# Magnetic properties of a spin-orbit entangled $J_{\mathrm{eff}}$ $=1/2$
three-dimensional frustrated rare-earth hyperkagome

B. Sana  Department of Physics, Indian Institute of Technology Madras,
Chennai, 600036, India
M. Barik  Department of Physics, Indian Institute of Technology Madras,
Chennai, 600036, India
M. Pregelj  Jožef Stefan Institute, Jamova cesta 39, 1000 Ljubljana, Slovenia
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska u. 19,
1000 Ljubljana, Slovenia  U. Jena  Department of Physics, Indian Institute of
Technology Madras, Chennai, 600036, India
M. Baenitz  Max Planck Institute for Chemical Physics of Solids, Nöthnitzer
Strasse 40, 01187 Dresden, Germany  J. Sichelschmidt  Max Planck Institute for
Chemical Physics of Solids, Nöthnitzer Strasse 40, 01187 Dresden, Germany  K.
Sethupathi  Department of Physics, Indian Institute of Technology Madras,
Chennai, 600036, India
Quantum Centre of Excellence for Diamond and Emergent Materials, Indian
Institute of Technology Madras, Chennai, 600036, India  P. Khuntia
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Madras, Chennai, 600036, India
Quantum Centre of Excellence for Diamond and Emergent Materials, Indian
Institute of Technology Madras, Chennai, 600036, India

###### Abstract

The interplay between competing degrees of freedom can stabilize non-trivial
magnetic states in correlated electron materials. Frustration induced strong
quantum fluctuations can evade long-range magnetic ordering leading to exotic
quantum states such as spin liquids in two-dimensional spin-lattices such as
triangular and kagome structures. However, the experimental realization of
dynamic and correlated quantum states is rare in three-dimensional (3D)
frustrated magnets wherein quantum fluctuations are less prominent. Herein, we
report the crystal structure, magnetic susceptibility, electron spin resonance
and specific heat studies accompanied by crystal electric field calculations
on a 3D frustrated magnet Yb3Sc2Ga3O12. In this material, Yb3+ ions form a
three-dimensional network of corner-sharing triangles known as hyperkagome
lattice without any detectable anti-site disorder. Our thermodynamic results
reveal a low-energy state with $J_{\mathrm{eff}}$ $=1/2$ degrees of freedom in
the Kramers doublet state. The zero field cooled and field cooled magnetic
susceptibility taken in 0.001 T rules out the presence of spin-freezing down
to 1.8 K. The Curie-Weiss fit of magnetic susceptibility at low temperature
yields a small and negative Curie-Weiss temperature indicating the presence of
a weak antiferromagnetic interaction between $J_{\mathrm{eff}}$ $=1/2$ (Yb3+)
moments. The Yb-electron spin resonance displays a broad line of asymmetric
shape consistent with the presence of considerable magnetic anisotropy in
Yb3Sc2Ga3O12. The crystal electric field calculations suggest that the ground
state is well separated from the excited states, which are in good agreement
with experimental results. The absence of long-range magnetic ordering
inferred from specific heat data indicates a dynamic liquid-like ground state
at least down to 130 mK. Furthermore, zero field specific heat shows a broad
maximum around 200 mK suggesting the presence of short-range spin correlations
in this three-dimensional frustrated antiferromagnet.

††preprint: APS/123-QED

##  I Introduction

Geometrically frustrated magnets have drawn significant attention due to their
emergent magnetic phenomena arising from competing interactions and quantum
fluctuations [1, 2, 3, 4]. Two-dimensional (2D) frustrated magnets are prime
contenders to host exotic quantum states with fractional quantum numbers
following the seminal proposal of quantum spin liquid state on a triangular
lattice antiferromagnet by P. W. Anderson [5]. Frustrated magnets with next
nearest-neighbor exchange interaction, bond dependent Ising interactions and
magnetic anisotropy are ideal candidates in this context [6, 2]. A quantum
spin liquid (QSL) is characterized by the absence of long-range magnetic order
despite strong exchange interaction between spins, the absence of local order
parameter and spins not freezing down to absolute zero. The spins maintain a
highly entangled state in the QSL state, which harbors exotic fractional
excitations such as spinons and Majorana fermions that are essential
ingredients for future quantum information processing [6, 3, 4, 7, 8, 9].
Geometrically frustrated triangular, kagome and hyperkagome lattices, wherein
frustration induced strong quantum fluctuations are at play, offer a promising
platform for the experimental realization of QSL. The identification of
associated non-trivial quasi-particles and their interactions in the QSL state
is an attractive track in quantum condensed matter. The exotic excitations in
the QSL state are completely different from spin-wave excitations observed in
conventional magnets with static order [6, 1, 2]. There are reports on a quite
good number of promising two-dimensional quantum spin liquid candidates,
however a very few 3D quantum spin liquids are studied so far [10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 8, 7, 21, 22, 23, 23, 24, 25, 26, 27, 28, 29, 30,
31].

Remarkably, an excellent three-dimensional lattice to realize exotic magnetic
ground state is rare-earth pyrochlore oxides, R2B2O7, where R3+ and B4+ are
generally trivalent rare-earth and tetravalent transition-metal ions,
respectively. In this rare-earth magnet family, the ground state of some
Kramers ion is a well-separated crystal electric field doublet with
interacting $J_{\mathrm{eff}}$ $=1/2$ pseudo-spins at low temperatures.
Depending upon how the ground state doublet transforms under time-reversal
symmetry and local D3d crystal field symmetry of the rare-earth site, this
pyrochlore can harbor dipole-octupole ground state doublet, which supports
exotic U(1) quantum spin liquid ground state [32, 33]. For example, in the
quantum spin liquid candidate Ce2Zr2O7, $x$ and $z$ components of pseudo-
spin-1/2 degrees of freedom of Ce3+ ion behave as dipoles, whereas only the
$y$ component behaves as an octupole under the time-reversal symmetry and
lattice symmetry [33]. Coupling between magnetic field and spinons enables to
control spinon excitations by applying an external magnetic field in this
octupolar spin liquid candidate [32]. Among 3D spin liquid candidates, three-
dimensional network of corner-sharing triangles known as hyperkagome lattice
has recently drawn significant attention due to their rich ground state
properties. One exemplar is PbCuTe2O6, wherein spin $S=1/2$ transition metal
Cu2+ ions constitute a hyperkagome lattice. The local probe techniques nuclear
magnetic resonance (NMR) and muon spin resonance ($\mu$SR) revealed a dynamic
ground state down to 20 mK in this hyperkagome [25, 24]. Diffusive continua in
magnetic excitation spectra revealed by neutron scattering experiment provides
a strong evidence of fractional spinon excitations in this 3D frustrated
magnet [26]. Another geometrically frustrated effective spin
$J_{\mathrm{eff}}$ $=1/2$ 3D spin liquid candidate is Na4Ir3O8 [34, 35].
Despite strong antiferromagnetic interaction between Ir4+ ($J_{\mathrm{eff}}$
$=1/2$) moments, which was confirmed by a large Curie-Weiss temperature of
$-650$ K, there is no sign of magnetic ordering down to a few Kelvin. A
theoretical study proposed the existence of a Z2 quantum spin liquid state in
the quantum regime (i.e, small spin) of hyperkagome lattice Na4Ir3O8 [36].
However, a small splitting of zero field cooled and field cooled magnetization
data suggests the role of static moments below 7 K, which imposes a strong
constraint for the unambiguous identification of spin liquid ground state in
this material [37]. The rare realization of 3D QSL is due to the fact that the
Neél order or spin freezing is energetically favorable over the QSL state in
the case of a 3D frustrated spin-lattice for $S>1/2$. However, frustration
induced strong quantum fluctuations in the $S=1/2$ or $J_{\mathrm{eff}}=1/2$
system makes it a potential candidate to host a dynamic ground state [5, 38,
30]. It is highly relevant to explore whether the disorder interaction or
exchange anisotropy or lattice imperfections account for spin-freezing in this
type of 3D spin liquid candidates.

In this context, synthesis and investigation of new three-dimensional
frustrated spin-lattices, wherein interplay between competing degrees of
freedom and spin correlations could lead to exotic quantum states, are highly
needed. Considerable efforts have been devoted in this direction, for
instance, disorder-free 4$f$-based hyperkagome lattice, wherein spin
frustration and anisotropic interactions governed by spin-orbit interaction
stabilizing an effective low energy $J_{\mathrm{eff}}$ $=1/2$ ground state,
offers an alternate route for the realization of elusive 3D spin-liquids.
Hyperkagome lattice-based material Li3Yb3Te2O12 wherein Yb3+ spins decorated
on a corner-shared frustrated triangular network shows a dynamical ground
state down to 38 mK temperature [39]. Magnetic Yb3+ ions show short-range spin
correlations with $J_{\mathrm{eff}}$ $=1/2$ degrees of freedom in the ground
state. One promising family of compound to study rare-earth hyperkagome
lattice is lanthanide garnets with general formula R3A2X3O12, R = rare-earth
elements; A = Ga, Sc, In, Te; X = Ga, Al, Li [40]. In this series, rare-earth
ions occupying two interpenetrating frustrated hyperkagome networks have a
rich potential to harbor myriads of novel physical phenomena. For example,
antiferromagnetic spin correlation and local magnetic anisotropy lead to a
non-trivial magnetic structure with long-range hidden order state in Gd3Ga5O12
[41, 42]. Spin clusters behave like a single object due to strong spin
correlation and these spin clusters show long-range non-dipolar order although
the individual spin maintains a dynamic liquid-like state [43, 41]. On the
other hand, emergent magnetic behavior with short-range spin correlations was
observed in $J_{\mathrm{eff}}$ $=1/2$ Yb3+ containing garnet Yb3Ga5O12 [44].
The specific heat data show a $\lambda$-like anomaly which was attributed to
long-range magnetic order [45]. However, muon spin resonance and Mössbauer
spectroscopy measurements indicate the absence of long-range magnetic order
[45, 46]. In an opposite scenario, antiferromagnetically ordered state
develops below in the sub-Kelvin temperature range in other members of the
garnet family [47, 48]. Though there are several studies on gallium garnets,
magnetic properties of scandium gallium garnets have yet to be investigated so
far. Recent study on R3Sc2Ga3O12 (R = Tb, Dy, Ho) reveals that most of these
garnets undergo a long-range antiferromagnetic ordering state at low
temperature [40]. The magnetic properties of this family of materials is
highly sensitive to external stimuli such as chemical pressure, temperature,
or an applied magnetic field. It is pertinent to test the effect of external
perturbations such as non-magnetic or magnetic ions on the anisotropy,
exchange interactions and hence the underlying spin Hamiltonian in this
interesting class of three-dimensional frustrated magnets. The magnetic moment
of Yb3+ ion is relatively weak compared to other R3+ ions. Being a Kramers ion
Yb3+, a combination of crystal electric field and spin-orbit coupling could
lead to a low energy $J_{\mathrm{eff}}$ $=1/2$ state in the Yb variant of this
promising garnet series. This suggests that Yb3Sc2Ga3O12 is of special
interest to look for spin-orbit driven ground state.

Herein, we report the synthesis, magnetization, electron spin resonance (ESR),
and thermodynamic studies on a garnet Yb3Sc2Ga3O12, wherein Yb3+ ions
constitute a three-dimensional frustrated corner-sharing network of triangles,
namely, a hyperkagome spin-lattice. The combination of crystal electric field
and spin-orbit coupling stabilizes a low energy pseudo-spin $J_{\mathrm{eff}}$
$=1/2$ state of the Kramers doublets of Yb3+ ion at low temperature. The
absence of ZFC-FC splitting of magnetic susceptibility in 0.001 T rules out
spin-freezing down to 1.8 K in this material. The low-temperature Curie-Weiss
fit yields a weak antiferromagnetic interaction between Yb3+
($J_{\mathrm{eff}}$ $=1/2$) moments in the spin-lattice. The magnetic ground
state is investigated via thermodynamic measurements which reveal that this
garnet remains in a quantum disordered ground state at least down to 130 mK.
The specific heat in zero field shows a broad maximum around 200 mK implying
the presence of short-range spin correlations in this three-dimensional
frustrated spin-lattice. The crystal electric field calculations infer that
the lowest Kramers ground state is well separated from the excited states and
imply considerable magnetic anisotropy which is consistent with experiments.

##  II Experimental details

Polycrystalline sample of Yb3Sc2Ga3O12 was prepared via the sol-gel method.
Rare-earth oxide Yb2O3 was preheated at 900∘C for 6 hours prior to use to
remove moisture and carbonates. Stoichiometric amount of Yb2O3 (Alfa Aesar,
99.998%), Sc2O3 (Alfa Aesar, 99.9%) and Ga2O3 (Alfa Aesar, 99.999%) were
dissolved in hot nitric acid separately in three beakers. The concentration of
these acidic solutions was reduced by adding and evaporating deionized water.
These three nitrate solutions were then mixed together and polyethylene glycol
was added to the solution. This solution was kept in between 90∘C to 120∘C on
a magnetic stirrer for several hours. The gel was dried yielding ash-like
powders, which were preheated at 800∘C for 6 hours, and finally a white color
polycrystalline sample was obtained. This is followed by the heat treatment of
the resulting sample at 1000∘C, 1100∘C and 1200∘C for a few days. Before each
heat treatment, the sample was grounded and pelletized to ensure better
homogeneity. Rigaku x-ray diffractometer was deployed to check the phase
purity at room temperature using Cu $K_{\alpha}$ radiation. Quantum Design,
SQUID-VSM was used to perform magnetization measurements in the temperature
range 1.8 K $\leq T\leq$ 350 K under magnetic field 0 T $\leq\mu_{0}H\leq$ 5
T. Electron spin resonance (ESR) experiments were performed at 9.4 GHz (X-band
frequency) on a high quality polycrystalline sample of Yb3Sc2Ga3O12 material
for temperatures down to 4 K. Specific heat measurements were carried out with
a Quantum Design, Physical Properties Measurement System (QD, PPMS) in
magnetic fields up to 7 T and in the temperature range 1.8 K $\leq T\leq$ 200
K. The low-temperature specific heat measurement down to 130 mK was performed
using a dilution refrigerator set up attached to the Dynacool PPMS from
Quantum Design, USA. Fitting and modeling of the crystal-electric-field (CEF)
effects were preformed using PHI software[49].

Figure 1: (a) Crystal structure of Yb3Sc2Ga3O12. (b) Two interpenetrating
hyperkagome network of Yb3+ ions. Figure 2: Rietveld refinement of powder XRD
data taken at room temperature. Figure 3: (a) The temperature dependence of
magnetic susceptibility in different applied magnetic fields reordered in ZFC
mode. (b) The temperature dependence of zero field cooled and field cooled
magnetization data taken in 0.001 T. (c) Curie-Weiss fit to inverse
susceptibility data taken in 0.1 T. Blue filled circle denotes experimental
data, whereas solid and dashed lines denote Curie-Weiss fits. Inset shows the
same fit at higher temperature range 200 K $\leq T\leq$ 350 K. (d) Magnetic
isotherms at different temperatures. Magnetization taken in ramping up and
down modes remain identical that add further credence to the absence of spin
glass physics. From the slope of linear fit (solid red line) to high field
data taken at 1.8 K a Van Vleck contribution to susceptibility was extracted.
Blue lines are Brillouin fits for $J_{\mathrm{eff}}$ $=1/2$ at different
temperatures.

###  II.1 Crystal structure

For the Rietveld refinement of Yb3Sc2Ga3O12 hyperkagome material, initial
crystallographic parameters were taken from its iso-structural compound
Gd3Sc2Ga3O12 which crystallizes in the cubic space group $Ia\bar{3}d$ [40].
Rietveld refinement was performed using FullProf Suite on powder XRD data
recorded at room temperature and the refined parameters are
$R_{\mathrm{p}}=8.3\%$, $R_{\mathrm{wp}}=9.2\%$ and $\chi^{2}=5.4$ [50] (see
Fig. 2). Refined atomic coordinates are presented in Table 1. The lattice
constant $a$ turned out to be 12.391 Å, which is smaller than that found in
other rare-earth-based iso-structural compounds (R3Sc2Ga3O12, R = Gd, Tb, Dy
and Ho) [40]. This reduction is due to the small ionic radius of Yb3+ ion
compared to other rare-earth ions. A tiny impurity peak at around
$2\theta\approx 31^{\circ}$ indicates a minor (1.5%) phase of unreacted Sc2O3
which is non-magnetic and its impact on the overall magnetic behavior of the
oxide is negligible. The crystal structure was drawn by using VESTA, which is
shown in Fig. 1a [51]. The two inter-penetrating hyperkagome network build up
of Yb3+ ions is shown in Fig. 1b. Each Yb atom is surrounded by eight oxygen
atoms forming a YbO8 polyhedron. Four of them are at a distance 2.4 Å and the
rest are at 2.455 Å forming a dodecahedron structure. The nearest-neighbor
Yb$-$Yb bond length is 3.794 Å, whereas the second nearest-neighbor bond is
5.796 Å. There are two Yb$-$O$-$Yb bridging exchange paths between two
nearest-neighbor Yb atoms both making an angle 102.8°. The presence of corner-
shared triangles might induce magnetic frustration in this material.

Table 1: Structural parameters of Yb3Sc2Ga3O12 determined from the Rietveld refinement of powder x-ray difraction pattern taken at room temperature using Fullprof software. Space group $Ia\bar{3}d$ and cell parameters $a=b=c=$ 12.391 Å, $\alpha$ = $\beta$ = $\gamma$ = 90∘, $\chi^{2}=5.4$. Atoms | Wyckoff | x | y | z | Occ.
---|---|---|---|---|---
Ga | 24d | 0 | 0.25 | 0.375 | 1
Sc | 16a | 0 | 0 | 0 | 1
Yb | 24c | 0 | 0.25 | 0.125 | 1
O | 96h | -0.0316 | 0.0561 | 0.1508 | 1

###  II.2 Magnetization

Magnetic susceptibility measurements were carried out in the temperature range
1.8 K $\leq T\leq$ 350 K under several applied magnetic fields. Fig. 3a
represents the temperature dependence of magnetic susceptibility data taken in
applied magnetic fields ranging from 0.01 T $\leq T\leq$ 5 T. The absence of
an anomaly in magnetic susceptibility data indicates that there is no magnetic
phase transition down to 1.8 K. The absence of any splitting in zero field
cooled (ZFC) and field cooled (FC) data taken in 0.001 T rules out spin
freezing down to 1.8 K (see Fig. 3b). This is further confirmed by the absence
of hysteresis in the magnetization isotherm taken at 1.8 K (see Supplementary
Fig.1b). High temperature inverse susceptibility data were well reproduced
using the Curie-Weiss formula, $\chi(T)=\frac{C}{T-\theta_{\mathrm{CW}}}$ in
the temperature range 200 K $\leq T\leq$ 350 K (see Fig. 3c). The fit yields a
negative $\theta_{\mathrm{CW}}\approx-114\pm 1$ K, which is attributed to
crystal electric field effects and a change in the population of crystal
electric field levels can induce a strong curvature in $\chi^{-1}(T)$ curve
(here below 150 K). The Curie-Weiss constant $C$ turns out to be 2.49
cm${}^{3}\cdot$K/mol resulting in effective moment $\mu_{\mathrm{eff}}=4.5$
$\mu_{\mathrm{B}}$, close to the free ion effective moment (4.54
$\mu_{\mathrm{B}}$) of Yb3+ (4$f^{13}$, 2F7/2; $J$ = 7/2) ion
($g\sqrt{J(J+1)}\mu_{\mathrm{B}}$, $g-$Landé factor). In the presence of
crystal electric field (CEF), the eight-fold degenerate ground state of Yb3+
ion splits into four Kramers doublets [52]. In this scenario, Curie-Weiss
temperature obtained from high temperature fit is inadequate to describe the
magnetic exchange interaction as some of the Kramers doublets with higher
energy might be populated and Curie-Weiss temperature is dominated by these
crystal field levels (see Supplementary text). The deviation of the
temperature dependence of inverse magnetic susceptibility around 150 K
indicates the presence of another energy scale of interaction between Yb3+
moments at low temperature. In principle, the correlation between 4$f$ moments
develop at very low temperature and this correlations accompanied by competing
degrees of freedom account for the ground state properties in rare-earth-based
frustrated quantum magnets [23]. In order to get an idea regarding the
exchange interaction between Yb3+ moments, we have fitted the temperature
dependence of inverse magnetic susceptibility with Curie-Weiss formula in the
temperature range $1.8\leq T\leq 10$ K (see Fig. 3c). A weak Curie-Weiss
temperature of $-0.2(1)$ K and an effective moment of $2.9(3)$
$\mu_{\mathrm{B}}$ were obtained from the best fit (see Supplementary Fig.
1a). The negative and small Curie-Weiss temperature indicates the presence of
weak antiferromagnetic interaction between the Yb3+ spins. Small Curie-Weiss
temperature is usually found in rare-earth magnets [44, 53, 23]. From the
value of the effective moment, the $g$ factor turns out to be $3.3(3)$. The
reduced effective moment compared to that expected for Yb3+ free ion is
attributed to the $J_{\mathrm{eff}}$ $=1/2$ Kramers doublet ground state at
low temperatures [54, 53, 55, 23].

Table 2: Comparison of lattice parameter, Curie-Weiss temperature ($\theta_{\mathrm{CW}}$), ordering temperature of scandium garnet R3Sc2Ga3O12.  R | Gd | Tb | Dy | Ho | Yb
---|---|---|---|---|---
$a$ (Å) | 12.573 | 12.539 | 12.502 | 12.475 | 12.391
$\theta_{\mathrm{CW}}$ (K) | $-2.2$ | $-1.2$ | $-0.8$ | $-2.9$ | $-0.2(1)$
$T_{\mathrm{N}}$(K) |  $<$0.4 | 0.7 | 1.11 | 2.4 |  $<$0.13
Reference | [40] | [40] | [40] | [40] | This work

Fig. 3d depicts the magnetization isotherms at different temperatures. In
higher field and low temperature, magnetization increases linearly due to the
Van-Vleck paramagnetism. From the linear behavior of the curve, we have
determined a Van-Vleck term of $\chi_{vv}=0.0059$
$\mu_{\mathrm{B}}/\mathrm{T}=0.003$ cm3/mol. The saturation magnetic moment
was found to be 1.43 $\mu_{\mathrm{B}}/$Yb3+ after subtracting the Van Vleck
term, which is consistent with
$M_{\mathrm{s}}=g_{\mathrm{avg}}J_{\mathrm{eff}}(1/2)\mu_{\mathrm{B}}$. The
fit of the $M(H)$ data taken at 1.8 K with Brillouin function also yields a
$g$ value of $\approx 3.0(2)$ close to the value obtained from Curie-Weiss fit
to low-temperature inverse magnetic susceptibility data (see Fig. 3d).
However, the Brillouin fit deviates slightly, probably indicating the presence
of a weak antiferromagnetic interactions between Yb3+ moments at low
temperature. The data taken at 20 K are well fitted with the Brillouin
function as the material is in the paramagnetic region. In these fittings, the
$J$ value was fixed to $1/2$ to account for the $J_{\mathrm{eff}}$ $=1/2$
ground state. The dipolar interaction energy and dipolar Curie-Weiss
temperature were deduced from the formula
$E_{\mathrm{d}}=\left(\frac{8}{\sqrt{6}}\right)^{3}\frac{\left(g\mu_{\mathrm{B}}S\right)^{2}}{a^{3}}$
and
$\theta_{\mathrm{CW}}^{\mathrm{dip}}=32\pi\frac{\left(g\mu_{\mathrm{B}}\right)^{2}S(S+1)}{3a^{3}k_{\mathrm{B}}}$
which was used for the isostructural compound Yb3Ga5O12 [55]. Here $a$ is the
lattice parameter. $E_{\mathrm{d}}$ and $\theta_{\mathrm{CW}}^{\mathrm{dip}}$
are turned out to be 31 mK and 90 mK, respectively. Different magnetic
interactions contributes additively to the Curie-Weiss temperature (i.e,
$\theta_{\mathrm{CW}}=\theta_{\mathrm{CW}}^{\mathrm{dip}}+\theta_{\mathrm{CW}}^{\mathrm{ex}}$)
[56, 57], the Curie-Weiss temperature due to exchange interaction (i.e,
$\theta_{\mathrm{CW}}^{\mathrm{ex}}$) turned out to be $\sim-0.3(1)$ K. The
exchange interaction $J_{\mathrm{ex}}$ could be obtained using the formula
$J_{\mathrm{ex}}=\frac{3k_{\mathrm{B}}\theta_{\mathrm{CW}}^{\mathrm{ex}}}{zS(S+1)}$
[58], where $z$ represents the number of nearest-neighbors. Using $z=4$ for
hyperkagome structure and effective spin $S\equiv J_{\mathrm{eff}}=1/2$, we
found $J_{\mathrm{ex}}=-0.3(1)$ K. The presence of weak exchange interaction
is usual in rare-earth oxides which is attributed to the fact that the $4f$
orbitals of rare-earth ions are localized and well shielded yielding weak
overlap between two nearest-neighbor orbitals.

###  II.3  Electron Spin Resonance ( ESR)

Electron spin resonance (ESR) is an excellent low energy, local probe to
determine the effective $g$–factor as well as characterizing single–ion and
exchange anisotropies, which are crucial to establish realistic microscopic
Hamiltonian of frustrated quantum materials. It was shown that ESR can shed
microscopic insights into the QSL state and associated exotic excitations in
frustrated magnets with significant spin-orbit interaction [59]. Fig. 4
depicts ESR signals of a polycrystalline sample of Yb3Sc2Ga3O12 at
representative temperatures. Due to the rather large linewidth, Yb containing
secondary phases may be excluded as origin of the signal. We described the
line by a Lorentzian shape including also the negative resonance field
component due to the influence of the counter-rotating component of the
linearly polarized microwave field. Such a fit yields for the 20 K data a
resonance field of 0.18 T (which corresponds to a $g$-value of 3.6) and a
linewidth of 0.2 T. The large line width as compared to the resonance field
pose a strong constraint to precisely quantify temperature dependencies of ESR
parameters in the measured temperature range $4\leq T\leq 175$ K.
Nevertheless, a clear tendency of line broadening towards low and high
temperatures as well as a down shift of the resonance field towards low
temperature can be identified. This behavior may be due to a relaxation via
the first excited crystal electric field level of the Yb3+ ion and the slowing
down of Yb3+ spin fluctuations in the low temperature regime [60]. The strong
deviation from a symmetric Lorentzian line shape points to off-diagonal terms
in the dynamic susceptibility. This is typical for systems with anisotropic
spin lattices and becomes even more pronounced when very broad lines are
present [61]. A detailed discussion in this direction would be reasonable only
if ESR data of single crystals would be available.

Figure 4: ESR spectra of a polycrystalline sample of Yb3Sc2Ga3O12 at indicated
temperatures and X-band frequency. $dP/dH$ denotes the first field derivative
of the absorbed microwave power. Line denotes fitting of the 20 K data with a
Lorentzian shape with a corresponding $g$-value of 3.6. Figure 5: (a)
Temperature dependence of specific heat ($130$ mK $\leq T\leq 200$ K) for
Yb3Sc2Ga3O12 in zero field. Nuclear Schottky contribution to the specific heat
for 3 $\times$ 0.1431 mol of 171Yb and 3 $\times$ 0.1613 mol of 173Yb are
denoted by blue and pink line, respectively. (b) Low temperature specific heat
($130$ mK $\leq T\leq 20$ K) in several applied magnetic fields. Corresponding
Schottky fits are denoted by black solid lines. Phonon contributions were
taken into account while fitting by adding $\alpha T^{3}$ term indicated by
red solid line. (c) Linear behavior of Schottky gap ($\Delta/k_{\mathrm{B}}$)
with applied magnetic field and the solid red line is a linear fit. From the
slope of the fit we found $g=3.3(1)$ and the intercept turns out to be
$0.2(1)$ K. (d) Magnetic entropy in applied magnetic fields. Corresponding
paramagnetic entropy in applied fields are denoted by red solid lines.
Experimental entropy is slightly smaller than the ideal paramagnet which
suggests the existence of small exchange interaction and/or magnetic
frustration. For higher fields, it saturates at 97 % of the full entropy
($R\ln 2$ ) expected for magnetic materials with $J_{\mathrm{eff}}$ $=1/2$
degrees of freedom consistent with magnetization results.

###  II.4 Specific heat

Specific heat provides an excellent probe to gain insights into the ground
state properties of this rare-earth hyperkgome lattice. Fig. 5a shows the
specific heat as a function of temperature down to 130 mK in zero field. The
absence of any anomaly in zero field, thus confirms the evasion of long-range
magnetic ordering down to at least 130 mK despite a weak antiferromagnetic
exchange interaction between $J_{\mathrm{eff}}$ $=1/2$ moments. The peak in
specific heat due to the nuclear Schottky effect usually occurs below 10-2 K
[62, 63, 64]. To calculate the expected nuclear specific heat contribution, we
have considered the fact that the natural abundance of 171Yb ($I=1/2$) and
173Yb ($I=5/2$) are 14.31% and 16.13%, respectively (see Fig. 5a) [65, 66,
67]. The energy separation between two consecutive levels for 171Yb and 173Yb,
are 63.9 mK and 17.55 mK, respectively, which are taken from that reported for
the isostructural compound Yb3Ga5O12 [65]. The quadrupole coupling constant
$P$ (see supplementary equation 3) is zero for 171Yb and non-zero for 173Yb.
However, we have neglected the quadrupole term in our calculation ( see SI for
details) due to its insignificant impact on shifting the Schottky peak. Upon
the application of an external magnetic field, a broad maximum appears at low
temperature in specific heat data of Yb3Sc2Ga3O12, that shifts significantly
to higher temperature upon increasing magnetic field (see Fig. 5b), which is
associated with the electronic Schottky anomaly originating from the Zeeman
splitting of the lowest lying energy levels. To evaluate Schottky
contribution, field dependent $C_{\mathrm{p}}(H)$ data were fitted with the
equation $C_{\mathrm{p}}=C_{\mathrm{L}}+C_{\mathrm{S}}(\Delta)$ (see Fig. 5c),
where $C_{\mathrm{S}}(\Delta)$ is the specific heat with energy level
splitting $\Delta$ due to an applied magnetic field $\mu_{\mathrm{0}}H$ [68,
69]. $C_{\mathrm{L}}$ is the lattice contribution to the specific heat. We
have fitted the low-temperature zero-field specific heat data in the
temperature range 4 K $\leq T\leq$ 22 K with the equation relevant for lattice
specific heat in the simplest approximation

$C_{\mathrm{L}}=\alpha T^{3}$ (1)

and then extrapolated it to lower temperature. The Schottky specific heat
$C_{\mathrm{S}}(\Delta)$ is given by

$C_{S}(\Delta)=3fR\left(\frac{\Delta}{k_{\mathrm{B}}T}\right)^{2}\frac{\mathrm{exp}({\frac{\Delta}{k_{\mathrm{B}}T})}}{\left[1+\mathrm{exp}{(\frac{\Delta}{k_{\mathrm{B}}T})}\right]^{2}}.$
(2)

Figure 6: (a) The temperature dependence of $\chi T$ emphasizing the $\chi$
behavior at low temperatures. (b) The temperature dependence of specific heat
measured in different magnetic fields. (c) Magnetization as a function of
external magnetic field at several temperatures. Lines correspond to the CEF
model presented in the text.

The factor 3 indicates that there are 3 Yb ions in a formula unit. Here, $f$
denotes the fraction of Yb3+ spins participating in the splitting of the
ground state. This value is close to one ($f>0.9$ for fields
$\mu_{\mathrm{0}}H>3$ T) indicating that almost all the spins are contributing
to the specific heat. It is worth mentioning that, for transition metal
oxides, $f$ denotes the fraction of orphan or defect spins (5-10 %) that are
present in the system owing to disorder [70, 12]. It is found that the value
of $\Delta$ increases with an external applied magnetic field typical for many
frustrated magnetic materials [39, 64]. The Landé-$g$ factor turns out to be
3.3(1) from the slope of the linear fit to $\Delta/k_{\mathrm{B}}$ vs.
$\mu_{\mathrm{0}}H$ plot (see inset of Fig. 5c), which is consistent with the
result obtained from magnetization data. We get a small zero-field splitting
of 0.2(1) K from the intercept of the extrapolated linear fit. However, the
deviation of the zero field data from Schottky specific heat indicates that
the peak around 200 mK might be due to the presence of short-range spin
correlation between Yb3+ ions in this 3D spin-lattice. It is worthwhile to
mention that a crystal electric field can also split the energy levels and
give rise to Schottky type anomaly. However, the energy difference from the
ground state to the first excited state in most of the rare-earth materials is
of the order of 5 meV or more, which corresponds to a temperature of 58 K [47,
71, 72, 23]. Magnetic specific heat $C_{\mathrm{m}}$ was obtained after
subtracting the lattice ($C_{\mathrm{L}}$) and nuclear contribution
[$C_{\mathrm{N}}(^{171}$Yb + 173Yb)] to specific heat from the raw specific
heat data $C_{\mathrm{p}}$ in the entire temperature range.

Table 3: CEF parameters (in meV) for CEF Hamiltonian of Eq. (3) and $g$ factors of the ground state for the CEF model.  $B_{2}^{0}$ | $B_{2}^{2}$ | $B_{4}^{0}$ | $B_{4}^{2}$ | $B_{4}^{4}$ | $B_{6}^{0}$ | $B_{6}^{2}$ | $B_{6}^{4}$ | $B_{6}^{6}$ | $g_{x}$ | $g_{y}$ | $g_{z}$
---|---|---|---|---|---|---|---|---|---|---|---
$-$0.06229 | 0.3036 | 0.08183 |  $-$0.04343 |  $-$0.03914 | 0.0000166 |  $-$0.01486 | 0.02468 |  $-$0.009285 | 2.93 | 2.25 | 4.24

However, we did not subtract the nuclear contribution for higher field data as
our lowest accessible temperature in applied field is 0.5 K, where the nuclear
contribution is almost negligible [i.e., 0.026 J/(mol$\cdot$K)]. The magnetic
entropy can shed insights into the ground state properties of frustrated
magnetic materials. We have extracted the entropy by integrating
$C_{\mathrm{m}}/T$ from the lowest accessible temperature (130 mK for zero
field) to 20 K and the results are shown in Fig. 5d. Due to experimental
limitations in accessing temperature below 130 mK, the zero field specific
heat is not fully recovered yielding a lower value of entropy than that
expected for spin half magnetic materials. The zero-field entropy does not
saturate to $R\ln 2$ even up to 20 K. This residual entropy below 130 mK might
indicates the role of magnetic frustration at very low temperature. However,
in the application of a magnetic field, the entropy is dominated by the
Schottky effect in Kramers doublet systems. The obtained high temperature
$\theta_{\mathrm{CW}}$ ($-114$ K), which indicates crystal field splitting of
the ground state, is very large. We expect that the excited state doublets are
well separated from the ground state so that we can describe the low-
temperature properties by accounting only the lowest $J_{\mathrm{eff}}=1/2$
ground state. The entropy taken in 3 T saturates with a value 5.6
J/(mol$\cdot$K) which is close to the full entropy of $R\ln 2$ [ i.e, 5.76
J/(mol$\cdot$K)] for $J_{\mathrm{eff}}$ $=1/2$ spin state. This suggests that
the ground state of this three-dimensional spin-lattice could be mapped with a
low energy effective $J_{\mathrm{eff}}$ $=1/2$ state at low temperature, which
is consistent with magnetization results [73, 74].

###  II.5 Crystal electric field (CEF) calculations

We performed a combined fit of specific heat data measured in 1, 2, 3, 5, and
7 T as well as magnetic susceptibility data measured in 0.01, 0.1, 1, 3, and 5
T. According to the Hund’s rule, the ground state multiplet of the Yb3+ ion is
${}^{2}F_{7/2}$, which is in a crystal field split into four Kramers doublets
composed of $|\pm m_{J}\rangle$ states [$m_{J}$ = $(2n-1)/2$, where $n$ =
1-4]. The composition of the four Kramers doublets directly depends on the CEF
Hamiltonian, which can be written as

$H_{\text{CEF}}=\sum_{i,j}B_{j}^{i}O_{j}^{i},$ (3)

where $O_{j}^{i}$ are Stevens operators [75] and $B_{j}^{i}$ are the
corresponding scaling parameters. The relevant $B_{j}^{i}$ (Table 3) are
determined by point symmetry at the Yb3+ site and in general comply with those
determined for the isostructural compound Yb3Ga5O12 [76].

Table 4: CEF energy levels (in meV) for the CEF model.  $E_{0}$ | $E_{1}$ | $E_{2}$ | $E_{3}$
---|---|---|---
0 | 69 | 96 | 143

Indeed, we obtained a very good agreement with the data measured in the
applied magnetic field [lines in Fig. 6(a) and (b)] for the derived CEF
parameters (Table 3), when considering additional $T^{3}$ term for the
specific heat, accounting for the phonon contribution at low temperatures.
Finally, we have calculated the corresponding magnetization curves $M(H)$
[lines in Fig. 6(c)]. The agreement of the simulation and experiment is quite
reasonable for both experiments. The derived magnetic anisotropy of the CEF
ground state is determined by the $g$ factor, also given in Table 3, implying
a rather uniaxial anisotropy, as $g_{z}$ is considerably larger than $g_{x}$
and $g_{y}$. In fact, the derived $g$ values are in agreement with the average
$g=3.6$ derived from the ESR data measured at $20$ K. All energy levels are
summarized in table Table 4, indicating that the ground state is well
separated from the excited states, as found also in Yb3Ga5O12 [76]. We note
that, the broad maximum in specific heat around 200 mK in zero field is
clearly not related to CEF effects and as suggested above most likely
associated with the presence of short-range spin correlations.

##  III Discussion

The intertwining of spin-orbit coupling and electron correlation in frustrated
magnets can stabilize exotic quantum states. In this context, rare-earth-based
$4f$ frustrated magnets offer an exciting platform. In these materials,
anisotropy and crystal electric field generated by the neighboring ligands
govern the ground state properties. The choice of a suitable chemical element
has a great role in tuning the crystal electric field, anisotropy and exchange
interaction and hence the ground state properties of rare-earth-based
frustrated magnet. The hyperkagome material Yb3Sc2Ga3O12 belongs to the garnet
family with cubic space group $Ia\bar{3}d$ and lattice constant $a=12.391$ Å.
Unlike R3Sc2Ga3O12 garnets (R = Tb, Dy, Ho), which orders at low temperature,
Yb3Sc2Ga3O12 does not show any long-range magnetic ordering at least down to
130 mK. This might be due to the low magnetic moment of Yb3+ and low exchange
energy compared to other R3Sc2Ga3O12 compounds [40]. Furthermore, the absence
of ZFC-FC splitting rules out spin-freezing in this material. The small
negative Curie-Weiss temperature indicates the presence of a weak
antiferromagnetic interaction which arises mainly due to super-exchange
interaction. The $\theta_{\mathrm{CW}}$ value in a three-dimensional
hyperkagome lattice Li3Yb3Te2O12 with a similar oxygen environment is almost
the same order of magnitude at high temperature as well as at low temperature
as compared to the present hyperkgome material [39]. ESR spectra taken on
polycrystalline samples of Yb3Sc2Ga3O12 are found to be relatively broad with
a highly asymmetric shape which is related to magnetic anisotropy and most
likely also the distribution of Yb3+ moments [61]. The specific heat
experiment which is very sensitive to probe magnetic phase transition rules
out a long-range magnetic order down to 130 mK in this frustrated magnet. The
broad maximum around 200 mK in specific heat data taken in zero field is
ascribed to the development of short-range spin correlations at low
temperature. The development of short-range spin correlations suggesting the
dominance of quantum effects at low temperature that lead to a dynamic liquid-
like state in this 3D frustrated antiferromagnet. Similar broad maximum in
specific heat at low temperature was also observed in its sister compound
Yb3Ga5O12 [65, 45]. The low temperature magnetic behavior can be described by
$J_{\mathrm{eff}}$ $=1/2$ Kramers doublet ground state as the energy levels of
the excited Kramers doublets is well above the ground state, which is
confirmed by our CEF calculations. This family of R3Sc2Ga3O12 rare-earth
magnetic materials with a suitable choice of rare-earth and non-magnetic
cations has a great potential to host non-trivial quantum states with exotic
excitations.

##  IV Conclusion

In summary, we have successfully synthesized and characterized a three-
dimensional hyperkagome garnet, namely Yb3Sc2Ga3O12. Magnetization data
indicate the absence of long-range magnetic ordering and spin freezing down to
1.8 K. Specific heat measurements reveal that there is no long-range magnetic
ordering down to 130 mK. A weak antiferromagnetic exchange interaction is
present in this system, which is typical for rare-earth-based magnetic
materials due to the localized 4$f$ moments. In addition, the absence of ZFC-
FC splitting rules out the formation of a glassy phase in this
antiferromagnet. Our thermodynamic results indicate that the Yb3+ ions are in
the Kramers doublet state with an effective $J_{\mathrm{eff}}$ $=1/2$ spin
degrees of freedom at low temperature. Furthermore, ESR results indicate the
existence of magnetic anisotropy. The presence of a broad maximum around 200
mK in the temperature dependence of specific heat suggests that a short-range
spin correlation phenomena at play in this frustrated magnet. Our CEF modeling
suggests that the ground state is well separated from the excited states and
yields a very good agreement with thermodynamic and ESR results. Given the
fact that the exchange interaction is weak in 4$f$ systems, further low-
temperature thermodynamic and muon spin relaxation experiments are required to
track the ground state properties unambiguously. Microscopic experiments such
as neutron scattering are desired to shed insights into the spin-orbit driven
anisotropy driving novel ground state arises from the crystal-field splitting
of the rare-earth multiplet. Understanding the spin anisotropy and low energy
spin excitation spectra sets an interesting track to reveal interesting
insights into the ground state properties of this novel frustrated magnet.
This family of rare-earth hyperkagome R3Sc2Ga3O12 (R $=$ rare-earth) offers a
promising platform for the experimental realization of unconventional ground
states borne out of spin correlation, frustration and spin-orbit driven
anisotropy.

##  V Acknowledgment

PK acknowledges the funding by the Science and Engineering Research Board, and
Department of Science and Technology, India through Research Grants. MP
acknowledges the funding by the Slovenian Research Agency (project J2-2513,
and program No. P1-0125).

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