
reconfirmed by energy dispersive x-ray (EDAX) from ZEISS Ultra Plus.
Magnetization and physical property measurements were performed using a
Quantum Design (MPMS-XL) SQUID magnetometer and a Physical Property
Measurement System (PPMS) respectively.

##  III Results and Discussion

Figure 1:  Main panel: Fit to the Rietveld refinement of the synchrotron data
at 295 K for Ba3NaIr2O9. The compound crystallizes in a 6H-hexagonal
perovskite with space group P63/mmc (194). The calculated and the observed
diffraction profiles are shown in red and black respectively. The vertical
green lines indicates the Bragg positions and the brown line at the bottom is
the difference between observed and calculated intensities. Inset: Enlarged
view of the higher angle peaks and the corresponding fit. Figure 2: (a) A
schematic representation of the crystal structure of Ba3NaIr2O9 using Vesta.
Here pink and green octahedra represents Iridium and Sodium respectively and
the Barium atoms are represented in Blue. (b) The projection of the structure
along the c-axis is shown. The Iridium octahedra form a hexagonal ring
surrounded by Sodium. (c) Scanning electron micrograph of the compound showing
hexagonal facets.

A Rietveld fit to the synchrotron diffraction data obtained at 300 K is shown
in Fig. 1 where Ba3NaIr2O9 is seen to stabilize in the high symmetry hexagonal
($P6_{3}/mmc$) symmetry, and the lattice parameters are deduced to be a = b =
5.86282(3)$\AA$, c = 14.61922(10)$\AA$ and $\alpha=\beta$ = 90∘; $\gamma$ =
120∘. This is in good agreement with previous reportsLightfoot and Battle
(1990); Rijssenbeek  _et al._ (1999); Doi  _et al._ (2001); Lufaso and zur
Loye (2005). The room temperature structure is illustrated in Fig. 2(a) where
face sharing octahedra (in pink) forms a Ir2O9 dimer and are connected via
corners to NaO6 octahedra (in green). Fig. 2(b) represents the projection
along the crystallographic $c$-axis where IrO6 octahedra forms a hexagonal
ring around the NaO6 octahedra. Since Na is in the +1 oxidation state, Ir is
forced to stabilize in an atypical high oxidation state of +5.5. EDAX
measurements were also used to confirm the stoichiometry. Since it is
difficult to quantify the lighter elements (Na and O) using this technique,
the atomic percentage ratio between heavy elements Ba and Ir was compared. The
Ba:Ir ratio obtained from EDAX was observed to be 1.54 which is very close to
the stoichiometric ratio of 3:2=1.5 expected from the chemical formula. A
scanning electron micrograph image is shown in Fig. 2(c) where hexagonal
facets - a reflection of the underlying crystallographic symmetry - can be
clearly seen.

Figure 3: (a) Temperature evolution of synchrotron peaks at 5 K (black) and
300 K (blue). The lattice strain manifests in the form of broadening of
diffraction peaks as evident by the highly anisotropic peak profile at 5K.
(b,c,d) Attempts to fit the synchrotron diffraction data at 5 K using various
refinement models as indicated.

A comparison of the temperature dependence of a few representative x-ray
diffraction peaks as measured at the extreme temperatures of 5 K and 300 K is
shown in Fig. 3(a). As the temperature is lowered, the diffraction peaks shift
to higher angles and also becomes anisotropic. The modification of the peak
profile could either signal the presence of strain in the lattice or a
transformation to a lower symmetry phase. The former could be a consequence of
the large ionic radii which Na possesses, whereas the latter has been reported
in a number of triple perovskite iridates earlier. Since there were no
additional peaks visible in the low temperature scan, the data was initially
fit using a hexagonal model alone. These attempts were not successful, as is
shown in the Fig. 3(b). Addition of strain using the broadening model
available in FullProf made the fit better as can be seen in Fig. 3(c). This
method is based on Stephens model Stephens (1999) of anisotropic broadening,
where the refinement of microstrain covariance parameters S400, S004 and S112
corresponds to strain along the 100, 001 and 101 hkl planes. Though strain
does appear to have an impact on the low temperature phase, the fitting was
still not satisfactory enough, which hints at the possible presence of an
additional low symmetry phase at low temperatures.

Figure 4: A schematic representation of the crystal structure of Ba3NaIr2O9
using Vesta for the orthorhombic phase. Here pink and green octahedra
represents Iridium and Sodium respectively. The Barium atoms are not shown for
clarity. The yellow dotted line shows the hexagonal arrangement for Iridium
octahedra.

To identify the possible symmetry of the additional low temperature phase,
existing literature in the ruthenate triple perovskite family was referred to,
where multiple scenarios ranging from monoclinic ($P2/c$, $C2/c$) to
orthorhombic ($Cmcm$), or even different structural models for the same
compounds Kimber  _et al._ (2012); Stitzer  _et al._ (2002) have been
reported. After exploring all these possible options, the orthorhombic (space
group-Cmcm (63)) phase Stitzer  _et al._ (2002) resulted in the best fit, with
Rwp and Rp values of 3.24 and 2.47 respectively. The generated pattern was
seen to match well with the high resolution synchrotron data as shown in Fig.
3(d). The lattice parameters obtained from the fit for the additional
orthorhombic phase at 5K are a= 11.6574(11)$\AA$, b=20.1975(21)$\AA$,
c=14.5773(03)$\AA$ and $\alpha=\beta=\gamma$= 90∘. Fig. 4 depicts this
orthorhombic phase as viewed along the crystallographic $c$-axis. The yellow
dotted line indicates the hexagonal arrangement formed by Ir octahedra. The
high temperature hexagonal structural symmetry allows for only one
crystallographic position (4f) for Iridium. Therefore, given the presence of
mixed valent state Ir5.5, this position is highly disordered. On the other
hand, the low temperature C-centred orthorhombic symmetry is a 2a x 2b
primitive hexagonal pseudo-cell (or an orthohexagonal cell) and allows for
three different crystallographic sites for Ir (8f,8f,16h) making it possible
for the charge to be redistributed at these distinct cation sites. In
addition, Na also now has 3 unique Wyckoff positions (4a, 4b 8d) allowing for
the movement of Iridium while still maintaining the orthorhombic crystal
framework. This is a complex low symmetry where each element has multiple
unique positions, the details of which are given in Kimber  _et al._ (2012).
There have been prior reports of orthorhombic phases with only one
crystallographic position for Ir, but attempts to fit the low temperature
profile of Ba3NaIr2O9 using this symmetry were not successful. Interestingly,
in the ruthenium analogue, this need for multiple Ru positions was attributed
to the presence of a charge ordered state.

Figure 5: (a) The variation of the phase fraction of the hexagonal P63/mmc
with temperature. As the temperature reduces, the hexagonal phase converts
slowly to orthorhombic phase, nucleating at 50K and reached 80$\%$ of the
total volume fraction at 5 K. Temperature evolution of the (b) volume, (c)
ratio of lattice parameters c/a for the hexagonal symmetry. A slight variation
in both the parameters are observed marking the onset of the lower symmetry
orthorhombic phase. (d) The temperature dependence of the microstrain
parameters SHKL for three different hkl is depicted. The sharp change in S400
and S004 close to the structural transformation temperature is consistent with
distortions of the lattice with the onset of orthorhombic symmetry.

It is observed that down to 50 K, a single structural hexagonal model with
strain parameters is sufficient for the fitting. As a function of reducing
temperatures, the phase fraction of hexagonal symmetry is invariant till 50 K,
below which the orthorhombic symmetry is seen to stabilize, reaching 20$\%$ of
the total volume fraction at 5 K (Fig. 5(a)). The temperature dependence of
volume and $c/a$ ratio for the primary hexagonal phase are depicted in in Fig.
5(b) and Fig. 5(c) respectively. Clearly, below 50 K, the $c/a$ ratio shows a
change in slope associated with onset of the partial structural phase
transformation. The evolution of the secondary orthorhombic phase is also
evident in the temperature dependence of the microstrain covariance parameters
as in depicted in Fig. 5(d). The strain parameters S400 and S004 show a sharp
change close to the structural transformation temperature and remains almost
constant below it, whereas the parameter S112 increases dramatically. These
changes in the microstrain parameters are indicative of deviations in the
$\alpha$ and $\beta$ angles of the hexagonal lattice framework, and consistent
with a distortion towards an orthorhombic symmetry. It is interesting to note
that the emergence of the secondary orthorhombic phase at low temperatures is
not associated with the observation of a splitting of the hexagonal peaks, as
was observed in an earlier report on the same system zur Loye  _et al._
(2009). We believe that this is due to the excess broadening of the
diffraction peaks due to strain. This incipient strain not only masks the peak
splitting expected due to the orthorhombic distortion, but also results in an
incomplete conversion of the high temperature hexagonal phase to the lower
symmetry orthorhombic one.

Fig. 6(a) shows the temperature dependence of the magnetic susceptibility of
Ba3NaIr2O9 as measured at an applied field of 500 Oe. The susceptibility
increases with decrease in temperature with the zero field cooled (zfc) and
field cooled (fc) curves diverging close to 6K as shown in Fig. 6(b). This is
at variance with what has been reported in early single crystalline specimens
of this system, where features in the magnetization was observed at 75 K and
50 K zur Loye  _et al._ (2009); Kim  _et al._ (2004). The temperature
dependence of the heat capacity as measured from 2-250 K is depicted in Fig.
6(c). Clearly, the low temperature anomaly observed in magnetization is absent
here which implies that the change in entropy is rather small. Fig. 6(d) shows
the temperature dependence of reciprocal magnetic susceptibility (1/$\chi$).
Interestingly, a linear region was observed well in excess of 200 K, and hence
only the temperature range 260- 300 K was chosen to fit the inverse magnetic
susceptibility using the Curie-Weiss law. An effective magnetic moment value
3.42(5)$\mu_{B}$ per formula unit and a Weiss temperature ($\theta_{c}$) of
-285.36(1.1) K were obtained, with the latter being indicative of the extent
of frustration in this system, since we only observe a feature in
magnetization at 6 K.

Since Iridium is the only magnetic ion in this system, the magnetic moment
arises from the charge balance between Ir(V) (5d4) and Ir(VI) (5d3). Based on
these oxidation states and the octahedral coordination environments, the
theoretical spin-only moment for non-interacting mixed valent Ir5+ (S=1, 2.83
$\mu_{B}$) and Ir6+ (S=3/2, 3.87 $\mu_{B}$) is 6.7$\mu_{B}$ per formula unit.
These calculated moments are significantly larger from the experimentally
determined value 3.42(5)$\mu_{B}$ per formula unit. However, the
experimentally obtained value is close to the reported magnetic moments
3.6$\mu_{B}$ per formula unit for Ba3NaIr2O9 and 3.93$\mu_{B}$ per formula
unit for Ba3LiIr2O9, both having Ir in a similar 5.5 charge state Kim  _et
al._ (2004). Such reduction in moment is a peculiar feature seen in iridates
and has been reported for a wide range of iridium based oxides Nag  _et al._
(2018); Boseggia  _et al._ (2013); Ming  _et al._ (2018); Rau  _et al._
(2016).

The strong suppression of the magnetic moment here is ascribed to the joint
effect of spin orbit interaction and strong covalency, resulting in the
formation of metal-metal bonds. They act against the intraatomic Hund’s rule
exchange interaction to reduce the total magnetic moment on the Iridium dimer.
This was further confirmed by our synchrotron measurements where an anomalous
shortening of Ir-Ir bond distance in the +5.5 valence state (2.73$\AA$) as
compared to the +5 state (2.75$\AA$) corroborates the formation of the metal-
metal bonds. Na being non-magnetic, the inter and intra dimer interactions
between Ir ions drives the magnetic ground state of the system. In the absence
of a superexchange path for inter dimer interactions (J2 and J3), the extended
superexchange pathways Ir-O-O-Ir could possibly influence the magnetic
exchange interactions. The Ir dimers are separated from each other via non
magnetic Na octahedra (green) as shown in Fig. 7. The next nearest neighbour
inter dimer Ir (5.8619(8) and 5.6911(12)) are connected via 2 oxygens from the
Na octahedra as shown by the dotted lines. Thus, in addition to metal-metal
bonds, the presence of super exchange and extended super exchange interactions
pathways lead to complex magnetic exchange interactions.

Figure 6: (a) Zero field cooled temperature dependent magnetization measured
at 500 Oe for Ba3NaIr2O9. (b) the FC and ZFC curves show divergence close to
6K, corresponding to a cluster glass transition. (c) Heat capacity as a
function of temperature measured in zero magnetic field shows no discernible
anomaly in the entire range of measurement. (d) log-log plot of temperature
dependence of the inverse magnetic susceptibility data as measured at 500 Oe.
The solid red line is a guide to the eye to show the deviation from Curie-
Weiss law, which starts close to 175 K. (e) Thermo-remnant magnetization (TRM)
measured at 1 kOe with two systematic jumps corresponding to the onset of the
co-operative paramagnetic regime, and the cluster glass state respectively.
Figure 7:  A schematic representation of the crystal structure of Ba3NaIr2O9
using Vesta. The projection of the structure perpendicular to the c-axis is
shown. Here pink and green octahedra represents Iridium and Sodium
respectively and the Barium atoms are not shown for clarity. The Iridium
dimers (pink) are separated by Sodium octahedra (green) along the c-axis where
the extended super exchange between Ir dimers is mediated by oxygens in Na
octahedra as shown by dotted line.

Though heat capacity measurements did not show evidence of any long range
ordered state, a Curie Weiss fit of the inverse magnetic susceptibility was
valid only in temperatures in excess of 260 K, indicating the presence of an
extended regime of short range magnetic correlations. To gain further insight
in to the extent of this regime, we performed temperature dependent
measurements of the Thermo-remnant magnetization (TRM), which has proven to be
an effective tool in the investigation of magnetically frustrated systems. A
TRM measurement as performed on the Ba3NaIr2O9 system in a cooling field of 1
kOe is depicted in Fig. 6(e). Two precipitous jumps are clearly observed - one
below 10 K, which corresponds to the low temperature magnetic transition
observed in the ZFC-FC measurements, and one just below 175 K, which roughly
corresponds to the region where the inverse magnetic susceptibility deviates
from the linear Curie-Weiss fit. In the absence of long range order, this
feature at high temperature could be ascribed to the onset of a cooperative
paramagnetic regime. First coined by Villain Villain (1979), cooperative
paramagnetism was used to describe the low temperature dynamics of a classical
Heisenberg spins on a corner sharing tetrahedral framework, and is a defining
feature of systems with high geometric frustration. Cooperative paramagnetism
is seen in many transition metal oxides which crystallizes in magnetic spin
configurations that are geometrically or topologically prone to frustration
due to underlying lattices based upon corner, edge or face sharing triangles
or tetrahedra. A wide range of systems including pyrochlore, spinels, and
jarosites are now known to exhibit this phenomena Lee  _et al._ (2010); Ueland
_et al._ (2010); van Duijn  _et al._ (2008). This state can also be looked
upon as being analogous to the Griffiths phase Yamamoto  _et al._ (2020), with
the notable difference that the low temperature magnetic ground state instead
of being magnetically ordered, now undergoes a glass-like dynamical phase
transition. We believe that the nucleation of finite size correlated regions
within the antiferromagnetic matrix starts to develop close to 175 K. As the
temperature reduces, magnetic frustration develops due to competing intra
dimer (nearest neighbour J1) and inter dimer (next- nearest neighbour J2 and
J3) interactions. The absence of conventional long range antiferromagnetic
order is due to the interplay between frustration and quenched disorder. As
proposed by Imry and Ma Imry and Ma (1975), a random quenched disorder
inhibits the transition to a long range magnetically ordered state but instead
favours the nucleation of correlated magnetic clusters Pal  _et al._ (2019).

Figure 8: Main panel: Resistivity ($\rho$) plotted as a function of
temperature. Inset: Resistivity (ln$\rho$) as a function of temperature
(T-0.33). The red line is the fit to the Mott variable range hopping (VRH)
model.

Interestingly, the high temperature magnetic feature ($\sim$175 K) which we
observe in the TRM measurements is not easily discernible in other
measurements, and hence has gone unreported in prior single crystal
measurements of Ba3NaIr2O9 as well Kim  _et al._ (2004). This is a consequence
of the fact that the magnetic susceptibility of the paramagnetic matrix
($\chi{{}_{PM}}$) would be of the same order (or even larger) than that of the
antiferromagnetic clusters ($\chi{{}_{AFM}}$), making it difficult to
unambiguously determine the contribution of the antiferromagnetic clusters in
traditional in-field magnetic measurements. On the other hand, since TRM is a
zero-field measurement, the contribution of the paramagnetic magnetic
susceptibility is likely to be suppressed, allowing for one to identify more
clearly the temperature regime at which the antiferromagnetic clusters begin
to nucleate. Interestingly, the ruthenate analogue of this triple perovskite
was reported to exhibit the opening of a charge gap at 210 K Kimber  _et al._
(2012), though we do not observe any evidence of a similar phenomena in its
Iridium counterpart investigated here. The magnetic transition in these family
of oxides is typically associated with a symmetry breaking lattice distortion
which alters the exchange parameters, thereby neutralizing the frustration. In
the case of Ba3NaIr2O9, the interesting capacity of the system to accommodate
strain impedes a traditional structural transformation. Therefore, rather than
observing a first order transition from hexagonal symmetry to an orthorhombic
one, we observe a slowly evolving strained lattice gradually transforming to a
lower symmetry where the major phase still retains the original high
temperature symmetry. A strained lattice of this nature is probably closer to
the reports of the triple perovskite family when subjected to external
pressure Zhao  _et al._ (2009); Senn  _et al._ (2013). For instance on
application of pressure, the Ba3NaRu2O9 transforms to a new phase,
3C1:2-Ba3NaRu2O9, where the charge gap completely disappears and Pauli
paramagnetism emerges, possibly as a consequence of strong electron-electron
correlations. The Ba3CaRu2O9 system has also been reported to exhibit excess
strain in the lattice, in the form of peak broadening as the temperature was
lowered. Therefore, the ground state in Ba3NaIr2O9 is clearly influenced by
the complex interplay of a mixed valent Ir, frustration, phase coexistence and
strain.

The temperature dependence of the electrical resistivity is shown in the Fig.
8. The system is semiconducting in nature, with the magnitude of resistivity
changing by 4 orders of magnitude from its room temperature value. Attempts to
fit using the Arrhenius model and Efros-Shklovskii variable range hopping (ES-
VRH) model were unsuccessful. A better fit was obtained by using the Mott
variable-range hopping (VRH) model which is given by:

$\rho\ \propto\ exp((T_{0}/T)^{\nu})$

where the best fits were obtained for $\nu$=1/3 indicating variable range
hopping in two dimensions Mott and Davis (2012). The magnetic Ir dimers are
connected via non-magnetic Na octahedra, generating a pseudo 2D structure.
Thus, the crystal structure of this triple perovskite can be expressed by
alternate stacking of two kinds of 2-D layers which consist of the NaO6
octahedra and the Ir2O9 polyhedra. This may account for the observed 2-D
resistivity behaviour. The resistivity of Ba3NaIr2O9 as a function of ln$\rho$
vs T-0.33 is shown in the inset of Fig. 8. The localization of states due to
strong Coulomb interactions and slight structural disorder would be consistent
with variable range hopping behaviour. The resistivity of the ruthenate
analogues of the triple perovskites Ba3MRu2O9 (R=Fe,Co,Ni,Cu,In,Ln), all
follows the same characteristics Rijssenbeek  _et al._ (1998); Hinatsu and Doi
(2003).

Figure 9: Normalised Isothermal Remanent Magnetization (IRM) at 2K in cooling
fields of 500 Oe and 1 T (inset), fitted using the Kohlrausch Williams Watt
(KWW) stretched exponential (red) and a power law (blue). Figure 10:
Isothermal magnetization measured at 2 K for Ba3NaIr2O9 after cooling in
different magnetic fields. A systematic shift of the hysteresis loop as a
function of the magnitude of the cooling field can be clearly seen indicating
the presence of mixed magnetic interactions.

In addition to frustration, the presence of mixed valent states of Iridium and
phase co-existence sets the stage for inhomogeneous magnetic exchange
interactions. The observation of an anomaly below 6 K in the M(T) curves
indicates the emergence of glassy dynamics owing to this frustration. However,
the signal was too small for us to clearly identify a frequency dependent peak
in the ac susceptibility measurements. Another method to probe glassy dynamics
is to use the time evolution of Isothermal Remanent Magnetization (IRM). This
involves cooling the sample from 300 K to 2 K in the presence of a magnetic
field, after which the magnetic field is switched off and the decay of
magnetization is measured as function of time. A special formulation of power
law is known to study the time dynamics of magnetization for glasses under
stress known as the Kohlrausch Williams Watt (KWW) stretched exponential
equation Edwards and Vilgis (1986); Ito  _et al._ (1986); Ghara  _et al._
(2014) given by :

$m(t)=m_{0}-m_{g}exp\\{-(t/\tau)^{\beta}\\}$

where m0 is related to initial remanent magnetization, mg is magnetization of
glassy component, $\tau$ and $\beta$ are the characteristic relaxation time
constant and stretching exponent respectively. Here m(t) is representative of
the sum of many exponential decays weighted by a distribution of individual
relaxation times, with the magnitude of $\beta$ indicating the breadth of that
distribution Sidebottom  _et al._ (1995). The value of $\beta$ has been
reported to lie between 0 and 1 for a wide range of disordered systems. The
normalized magnetization m(t) = (Mt/Mt=0) as measured in Ba3NaIr2O9 at 2 K
with cooling fields of 500 Oe (main panel) and 1T (inset) at 2K is plotted in
Fig. 9. As depicted by the blue curve, the fit to a simple power law was not
satisfactory. However, a good fit was obtained for the KWW model and the
resultant values of $\beta$ are 0.518(14) and 0.5464(68) for 500Oe and 1T
respectively. These values are in lines with the reported values for cluster
glass phase in many double perovskites Pal  _et al._ (2019); Anand  _et al._
(2019), and reinforces our contention that the low temperature magnetic ground
state is one which has magnetically frozen clusters.

Figure 11:  Main panel: The low temperature specific heat CP/T3 as a function
of temperature. The slight upturn in the low temperature range is a strong
indication of disorder in the system. Inset: The red line line depicts the fir
to the low temperature specific heat data using the Debye and Petit’s law.

The magnetic interactions in Ba3NaIr2O9 are predominantly antiferromagnetic,
though, signatures of the presence of mixed magnetic interactions are
suggested by a weak loop opening in magnetization isotherms at 2 K as shown in
Fig. 10. As revealed by our synchrotron studies, the low temperature structure
comprises of a highly strained lattice with two unique structural motifs
coexisting. This coupled with the existence of finite size antiferromagnetic
clusters allow for exchange bias, with the antiferro and ferro-magnetic
contributions arising from the magnetic order within ordered clusters and
uncompensated spins at the surface of these clusters respectively (Fig. 10).
The presence of a low temperature glass-like magnetic ground state is also
evidenced in a strong upturn in C/T3 vs T Fig. 11, with a clear deviation from
what is expected from Debye’s law. This excess entropy arises as a consequence
of the glassy dynamics Banerjee  _et al._ (2009), and appears to be a
signature common to structural and magnetic glasses. This is also evident on
plotting $C/T$ vs T2 curve, indicating the presence of an excess entropy that
releases as a consequence of short range ordering. The inset of Fig. 11 shows
the fit to the low temperature $C/T$ vs $T^{2}$ curve . The data is fitted
using the expression $C_{p}=\gamma T+\beta T^{3}$ where $\gamma$ and $\beta$
are related to the electronic and vibrational degrees of freedom respectively.
We also calculated the Debye temperature $\theta_{D}$, which is derived from
the expression, $\theta_{D}$ = (12$\pi^{4}$pR/5$\beta$)1/3, where R is the
ideal gas constant and p is the number of atoms per formula unit. The
calculated value of $\theta_{D}$ is 236.84K. The obtained values of $\gamma$
and $\beta$ are 77mJ/molK2 and 2.19mJ/molK4T respectively. The high value of
$\gamma$, unusual for insulating systems, can be attributed to the inherent
disorder which affects the spin, charge and orbital degrees of freedom. This
has been previously observed in insulating tripe perovskite iridate
(Ba3ZnIr2O9-25.9mJ/molK2), and manganites Hardy  _et al._ (2003); Nag  _et
al._ (2016). The high value observed here signifies the excess entropy
imparted by the frustration and disorder in this oxide owing to the mixed
valence state and stress. Interestingly, on the application of a moderate
magnetic field (0.5T), no change in the heat capacity was observed (not shown
here), which suggests against the presence of paramagnetic impurity centres.
Yamashita  _et al._ (2011); Schliesser and Woodfield (2015).

##  IV Summary

In summary, we report on the structure-property relationship in the mixed
valent geometrically frustrated triple perovskite iridate Ba3NaIr2O9. In
contrast to what is expected from purely structural considerations, this
system stabilizes in a high symmetry hexagonal symmetry at room temperatures.
On reducing the temperature, the lattice prefers to be strained rather than
distort to a low symmetry phase, as is the norm in this family of materials.
Though a low symmetry orthorhombic phase is finally nucleated below 50 K, this
conversion is only partial and the high symmetry hexagonal structure remains
the dominant one down to the lowest measured temperatures. Magnetic
measurements indicate an extended co-operative paramagnetic regime, which
finally freezes to a cluster glass-like phase at very low temperatures, as is
also evidenced from magnetization decay and specific heat data. This makes an
interesting addition to the family of triple perovskite iridates which exhibit
material sensitive physical properties.

##  V Acknowledgements

S.N. acknowledges DST India for support through the DST Nanomission Thematic
Unit Program, SR/NM/TP-13/2016. C.G. and S.N. thank the Department of Science
and Technology, India (SR/NM/Z-07/2015) for financial support and the
Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR) for managing
the project.

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