# Efficient adiabatic demagnetization refrigeration to below 50 mK with UHV
compatible Ytterbium diphosphates $A$YbP2O7 ($A=$Na, K)

U. Arjun1,2<EMAIL_ADDRESS>K. M. Ranjith3 A. Jesche2
<EMAIL_ADDRESS>F. Hirschberger2 D. D. Sarma1 P.
Gegenwart2<EMAIL_ADDRESS>1Solid State and
Structural Chemistry Unit, Indian Institute of Science, Bangalore-560012,
India  2Experimental Physics VI, Center for Electronic Correlations and
Magnetism, Institute of Physics, University of Augsburg, 86135 Augsburg,
Germany  3Laboratoire National des Champs Magnétiques Intenses-EMFL, CNRS,
Université Grenoble Alpes, 38042 Grenoble, France

###### Abstract

Attaining milli-Kelvin temperatures is often a prerequisite for the study of
novel quantum phenomena and the operation of quantum devices. Adiabatic
demagnetization refrigeration (ADR) is an effective, easy and sustainable
alternative to evaporation or dilution cooling with the rare and super-
expensive 3He. Paramagnetic salts, traditionally used for mK-ADR, suffer from
chemical instability related to water of crystallization. We report synthesis,
characterization as well as low-temperature magnetization and specific heat
measurements of two new UHV compatible candidate materials NaYbP2O7 and
KYbP2O7. Utilizing the PPMS at 2 K, the ADR of sintered pellets with Ag powder
admixture starting at 5 T yields base temperatures (warm-up times) of 45 mK
(55 min) and 37 mK (35 min) for NaYbP2O7 and KYbP2O7, respectively, slightly
advantageous to KBaYb(BO3)2 (45 mK and 40 min) studied under similar
conditions.

††preprint: APS/123-QED

##  I Introduction

Quantum effects are most evident at low-temperatures where the thermal
fluctuations are suppressed. Novel quantum effects such as Bose-Einstein
condensation [1], superconductivity [2], quantum Hall effect [3], quantum spin
liquid [4], quantum spin-ice [5], etc. have been experimentally discovered
only at temperatures close to absolute-zero. The global scarcity of helium [6,
7] and the increasing need for refrigeration for various technological
applications enhanced the importance of helium-free magnetic refrigeration
techniques. The isotope 3He, used in these refrigeration techniques, is
extremely scarce in availability and thus extraordinarily expensive due to its
increased demand in the defence sector for the production of neutron detectors
to combat nuclear terrorism [8, 9, 10].

The significant temperature changes in some magnetic materials due to the
magneto-caloric effect (MCE) while exposed to an external field [11] make them
useful for attaining low temperatures through adiabatic demagnetization
refrigeration (ADR) [12, 13, 14]. Currently milli-Kelvin (mK) ADR is
intensively used in satellites and other space technologies [15, 16, 17, 18,
19] and has already been shown to be potentially practical for quantum
computers [20]. More generally, MCE-based magnetic refrigeration significantly
reduces environmental impact and offers energy savings of nearly 30% compared
to conventional techniques that use refrigerant gases [21].

For ADR applications, replacing a 3He refrigerator or a 3He-4He dilution
refrigerator, materials with a significant entropy change $\Delta S$ in the
temperature range between 10 mK and 4 K are required. They should also exhibit
a strong MCE within the lowest possible applied magnetic field. In the multi-
stage process for attaining sub-Kelvin temperatures, the material is first
pre-cooled to a temperature of about $2$ K in a magnetic field of typically a
few teslas (e.g. using a pumped 4He bath or a pulse-tube cooler).
Subsequently, the thermal contact with the bath is cut by pumping out the He-
gas or opening a heat switch, and the cooling substance is demagnetized. In
this adiabatic process, the very low entropy of the cooling substance due to
pre-cooling in the magnetic field requires that the temperature of the cooling
substance is significantly lowered, ideally below $50$ mK.

A major disadvantage of ADR compared to 3He–4He dilution refrigeration has
been its incapability of continuous cooling, which is being addressed by the
recent developments in continuous ADR [16, 22]. Commercial continuous
refrigerators based on ADR are now available [23], suggesting that ADR has the
potential to become a major refrigeration technology.

Even several decades after the first realization of magnetic refrigeration
down to the mK range, hydrated paramagnetic salts with very low ordering
temperatures are commonly used as cooling substances in ADR refrigerators. In
the paramagnetic salts, the spins are very much diluted and are almost non-
interacting, resulting in a low volumetric cooling power and low thermal
conductivity. By contrast, systems with higher number of magnetic ions per
unit volume may exhibit stronger exchange interactions, resulting in magnetic
order. The magnetic ordering typically prevents the system from cooling down
adiabatically to lower temperatures.

Even in almost-ideal paramagnets, there may be a very weak magnetic
interaction $J$ through which the neighboring spins can interact weakly even
at $H=0$, resulting in a small internal field. This can lead to a tiny Zeeman
splitting ($\Delta_{0}$) and magnetic ordering at a temperature of the same
energy scale. Magnetic ordering suddenly drops the entropy to zero and limits
the final ADR temperature to $T_{f}\sim\Delta_{0}\sim J$, since the entropy
difference between $H=0$ and a finite $H$ is an important parameter for ADR
[24]. A perfect paramagnet with zero interaction and maximum entropy at $H=0$
close to $0$ K is an ideal refrigerant. But in real paramagnetic materials,
the presence of a finite weak interaction is inevitable.

Recently the disordered quantum magnets came into focus. Since entropy
accumulates near quantum critical points and/or due to magnetic frustration,
new strategies for obtaining efficient mK-ADR were explored [24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35]. In frustrated quantum magnets, the enhanced
quantum fluctuations suppress the long-range ordering, despite the strong spin
interactions, giving rise to a shift of entropy towards low temperatures, that
is crucial for obtaining lower ADR end temperatures [24]. For the efficiency
of the cooling substance, the available entropy change per volume is an
essential parameter [36]. The size of the spins and their density constitute
to the volumetric entropy density which should be as high as possible.

Figure 1: Crystal structure of NaYbP2O7 in the $ab$-plane and along the
$c$-direction. The YbO6 octahedra are linked via PO4 tetrahedra and span a
distorted triangular grid (left panel). One such triangular layer is
highlighted by a rectangular box in the right panel. Different Yb-Yb distances
are color-coded. Figure 2: Crystal structure of KYbP2O7. The YbO6 octahedra
are linked via PO4 tetrahedra forming uniform chains running along the
$c$-axis (highlighted by a rectangular box) which are interconnected by
alternating chains running in the $ac$-plane (highlighted by an ellipse in the
right panel). This forms a three-dimensional distorted hyper-honeycomb-like
network. Different Yb-Yb distances are color-coded.

Traditional mK-ADR substances such as CrK(SO4)${}_{2}~{}\cdot~{}12$H2O (CPA)
[37] incorporate water of crystallization for separation of magnetic ions.
Hence, the magnetic interactions are sufficiently small, resulting in very low
magnetic ordering temperatures, like 30 mK for CPA. However, these salts have
high water vapor pressure and, therefore, cannot be evacuated. This requires
vacuum-tight encapsulation of the active material in a protective container.
In order to ensure sufficient thermal coupling of very poorly thermally
conductive salts, it is also necessary to grow them as single crystals within
a very fine wire network [36]. This complex production of encapsulated cooling
units leads to high costs. Furthermore, they must not be heated due to the
lack of chemical stability. This means that they cannot be thoroughly baked
above $100^{\circ}$C for ultra-high-vacuum (UHV) applications.

Oxide quantum magnets, on the other hand, are stable upon heating and
evacuation. Recently, it was discovered that potassium-barium-ytterbium borate
KBaYb(BO3)2 can be used as an inert, easy-to-produce and inexpensive cooling
substance for UHV-compatible adiabatic demagnetization cooling [24]. Magnetic
order in KBaYb(BO3)2 is hindered by both geometrical frustration on the
triangular lattice as well as the statistical mixing of K+ and Ba2+, resulting
in uneven electric fields acting on the Yb3+ ions, eventually causing a wide
distribution of magnetic couplings. Isostructural KBaGd(BO3)2 shows magnetic
order below $T_{N}=263$ mK in zero field, reaches a minimal ADR temperature of
122 mK and warm-up time of 8 hours in the PPMS setup [38]. Dipolar and
magnetic exchange couplings are of similar magnitude in this system [38, 39].
Taking into consideration that dipolar and exchange interactions are
proportional to the square of the magnetic moment and using the measured
saturation moments for KBaYb(BO3)2 and KBaGd(BO3)2, suggests a magnetic order
at 9 mK in the former [38].

In this work, we report mK magnetization, specific heat and ADR measurements
on the geometrically frustrated quantum magnets NaYbP2O7 and KYbP2O7. Similar
to KBaYb(BO3)2 [24], the two materials are chemically stable and UHV-
compatible. A direct comparison of the ADR performance with KBaYb(BO3)2 under
similar conditions indicates for NaYbP2O7 a similar ADR end temperature but
30% longer hold time and for KYbP2O7 a 20% lower ADR end temperature at the
cost of a 12.5% shorter hold time. Thus, the two diphosphates are superior
candidates for UHV-compatible mK-ADR.

##  II Methods

Synthesis: Polycrystalline samples of $A$YbP2O7 ($A$= Na, K) are synthesized
by the conventional solid-state reaction technique by annealing the
stoichiometric mixture of Na2CO3/ K2CO3 (99.99%), Yb2O3 (99.99%), and NH4H2PO4
(99.99%) in an alumina boat at 650 ∘C for NaYbP2O7 and 600 ∘C for KYbP2O7 for
a duration of $48$ h with one intermediate grinding and pelletization. The
synthesis procedures for the KBaYb(BO3)2 pellet used for the comparison of ADR
performances are described in Ref. 24.

Powder X-Ray diffraction: Phase purities of the samples were confirmed by
powder x-ray diffraction (XRD, PANalytical powder diffractometer with
Cu$K_{\alpha}$ radiation, $\lambda_{\rm ave}=1.54182$ Å) at room temperature.
Rietveld refinement of the observed XRD patterns was performed using the
`FullProf` package [40] (see Fig. 3), taking the initial parameters from Ref.
41, 42.

DC magnetization: DC magnetization ($M$) was measured as a function of the
temperature ($T$) down to 0.4 K and the applied magnetic field ($H$) up to 7 T
in a MPMS-3 Quantum Design SQUID magnetometer with 3He option.

Specific heat: Specific heat $C_{\rm p}(T)$ was measured using the heat
capacity option of a Physical Property Measurement System (PPMS) manufactured
by Quantum Design. For the low temperature ($0.4$ K $\leq T\leq 2.2$ K)
$C_{\rm p}$ measurements the 3He option was used.

For strong thermal coupling, the specific heat measurements were performed on
pellets made from sodium/potassium diphosphates (grain size $10-50~{}\mu$m)
mixed with fine silver powder ($1~{}\mu$m) in a mass ratio of $1:1$. In order
to extract the specific heat of the sample, we subtracted the Ag-contribution
from the measured data. In a magnetic insulator, the specific heat $C_{\rm p}$
contains significant contributions from the phonon excitations ($C_{\rm ph}$)
and the magnetic lattice ($C_{\rm m}$). At high temperatures, $C_{\rm p}(T)$
is entirely dominated by $C_{\rm ph}$, while at low temperatures, it is
dominated by $C_{\rm m}$. In order to estimate the phonon part of the specific
heat, the $C_{\rm p}(T)$ data at high temperatures $(T>25$ K) were described
phenomenologically by a polynomial $C_{\rm ph}(T)=aT^{3}+bT^{5}+cT^{7}$, where
$a$, $b$, and $c$ are appropriate coefficients. Similar procedures have been
used previously and have shown to be efficient for estimating $C_{\rm ph}$ in
cases where heat capacity data of non-magnetic analogue compounds are not
available [43, 44, 45, 46]. The fit was extrapolated down to low-temperatures,
and the magnetic specific heat $C_{\rm m}$ was obtained by subtracting the
obtained $C_{\rm ph}$ values from the experimental $C_{\rm p}$ data. Magnetic
entropy $S_{\rm m}$ was estimated by integrating $C_{\rm m}(T)/T$ from $0.4$ K
to high-temperatures as $S_{\rm m}(T)=\int_{0.4~{}K}^{T}\frac{C_{\rm
m}(T^{\prime})}{T^{\prime}}dT^{\prime}$.

Adiabatic Demagnetization Refrigeration (ADR): Using an ADR material in a
commercial PPMS in order to achieve temperatures below $1.8$ K without using
3He presents an immediate practical application. For the cooling experiment,
we used a $\sim 3.5$ g cylindrical pellet of $6$ mm thickness and $15$ mm
diameter containing equal weights of $A$YbP2O7 (grain diameter $10–50~{}\mu$m)
and silver powder (grain diameter $1~{}\mu$m). Silver powder is used to
improve the thermal conductivity within the pellet as $A$YbP2O7 is insulating
in nature. Also, we sintered the pressed pellet at $600~{}^{\circ}$C to
further improve the thermal conductivity.

We performed the ADR experiments in the PPMS with a similar set-up as
described in [24]. A sample stage is constructed in which the pellet is
mounted on a plastic straw. The sample is thermally isolated from the heat
bath. An RuO2 resistor is glued as a thermometer on the pellet and is
connected using thin resistive manganin wires to minimize the heat flow. A
metallic cap is used as a shield to minimize the effect of surrounding thermal
radiation. PPMS is kept in the high-vacuum mode (pressure $<10^{-4}$ mbar),
and the bath temperature is set to $2$ K. Then a magnetic field of $5$ T is
applied. Through the weak thermal link via manganin wires, the pellet slowly
reaches $2$ K. After stabilizing at $2$ K, the magnetic field of $5$ T is
swept to zero at a rate of $0.15$ T min-1. At zero field the pellet reaches
the lowest temperature and then slowly up back to $2$ K by the slow heat flow
from the bath. We also performed ADR experiments with a KBaYb(BO3)2 pellet
from [24] utilizing the same thermometer and parameters as in case of
$A$YbP2O7. This allows a direct comparison of the ADR performances of two
diphosphates with that of the previously reported borate.

##  III Results

###  III.1 Structure

Both NaYbP2O7 and KYbP2O7 crystallize in a monoclinic lattice with the space
group $P21/c$ (No. 14) but form different structure types. The crystal
structure of NaYbP2O7 consists of distorted YbO6 octahedra, which are corner-
shared with PO4 tetrahedra, forming a magnetic Yb triangular layer in the
crystallographic $ab$-plane. These triangular planes are well separated by two
non-magnetic PO4 units and Na ions along the crystallographic $c$-direction
(see Fig. 1). In KYbP2O7, on the other hand, the YbO6 octahedra are connected
by PO4 tetrahedra forming uniform chains running along the $c$-axis and are
connected by alternating chains in the $ac$-plane. It forms a three-
dimensional distorted hyper-honeycomb-like network (see Fig. 2).

###  III.2 Powder x-Ray diffraction

Figure 3: Powder x-ray diffraction pattern (open red circles) for NaYbP2O7
(upper panel) and KYbP2O7 (lower panel) at room temperature. The solid line
represents the Rietveld refinement, with the vertical bars showing the
expected Bragg peak positions and the lower solid blue line representing the
difference between observed and calculated intensities.

Figure 3 shows the powder XRD pattern of NaYbP2O7 and KYbP2O7 at room
temperature, along with the Rietveld refinement. For NaYbP2O7, the refinement
uses the monoclinic space group $P2_{1}/c$ (No. 14) $[$settings $P2_{1}/n$
(unique axis-$b$)$]$, taking the initial parameters from Ref. 42. The goodness
of fit is $\chi^{2}=4.72$. The obtained lattice parameters are $a=9.0215(1)$
Å, $b=5.3599(1)$ Å, $c=12.7816(1)$ Å, $\beta=103.1655(1)^{\circ}$, and $V_{\rm
cell}\simeq 601.81(1)$ Å3. These values are in close agreement with reported
values [42]. For KYbP2O7, the refinement is performed using the monoclinic
space group $P2_{1}/c$ (No. 14) $[$settings $P2_{1}/c$ (unique axis-$b$)$]$,
taking the initial parameters from Ref. 41. The goodness of fit is
$\chi^{2}=8.78$. The obtained lattice parameters are $a=7.5500(1)$ Å,
$b=10.8306(1)$ Å, $c=8.5492(1)$ Å, $\beta=106.7200(1)^{\circ}$, and $V_{\rm
cell}\simeq 669.52(1)$ Å3. These values are in close agreement with reported
values [41]. The unit cell volume of NaYbP2O7 is small compared to KYbP2O7
because the ionic radius of Na+ (1.18 Å) is smaller than that of K+ (1.51 Å)
[47].

###  III.3 Magnetization

Figure 4: Inverse magnetic susceptibility ($1/\chi$) data measured at $1$ T as
a function of temperature ($T$) for NaYbP2O7 (upper panel) and KYbP2O7 (lower
panel). The solid line represents the high-temperature CW-fit according to Eq.
1. Figure 5: Isothermal magnetization $M(H)$ measured at $0.4$ K for NaYbP2O7
(panel (a)) and KYbP2O7 (panel (b)). Dashed lines represent linear
contributions (see text). The insets show the low-temperature inverse magnetic
susceptibility after subtracting $\chi_{\rm VV}$ along with the CW-fit for
NaYbP2O7 [inset of (a)] and KYbP2O7 [inset of (b)].

The temperature-dependent magnetic susceptibility $\chi(T)=M/H$ data of
NaYbP2O7 and KYbP2O7 were measured at applied fields $H=0.1$ T and $1$ T. No
evidence of any long-range magnetic order was observed down to $0.4$ K. At
high temperatures (above $150$ K), $1/\chi(T)$ can be well fitted with the
modified Curie-Weiss (CW) expression (see Fig. 4)

$\chi(T)=\chi_{0}+\frac{C}{T-\theta_{\rm CW}},$ (1)

where $\chi_{0}$ is the temperature-independent contribution consisting of the
diamagnetic susceptibility of core electron shells ($\chi_{\rm core}$) and the
van-Vleck paramagnetic susceptibility ($\chi_{\rm vV}$) of the open shells of
the Yb3+ ions. The second term in Eq. 1 is the CW law with the CW temperature
$\theta_{\rm CW}$ and Curie constant $C=N_{\rm A}\mu_{\rm eff}^{2}/3k_{\rm
B}$. Here, $N_{\rm A}$ is Avogadro’s number, $\mu_{\rm
eff}=g\sqrt{J(J+1)}$$\mu_{\rm B}$ is the effective magnetic moment, $g$ is the
Landé $g$-factor, $\mu_{\rm B}$ is the Bohr magneton, and $J$ is the effective
spin.

The fitting yields the parameters $\chi_{0}^{\rm HT}\simeq-1.73\times 10^{-4}$
cm3/mol, $C_{\rm HT}\simeq 3.20$ cm3K/mol, and $\theta_{\rm CW}^{\rm
HT}\simeq-92$ K for NaYbP2O7 and $\chi_{0}^{\rm HT}\simeq-2.48\times
10^{-4}$cm3/mol, $C_{\rm HT}\simeq 3.18$ cm3K/mol, and $\theta_{\rm CW}^{\rm
HT}\simeq-121$ K for KYbP2O7. The resulting effective moment $\mu_{\rm
eff}^{\rm HT}$ was calculated to be $\sim 5.06~{}\mu_{\rm B}$ and $\sim
5.05~{}\mu_{\rm B}$ for NaYbP2O7 and KYbP2O7, respectively. These values of
$\mu_{\rm eff}^{\rm HT}$ are in good agreement with the expected value of
$4.54~{}\mu_{\rm B}$ for Yb3+ ($J=7/2$, $g=8/7$) ion in the $4f^{13}$
configuration.

A change in slope was observed in $1/\chi(T)$ data below $50$ K, arising from
thermal depopulation of excited crystal electric field (CEF) levels. Similar
behavior was observed in several other Yb3+-based spin systems. Typically, the
combination of the spin orbit coupling and the CEF leads to a Kramers doublet
ground state for the Yb3+ ion, and the low temperature properties can be
described by an $S_{\rm eff}=1/2$ ground state [48, 49, 50, 51, 52, 53].

The magnetization isotherms $M(H)$ of KYbP2O7 (see Fig. 5a) and NaYbP2O7 (see
Fig. 5b) measured up to 7 T at $T=0.4$ K show saturation around 1.5 T. The
saturation values $M_{\rm sat}$ are estimated by fitting the high-field
regions above 5 T with straight lines reflecting $\chi_{0}H$ and extrapolating
the lines back to zero field. This yields $\chi_{0}\simeq 5.51\times 10^{-3}$
cm3/mol and $M_{\rm sat}\simeq 1.55~{}\mu_{\rm B}$ for NaYbP2O7 and
$\chi_{0}\simeq 4.11\times 10^{-3}$ cm3/mol and $M_{\rm sat}\simeq
1.47~{}\mu_{\rm B}$ for KYbP2O7. Using $\mu_{\rm sat}=g_{\rm eff}S_{\rm
eff}~{}\mu_{\rm B}$ gives $g_{\rm eff}$ values of $\simeq 3.10$ for NaYbP2O7
and $\simeq 2.94$ for KYbP2O7.

After subtracting $\chi_{0}$, the $\chi(T)$ data below $2$ K are fitted by the
CW law $\chi(T)=C/(T-\theta_{\rm CW})$. The fitting yields $\theta^{\rm
LT}_{\rm CW}\simeq(51\pm 4)$ mK and $\mu_{\rm eff}^{\rm LT}\simeq
2.68~{}\mu_{\rm B}$ for NaYbP2O7 and $\theta^{\rm LT}_{\rm CW}\simeq(18\pm 4)$
mK and $\mu_{\rm eff}^{\rm LT}\simeq 2.53~{}\mu_{\rm B}$ for KYbP2O7. The
obtained effective moments are in good agreement with $\mu_{\rm eff}=g_{\rm
eff}\sqrt{S_{\rm eff}(S_{\rm eff}+1)}~{}\mu_{\rm B}$ for a pseudo spin-$1/2$
ground state with $g_{\rm eff}$ values of $\sim 3.10$ for NaYbP2O7 and $\sim
2.92$ for KYbP2O7, consistent with those estimated from $M_{\rm sat}$.

Very small values of $\theta^{\rm LT}_{\rm CW}$ at low-$T$ suggest the lack of
significant exchange interactions between Yb3+ magnetic moments. According to
mean field theory, $\theta_{\rm CW}$ is the sum of all possible exchange
interactions [54]. Since there are many possible interaction pathways in the
structure, as shown by Fig. 1 and Fig. 2, it is also possible that the
competing ferromagnetic and antiferromagnetic exchange interactions cancel
each other, resulting in very small values of $\theta^{\rm LT}_{\rm CW}$.

###  III.4 Specific heat

Figure 6: Low-temperature magnetic specific heat $(C_{\rm m})$ of NaYbP2O7
(upper panel of (a)) and KYbP2O7 (upper panel of (b)) at several external
magnetic fields. Phonon contribution was subtracted from the raw data using a
polynomial fit. Magnetic entropy $(S_{\rm m})$ of NaYbP2O7 (lower panel of
(a)) and KYbP2O7 (lower panel of (b)) calculated by integrating $C_{\rm m}/T$.
At $0$ T and $1$ T fields, the entropy is vertically shifted to $2$ T and $5$
T values to match $R\ln 2$ expected for the Kramers doublet. Two arrows show
the ADR process.

The specific heat analyses of NaYbP2O7 and KYbP2O7 at different applied fields
are shown in various panels of Fig. 6, suggesting their ADR potentials. The
specific heat data of NaYbP2O7 and KYbP2O7 show no signatures of magnetic
long-range ordering down to $0.4$ K. In zero-field, the specific heat shows an
increase towards lower temperatures, suggesting the entropy accumulation
associated with the Kramers doublet of Yb3+. This doublet is split by the
magnetic field into $j_{z}=+1/2$ and $j_{z}=-1/2$ levels, causing a Schottky
anomaly that shifts toward higher temperatures with increasing field.

For NaYbP2O7 and KYbP2O7, the resulting magnetic entropy ($S_{m}$) saturates
about $5.3$ J/mol K and $5.6$ J/mol K, respectively (see Fig. 6). The values
agree quite well with the expected theoretical value $S_{\rm m}=R\ln(2J+1)$ of
5.76 J/mol K for the Kramers doublet with $J_{\rm eff}=1/2$. At $0$ T and $1$
T, the entire entropy of $R\ln 2$ associated with the lowest Kramers doublet
of Yb3+ could not be recovered as we did not measure below $0.4$ K. Higher
fields shift the entropy toward higher temperatures, so that at $2$ T, almost
complete entropy of $R\ln 2$ can be recovered above $0.4$ K. The entropy data
at $0$ T and $1$ T, are vertically shifted to match values at higher-fields
($2$ T and $5$ T). At $5$ T, only $\sim 10$% of $R\ln 2$ remains at $T=2$ K.
Therefore, by adiabatic demagnetization starting from $2$ K and $5$ T, very
low temperatures can be attained as indicated by arrows in Fig. 6.

In the zero-field specific heat data of NaYbP2O7 and KYbP2O7, a tiny anomaly
at $\sim 2.23$ K is found, arising from a Yb2O3 impurity contribution [55].
Integrating the entropy related to this impurity peak yields a negligible
fraction of $\sim 0.6\%$ of $R\ln 2$ related to the impurity contribution for
both compounds.

###  III.5 Adiabatic Demagnetization Refrigeration (ADR)

Figure 7: ADR performance comparison of NaYbP2O7 (red), KYbP2O7 (green), and
KBaYb(BO3)2 (brown) in the commercial physical property measurement system
(PPMS). The sample is slowly cooled to $T=2$ K at $H=5$ T through the weak
thermal link to the PPMS puck kept at $2$ K and the ’high vacuum mode’ of the
PPMS was employed in order to achieve thermal decoupling. Subsequently,
magnetic field is swept from $5$ T to zero at a rate of $0.15$ T min-1 and the
temperature change with time t is recorded. The blue curve shows the
respective magnetic field.

The comparison of the cooling performances of NaYbP2O7 (red) and KYbP2O7
(green) with KBaYb(BO3)2 (yellow) in the commercial PPMS are shown in Fig 7.
The sample temperatures are plotted on the left $y$-axis, while the external
magnetic field is plotted on the right $y$-axis. During demagnetization, the
lowest temperature attained by KYbP2O7 is $\sim 37$ mK while the lowest
temperatures attained by NaYbP2O7 and KBaYb(BO3)2 are both $\sim 45$ mK. A
spike in the sample temperature around $70$ mK is observed, possibly due to
flux pinning of the superconducting magnet.

The time required for NaYbP2O7 to relax back from the lowest temperature to
$2$ K is $\sim 55$ min, whereas the warming time for KYbP2O7 is shorter, i.e.,
about $\sim 35$ min. The warming time of NaYbP2O7 is much longer than that of
KBaYb(BO3)2 ($\sim 40$ min). Note that the hold-times at low temperatures
could be significantly enhanced by using a thermal shield and precooling ADR
station for all connected wires. But already in this simple setup, the warm-up
times are sufficient to use the setup for instance to perform temperature
dependent electrical resistance measurements on small samples. Importantly,
the obtained base temperatures are much lower as for the commercially
available ADR option for the PPMS, which uses CPA and reaches a minimum
temperature of $100$ mK [56].

##  IV Discussion

Table 1: Comparison of important parameters of different mK ADR materials: $T_{\rm m}$ is the magnetic ordering temperature, $S_{\rm GS}$ is the entropy of the ground-state multiplet, and $R$ is the universal gas constant. The abbreviations are MAS = Mn(NH4)2(SO4)${}_{2}\cdot 6$H2O (manganese ammonium sulfate), FAA = NH4Fe(SO4)$\cdot$$12$H2O (ferric ammonium alum), CPA = KCr(SO4)$\cdot 12$H2O (chromium potassium alum), CMN = Mg3Ce2(NO3)${}_{12}\cdot 24$H2O (cerium magnesium nitrate). |  | ADR materials |  |
---|---|---|---|---
Material | $T_{\rm m}$ | mag. ion/vol. | $S_{\rm GS}$ |  $S_{\rm GS}$/vol.
| (mK) | (nm-3) |  | [mJ/(K cm3)]
MAS [57] | $170$ | $2.8$ | $R\ln 6$ | $70$
FAA [57] | $30$ | $2.1$ | $R\ln 6$ | $53$
CPA [37] | $10$ | $2.2$ | $R\ln 4$ | $42$
CMN [58] | $2$ | $1.7$ | $R\ln 2$ | $16$
YbPt2Sn12 [28] | $250$ | $12.9$ | $R\ln 2$ | $124$
Yb3Ga5O12 [29] | $54$ | $13.2$ | $R\ln 2$ | $124$
KBaYb(BO3)2 [24] | $<22$ | $6.7$ | $R\ln 2$ | $64$
NaYbP2O7 | $<45$ | $6.6$ | $R\ln 2$ | $64$
KYbP2O7 | $<37$ | $6.0$ | $R\ln 2$ | $57$

The low-temperature properties of both NaYbP2O7 and KYbP2O7 can be well
described by an effective spin-$1/2$ ground state expected for Yb3+-based
quantum magnets. Very small CW temperatures obtained from the analysis of the
susceptibility data at low temperatures indicate the absence of significant
interactions between the Yb-moments, which is advantageous to achieve low
final temperatures in ADR measurements.

The compounds NaYbP2O7 and KYbP2O7 have proven to be very efficient for ADR
applications because they do not contain water molecules and are stable to
heating and evacuation. It can be seen from Fig. 6 that almost all of the
entropy ($R\ln 2$) of the lowest Kramers doublet can be used for cooling, even
at a moderate magnetic field of $5$ T, similar to the conventionally used
paramagnetic salts [14]. The crucial parameters determining the efficiency of
various known ADR materials are compared in Table 1. The full entropy of the
ground state is described as $S_{\rm GS}=R\ln(2J+1)$, and the entropy density
$S_{\rm GS}$/vol. is calculated by dividing $S_{\rm GS}$ by the unit cell
volume. A large value of $S_{\rm GS}$ is always beneficial because the
magnetic entropy changes ($\Delta S_{\rm m}$) of magnetic refrigerants act as
the driving force of ADR. However, it is worth noting that for practical
purposes, the relevant quantity is not the molar entropy but rather the
volumetric entropy density $S_{\rm GS}$ of the material.

As can be seen from the Table. 1, materials with high entropy density such as
YbPt2Sn or Yb3Ga5O12 exhibit magnetic orderings at $250$ mK and $54$ mK,
respectively, which limit their lowest attainable temperatures as the
entropies drop to zero below the ordering temperatures. The high-temperature
paramagnetic salts such as MAS and FAA, which have a larger magnetic entropy
$R\ln 6$ are also affected by their much higher magnetic ordering
temperatures. On the other hand, low transition temperature materials such as
CPA and CMN have low magnetic moment density and hence low entropy density. A
high entropy density usually contradicts a low magnetic ordering temperature.

In this context, it has been reported that the disordered quantum magnet
KBaYb(BO3)2 exhibits these two mutually exclusive criteria, such as a high
volumetric entropy density ($64$ mJ/(K cm3)) combined with a very low ordering
at approximately 9 mK [38], allowing ADR to well below 20 mK [24]. In this
compound, the magnetic frustration and the structural randomness help to
suppress the magnetic order. Because of its suitable properties, KBaYb(BO3)2
has been proven to be an excellent anhydrous ADR refrigerant. By demagnetizing
$H=5$ T at $2$ K in PPMS, a minimal temperature of $\sim 40$ mK was attained
with pellets mixed of KBaYb(BO3)2 powder with Ag powder [24]. Utilizing a
better adiabatic setup in the dilution refrigerator with feedback control of
the bath temperature following the sample temperature, KBaYb(BO3)2 cooled upon
demagnetization starting at 5 T from 2 K to well below 20 mK [24].

NaYbP2O7 and KYbP2O7 have also high magnetic ion densities ($\sim 6.6$ and
$\sim 6$ nm-3) and volumetric entropy densities ($\sim 64$ and $\sim 57$ mJ/(K
cm3)). These values are comparable to those of KBaYb(BO3)2 but are much higher
than those of the paramagnetic ADR salts for the mK application. The
comparative study of the ADR performance of AYbP2O7 and KBaYb(BO3)2 pellets
under exactly similar conditions revealed a significantly lower minimal
temperature of $\sim 37$ mK for KYbP2O7, whereas NaYbP2O7 reaches the same
minimum temperature of $\sim 45$ mK as KBaYb(BO3)2 but with the advantage of a
much longer warm-up time. Comparison with our previous work on KBaYb(BO3)2
[24] suggests that KYbP2O7 and NaYbP2O7 can also be cooled by demagnetization
to significantly lower temperatures of at least 20 mK in better adiabatic
conditions (as realized previously for the former utilizing a 3He-4He dilution
refrigerator).

Compared to commercially used conventional mK ADR coolants based on
paramagnetic salts, $A$YbP2O7 and KBaYb(BO3)2 are anhydrous compounds and
hence stable at high-vacuum and high temperatures up to at least
$600~{}^{\circ}$C. Therefore, an encapsulated installation is not required,
making them user-friendly and suitable for UHV applications. Finally,
excellent thermal contact can be achieved in easily prepared pellets by mixing
powder samples with silver powder in $1:1$ ratio. Overall, it has been shown
that all three cooling substances are very well suited for UHV-compatible ADR
down to $50$ mK.

##  V Conclusion

In summary, we have performed a comprehensive study on the low-temperature
properties of two Yb-based quantum magnets NaYbP2O7 and KYbP2O7 and compared
their mK-ADR performance with KBaYb(BO3)2. The low-temperature properties are
well described by very weakly interacting $J_{\rm eff}=1/2$ Kramers doublets
of the Yb3+ ions. ADR experiments in the PPMS under comparable conditions for
all three compounds confirm that all the three are highly suitable to achieve
temperatures below 50 mK. With respect to KBaYb(BO3)2, KYbP2O7 yields a $20\%$
lower temperature (but $12.5\%$ shorter hold time), while for NaYbP2O7 a
similar end temperature is combined with $30\%$ longer hold time compared to
KBaYb(BO3)2. Both diphosphates are thus excellent new UHV-compatible mK ADR
materials.

###### Acknowledgements.

UA would like to acknowledge DST, India, for financial support bearing
sanction (DST/INSPIRE/04/2019/001664). Work supported by the German Science
Foundation through projects 107745057 (TRR80) and 514162746 (GE 1640/11-1).
DDS thanks SERB, DST, and CSIR, Government of India, for financial support. We
note that a German patent for the usage of $AB$P2O7 ($A$=alkaline metal,
$B$=rare earth) for UHV compatible ADR to very low temperatures has been filed
by the University of Augsburg (file reference DE 10 2023 106 074.0, March 10,
2023).

## References

  * Zapf  _et al._ [2014] V. Zapf, M. Jaime, and C. Batista, Bose-Einstein condensation in quantum magnets, Rev. Mod. Phys. 86, 563 (2014).
  * Onnes [1911] H. K. Onnes, Akad. Van Wetenschappen. 14, 113 (1911).
  * Goerbig [2011] M. O. Goerbig, Electronic properties of graphene in a strong magnetic field, Rev. Mod. Phys. 83, 1193 (2011).
  * Zhou  _et al._ [2017] Y. Zhou, K. Kanoda, and T.-K. Ng, Quantum spin liquid states, Rev. Mod. Phys. 89, 025003 (2017).
  * Bramwell and Harris [2020] S. T. Bramwell and M. J. Harris, The history of spin ice, J. Phys. Condens. Matter 32, 374010 (2020).
  * Kelley [2003] L. M. Kelley, Challenging “helium scarcity”, Chemical Engineering 110, 8 (2003).
  * Olafsdottir and Sverdrup [2020] A. H. Olafsdottir and H. U. Sverdrup, Assessing the Past and Future Sustainability of Global Helium Resources, Extraction, Supply and Use, Using the Integrated Assessment Model WORLD7, Biophysical Economics and Sustainability 5, 1 (2020).
  * Kouzes and Ely [2010] R. T. Kouzes and J. H. Ely,  _Status summary of $\mathrm{{}^{3}He}$ and neutron detection alternatives for homeland security_ (2010).
  * Shea and Morgan [2011] D. A. Shea and D. Morgan, The helium-3 shortage: Supply, demand, and options for congress (LIBRARY OF CONGRESS WASHINGTON DC, 2011).
  * Cho [2009] A. Cho, Helium-3 shortage could put freeze on low-temperature research, Science 326, 778 (2009).
  * Weiss and Piccard [1917] P. Weiss and A. Piccard, Le phénomène magnétocalorique, J. Phys. Theor. Appl. 7, 103 (1917).
  * Debye [1926] P. Debye, Einige Bemerkungen zur Magnetisierung bei tiefer Temperatur, Ann. der Phys. 386, 1154 (1926).
  * Giauque and MacDougall [1933] W. Giauque and D. MacDougall, Attainment of temperatures below $1^{\circ}$ absolute by demagnetization of $\mathrm{Gd_{2}(SO_{4})_{3}\cdot 8H_{2}O}$, Phys. Rev. 43, 768 (1933).
  * Pobell [2007] F. Pobell,  _Matter and methods at low temperatures_ , 3rd ed. (Springer, 2007).
  * Shirron [2014] P. J. Shirron, Applications of the magnetocaloric effect in single-stage, multi-stage and continuous adiabatic demagnetization refrigerators, Cryogenics 62, 130 (2014).
  * Shirron  _et al._ [2004] P. Shirron, E. Canavan, M. DiPirro, J. Francis, M. Jackson, J. Tuttle, T. King, and M. Grabowski, Development of a cryogen-free continuous ADR for the constellation-X mission, Cryogenics 44, 581 (2004).
  * Shirron [2007] P. J. Shirron, Cooling capabilities of adiabatic demagnetization refrigerators, J. Low Temp. Phys. 148, 915 (2007).
  * Luchier  _et al._ [2010] N. Luchier, J. Duval, L. Duband, P. Camus, G. Donnier-Valentin, and M. Linder, 50 mK cooling solution with an ADR precooled by a sorption cooler, Cryogenics 50, 591 (2010).
  * Hagmann and Richards [1995] C. Hagmann and P. Richards, Adiabatic demagnetization refrigerators for small laboratory experiments and space astronomy, Cryogenics 35, 303 (1995).
  * Jahromi  _et al._ [2019] A. E. Jahromi, P. J. Shirron, and M. J. DiPirro, Sub-kelvin cooling systems for quantum computers, in  _2019 Cryogenic Engineering Conference and International Cryogenic Materials Conference (CEC/ICMC)_ , GSFC-E-DAA-TN70637 (2019).
  * Alahmer  _et al._ [2021] A. Alahmer, M. Al-Amayreh, A. O. Mostafa, M. Al-Dabbas, and H. Rezk, Magnetic refrigeration design technologies: State of the art and general perspectives, Energies 14 (2021).
  * Bartlett  _et al._ [2010] J. Bartlett, G. Hardy, I. Hepburn, C. Brockley-Blatt, P. Coker, E. Crofts, B. Winter, S. Milward, R. Stafford-Allen, M. Brownhill, _et al._ , Improved performance of an engineering model cryogen free double adiabatic demagnetization refrigerator, Cryogenics 50, 582 (2010).
  * [23] Kiutra, See e.g. Cooling Technology: Solid-state cryogen-free cooling, https://kiutra.com/technology/ .
  * Tokiwa  _et al._ [2021] Y. Tokiwa, S. Bachus, K. Kavita, A. Jesche, A. A. Tsirlin, and P. Gegenwart, Frustrated magnet for adiabatic demagnetization cooling to milli-kelvin temperatures, Commun. Mater. 2, 1 (2021).
  * Liu  _et al._ [2022] X.-Y. Liu, Y. Gao, H. Li, W. Jin, J. Xiang, H. Jin, Z. Chen, W. Li, and G. Su, Quantum spin liquid candidate as superior refrigerant in cascade demagnetization cooling, Commun. Phys. 5, 1 (2022).
  * [26] M. Kleinhans, K. Eibensteiner, J. Leiner, J. Spallek, A. Regnat, and C. Pfleiderer, Magnetocaloric properties of $\mathrm{(RE)_{3}Ga_{5}O_{12}}$ (RE= Tb, Gd, Nd, Dy), Phys. Rev. Appl. 19.
  * Wolf  _et al._ [2011] B. Wolf, Y. Tsui, D. Jaiswal-Nagar, U. Tutsch, A. Honecker, K. Remović-Langer, G. Hofmann, A. Prokofiev, W. Assmus, G. Donath, and M. Lang, Magnetocaloric effect and magnetic cooling near a field-induced quantum-critical point, Proc. Natl. Acad. Sci 108, 6862 (2011).
  * Jang  _et al._ [2015] D. Jang, T. Gruner, A. Steppke, K. Mitsumoto, C. Geibel, and M. Brando, Large magnetocaloric effect and adiabatic demagnetization refrigeration with $\mathrm{YbPt_{2}Sn}$, Nat. commun. 6, 1 (2015).
  * Brasiliano  _et al._ [2020] D. A. P. Brasiliano, J.-M. Duval, C. Marin, E. Bichaud, J.-P. Brison, M. Zhitomirsky, and N. Luchier, YbGG material for adiabatic demagnetization in the 100 mK–3 K range, Cryogenics 105, 103002 (2020).
  * Tokiwa  _et al._ [2016] Y. Tokiwa, B. Piening, H. S. Jeevan, S. L. Bud’ko, P. C. Canfield, and P. Gegenwart, Super-heavy electron material as metallic refrigerant for adiabatic demagnetization cooling, Sci. Adv. 2, e1600835 (2016).
  * Evangelisti  _et al._ [2011] M. Evangelisti, O. Roubeau, E. Palacios, A. Camón, T. N. Hooper, E. K. Brechin, and J. J. Alonso, Cryogenic magnetocaloric effect in a ferromagnetic molecular dimer, Angew. Chem. Int. Ed. 50, 6606 (2011).
  * Baniodeh  _et al._ [2018] A. Baniodeh, N. Magnani, Y. Lan, G. Buth, C. E. Anson, J. Richter, M. Affronte, J. Schnack, and A. K. Powell, High spin cycles: topping the spin record for a single molecule verging on quantum criticality, npj Quantum Mater. 3, 1 (2018).
  * Zhitomirsky [2003] M. Zhitomirsky, Enhanced magnetocaloric effect in frustrated magnets, Phys. Rev. B 67, 104421 (2003).
  * Wolf  _et al._ [2016] B. Wolf, U. Tutsch, S. Dörschug, C. Krellner, F. Ritter, W. Assmus, and M. Lang, Magnetic cooling close to a quantum phase transition—the case of $\mathrm{Er_{2}Ti_{2}O_{7}}$, J. Appl. Phys. 120, 142112 (2016).
  * Hu and Du [2008] Y. Hu and A. Du, Magnetization behavior and magnetic entropy change of frustrated Ising antiferromagnets on two-and three-dimensional lattices, J. Phys. Condens. Matter 20, 125225 (2008).
  * Wikus  _et al._ [2014] P. Wikus, E. Canavan, S. T. Heine, K. Matsumoto, and T. Numazawa, Magnetocaloric materials and the optimization of cooling power density, Cryogenics 62, 150 (2014).
  * Daniels and Kurti [1954] J. Daniels and N. Kurti, The thermal and magnetic properties of chromium potassium alum below 0.1 K, Proc. R. Soc. A 221, 243 (1954).
  * Jesche  _et al._ [2023] A. Jesche, N. Winterhalter-Stocker, F. Hirschberger, A. Bellon, S. Bachus, Y. Tokiwa, A. A. Tsirlin, and P. Gegenwart, Adiabatic demagnetization cooling well below the magnetic ordering temperature in the triangular antiferromagnet $\mathrm{KBaGd(BO_{3})_{2}}$, Phys. Rev. B 107, 104402 (2023).
  * [39] J. Xiang, C. Su, N. Xi, Z. Fu, Z. Chen, H. Jin, Z. Chen, Z.-J. Mo, Y. Qi, J. Shen, L. Zhang, W. Jin, W. Li, P. Sun, and G. Su, Dipolar spin liquid ending with quantum critical point in a Gd-based triangular magnet, arXiv:2301.03571 .
  * Rodríguez-Carvajal [1993] J. Rodríguez-Carvajal, Recent advances in magnetic structure determination by neutron powder diffraction, Phys. B: Condens. Matter 192, 55 (1993).
  * Horchani-Naifer and Férid [2007] K. Horchani-Naifer and M. Férid, Potassium ytterbium diphosphate, Acta Crystallographica Section E: Structure Reports Online 63, i33 (2007).
  * Férid  _et al._ [2004] M. Férid, K. Horchani-Naifer, and M. Trabelsi-Ayedi, Crystal structure of sodium ytterbium diphosphate $\mathrm{NaYbP_{2}O_{7}}$, Z. Kristallogr. NCS 219, 385 (2004).
  * Guchhait  _et al._ [2019] S. Guchhait, U. Arjun, P. K. Anjana, M. Sahoo, A. Thirumurugan, A. Medhi, Y. Skourski, B. Koo, J. Sichelschmidt, B. Schmidt, M. Baenitz, and R. Nath, Case study of bilayered spin-1/2 square lattice compound $\mathrm{VO(HCOO)_{2}\cdot(H_{2}O)}$, Phys. Rev. Materials 3, 104409 (2019).
  * Nath  _et al._ [2015] R. Nath, M. Padmanabhan, S. Baby, A. Thirumurugan, D. Ehlers, M. Hemmida, H.-A. Krug von Nidda, and A. A. Tsirlin, Quasi-two-dimensional S=1/2 magnetism of $\mathrm{Cu[C_{6}H_{2}(COO)_{4}][C_{2}H_{5}NH_{3}]_{2}}$, Phys. Rev. B 91, 054409 (2015).
  * Matsumoto  _et al._ [2000] T. Matsumoto, Y. Miyazaki, A. S. Albrecht, C. P. Landee, M. M. Turnbull, and M. Sorai, Heat capacities of the $\mathrm{S}$ = 1/2 two-dimensional Heisenberg antiferromagnet bis(2-amino-5-chloropyridinium) tetrabromocuprate(ii) $\mathrm{[(5CAP)_{2}CuBr_{4}]}$ and its diamagnetic analogue $\mathrm{[(5CAP)_{2}ZnBr_{4}]}$, J. Phys. Chem. B 104, 9993 (2000).
  * Lancaster  _et al._ [2007] T. Lancaster, S. J. Blundell, M. L. Brooks, P. J. Baker, F. L. Pratt, J. L. Manson, M. M. Conner, F. Xiao, C. P. Landee, F. A. Chaves, S. Soriano, M. A. Novak, T. P. Papageorgiou, A. D. Bianchi, T. Herrmannsdörfer, J. Wosnitza, and J. A. Schlueter, Magnetic order in the s=1/2 two-dimensional molecular antiferromagnet copper pyrazine perchlorate $\mathrm{Cu(Pz)_{2}(ClO_{4})_{2}}$, Phys. Rev. B 75, 094421 (2007).
  * Shannon [1976] R. D. Shannon, Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides, Acta Cryst. A32, 751 (1976).
  * Paddison  _et al._ [2017] J. A. Paddison, M. Daum, Z. Dun, G. Ehlers, Y. Liu, M. B. Stone, H. Zhou, M. Mourigal, _et al._ , Continuous excitations of the triangular-lattice quantum spin liquid $\mathrm{YbMgGaO_{4}}$, Nat. Phys 13, 117 (2017).
  * Bordelon  _et al._ [2019] M. M. Bordelon, E. Kenney, C. Liu, T. Hogan, L. Posthuma, M. Kavand, Y. Lyu, M. Sherwin, N. P. Butch, C. Brown, _et al._ , Field-tunable quantum disordered ground state in the triangular-lattice antiferromagnet $\mathrm{NaYbO_{2}}$, Nat. Phys 15, 1058 (2019).
  * Ranjith  _et al._ [2019a] K. M. Ranjith, S. Luther, T. Reimann, B. Schmidt, P. Schlender, J. Sichelschmidt, H. Yasuoka, A. Strydom, Y. Skourski, J. Wosnitza, _et al._ , Anisotropic field-induced ordering in the triangular-lattice quantum spin liquid $\mathrm{NaYbSe_{2}}$, Phys. Rev. B 100, 224417 (2019a).
  * Baenitz  _et al._ [2018] M. Baenitz, P. Schlender, J. Sichelschmidt, Y. Onykiienko, Z. Zangeneh, K. Ranjith, R. Sarkar, L. Hozoi, H. Walker, J.-C. Orain, _et al._ , $\mathrm{NaYbS_{2}}$: A planar spin-1/2 triangular-lattice magnet and putative spin liquid, Phys. Rev. B 98, 220409 (2018).
  * Schmidt  _et al._ [2021] B. Schmidt, J. Sichelschmidt, K. M. Ranjith, T. Doert, and M. Baenitz, Yb delafossites: Unique exchange frustration of $4f$ spin-$\frac{1}{2}$ moments on a perfect triangular lattice, Phys. Rev. B 103, 214445 (2021).
  * Ranjith  _et al._ [2019b] K. M. Ranjith, D. Dmytriieva, S. Khim, J. Sichelschmidt, S. Luther, D. Ehlers, H. Yasuoka, J. Wosnitza, A. A. Tsirlin, H. Kühne, and M. Baenitz, Field-induced instability of the quantum spin liquid ground state in the ${J}_{\mathrm{eff}}=\frac{1}{2}$ triangular-lattice compound $\mathrm{NaYbO_{2}}$, Phys. Rev. B 99, 180401 (2019b).
  * Domb and Miedema [1964] C. Domb and A. R. Miedema,  _Progress in Low Temperature Physics_ , edited by C. J. Gorter, Vol. 4 (Elsevier, North-Holland, Amsterdam, 1964) pp. 296–343.
  * Moon  _et al._ [1967] R. M. Moon, H. R. Child, W. C. Koehler, and L. J. Raubenheimer, Magnetic structure of $\mathrm{Er_{2}O_{3}}$ and $\mathrm{Yb_{2}O_{3}}$, J. Appl. Phys. 38, 1383 (1967).
  * [56] Quantum Design PPMS Platform Measurement Options, https://www.qdusa.com/siteDocs/productBrochures/1084-500.pdf .
  * Vilches and Wheatley [1966] O. E. Vilches and J. C. Wheatley, Measurements of the specific heats of three magnetic salts at low temperatures, Phys. Rev. 148, 509 (1966).
  * Fisher  _et al._ [1973] R. A. Fisher, E. W. Hornung, G. E. Brodale, and W. F. Giauque, Magnetothermodynamics of $\mathrm{Ce_{2}Mg_{3}(NO_{3})_{12}\cdot 24H_{2}O}$. II. The evaluation of absolute temperature and other thermodynamic properties of $\mathrm{CMN}$ to 0.6 mK, J. Chem. Phys. 58, 5584 (1973).