Sub-1 K Adiabatic Demagnetization Refrigeration with Rare-Earth Borates Ba₃XB₉O₁₈ and Ba₃XB₃O₉, X = (Yb, Gd)

Featured Application

The materials studied in this work can be utilized for adiabatic demagnetization refrigeration down to the milli-Kelvin range and offer multiple benefits over traditional hydrated paramagnetic salts.

Abstract

Adiabatic demagnetization refrigeration (ADR) is regaining relevance for refrigeration to temperatures below 1 K as global helium-3 supply is increasingly strained. While ADR at these temperatures is long established with paramagnetic hydrated salts, more recently, frustrated rare-earth oxides were found to offer higher entropy densities and practical advantages, since they do not degrade under heating or evacuation. We report structural, magnetic, and thermodynamic properties of the rare-earth borates Ba₃XB₉O₁₈ and Ba₃XB₃O₉ with X = (Yb, Gd). Except for Ba₃GdB₉O₁₈, which orders at 108 mK, the three other materials remain paramagnetic down to their lowest measured temperatures. ADR performance starting at 2 K in a field of 5 T is analyzed and compared to literature.

1. Introduction

Temperatures below one Kelvin are increasingly utilized not only in basic research but also in a growing number of industrial applications [1]. While dilution refrigerators (DRs) have long been the default option for obtaining and maintaining low temperatures [2,3], there have been recent advances and renewed interest in ADR [4,5,6,7]. These can be grouped into two broad categories: refrigerator technology and refrigerant technology.

In terms of refrigerator technology, there have been developments in continuous ADR that can maintain a set temperature by cycling multiple refrigerators connected via heat switches. While such setups are more complex than single-shot ADR, they offer the crucial benefit of maintaining continuous operation of the cold payload [8,9,10]. At the moment, continuous ADR is not yet competitive with DRs in terms of cooling power [11].

The field of refrigerant technology has long been limited by the performance characteristics of paramagnetic hydrated salts. Here, a large distance between magnetic moments, achieved by separating them with water molecules, achieves very weak magnetic interaction and correspondingly low ordering temperatures. As a consequence, the magnetic entropy density and the cooling power are low. Furthermore, due to their chemical instability, paramagnetic hydrated salts like ferric ammonium alum, cerium magnesium nitrate, and others need to be encapsulated for use in a vacuum environment, and even then they may never be heated above room temperature. Otherwise, the crystal water contained in these substances will quickly escape, and structural collapse will make them inoperative for ADR.

There have been three major development avenues for new refrigerants with higher entropy density and better chemical stability: Single-crystal refrigerants like gadolinium gallium garnet or ytterbium gallium garnet offer high entropy density and allow ADR to 1 and 0.2 K, respectively [12,13,14]. Intermetallic compounds like YbNi_1.6Sn have a much higher thermal conductivity because of the conduction electron contribution, while maintaining a high entropy density, and are thus a suitable alternative at temperatures down to $0.2\mathrm{K}$ [15,16,17,18]. The third alternative is rare-earth oxides in which geometrical frustration reduces the ordering temperature, allowing much lower ADR end temperatures in combination with a high entropy density [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]. These materials are chemically stable, the polycrystal synthesis can easily be upscaled, and excellent thermal and mechanical properties are obtained by compressing them with fine silver powder admixture, while maintaining a relatively high entropy density. Prototypes of this new class of oxide ADR materials include the borates KBaX(BO₃)₂ with magnetic triangular lattices of X = Yb, Gd [34,35]. While KBaGd(BO₃)₂ orders antiferromagnetically at 263 mK, the ADR experiment on a pellet in the Quantum Design Physical Property Measurement System (PPMS), starting at 2 K in a field of 5 T, revealed a minimal temperature of 122 mK [20]. Assuming a scaling of the ordering temperatures by the square of the ratio of the saturation moments (which is about 30), for isostructural KBaYb(BO₃)₂, a very low ordering near 9 mK has been deduced, and indeed, the system remains paramagnetic in ADR experiments down to 16 mK [19]. This is much lower than the minimal temperatures that can be obtained with the metallic ADR materials mentioned above. Both geometrical frustration due to the triangular magnetic lattice and K⁺/Ba²⁺ site randomness [34] may be important with respect to the outstanding ADR performance of the two materials [20].

Ba₃YbB₃O₉ is another triangular lattice ADR candidate material with a very similar intralayer Yb-Yb distance (5.43 Å) as in KBaYb(BO₃)₂, but an about 30% larger interlayer Yb-Yb distance [36,37,38,39], making it more two-dimensional. Furthermore, Ba₃YbB₃O₉ has no site randomness, but it does exhibit two inequivalent Yb sites. We also study the triangular lattice material Ba₃YbB₉O₁₈, with a significantly larger in-plane Yb-Yb distance of 7.18 Å, which arises from a larger separation of YbO₆ octahedra by the triangular arrangement of the BO₃ groups in the structure [39,40,41]. It is interesting to compare the ADR performance of these materials, together with their Gd counterparts, to that of KBaX(BO₃)₂ (X=Yb, Gd). In the Gd variants with classical spin $S=7/2$, the higher entropy density improves ADR performance at the expense of minimal ADR temperature, due to the enhanced magnetic couplings.

The rest of this paper is organized as follows: After the description of the utilized methods, we report below structural, magnetic, thermodynamic, and ADR properties of Ba₃XB₉O₁₈ and Ba₃XB₃O₉ with X = (Yb, Gd). No indication of magnetic ordering has been found in either of these Yb compounds down to the lowest obtained ADR temperatures of below 40 mK. In Ba₃GdB₃O₉, a broad maximum is found at low temperatures. In Ba₃GdB₉O₁₈, a Schottky-type broad peak, as well as a sharp phase transition, indicative of long-range magnetic order at lower temperatures, is observed in the heat capacity.

2. Materials and Methods

Polycrystals of Ba₃XB₉O₁₈ and Ba₃XB₃O₉ with X = (Yb, Gd) were prepared by solid-state reaction. The precursors were X₂O₃, BaCO₃, and H₃BO₃ with a purity of at least 99.9%. To compensate for losses, an excess of 2% wt. BaCO₃ and 4% wt. H₃BO₃ was used. All reagents were preheated, weighed, and thoroughly ground in an agate mortar. Subsequently, they were preheated in a furnace under a laboratory atmosphere at 700 °C over 12 h. After the first furnace run, the reagents were allowed to cool and then reground. A second oven run was performed under a laboratory atmosphere, but this time with an increased temperature of 900 °C over 24 h.

The products were ground again and analyzed for phase purity by powder X-ray diffraction in a PANAlytical Empyrean XRD (Almelo, Netherlands). To ensure good thermal contact, even at very low temperatures, the substances were mixed with 50% silver powder by weight and pressed into pellets for further analysis.

Pellets of 3 mm diameter were prepared for specific heat and magnetization measurements, while 15 mm diameter pellets were prepared for direct magnetocaloric studies.

Magnetization was measured in a Quantum Design MPMS3 SQUID magnetometer (San Diego, CA, USA); for measurements in the range $0.4\mathrm{K}<T<2\mathrm{K}$, the ³He option was utilized. The samples’ magnetic moment was measured under an isothermal field sweep with a vibrating-sample magnetometer (VSM). In order to achieve higher accuracy, DC magnetization measurements were conducted at the initial and final fields, and the VSM data was scaled to these DC measurements. Furthermore, the data was corrected for the calculated geometry factor of the cylindrical samples.

Specific heat was measured in a PPMS DynaCool by Quantum Design (San Diego, CA, USA). For measurements at lower temperatures, a ³He upgrade was utilized, for which a small piece was cut from the middle of a 3 mm pellet in order to limit the total specific heat and, thus, allow for more accurate and faster measurements.

The actual ADR performance of the samples was determined in a custom-built adiabatic demagnetization refrigeration setup inside a PPMS similar to the ones described in [19,20,21,25]. This setup comprises a standard PPMS puck that mounts a polyimide (PI) frame in which the pellet is supported by aromatic polyamide yarns. The suspended pellet is shielded from infrared radiation by a polished brass cap. Thermometry in this setup is performed by an ultra-low specific heat, low-time constant custom chip thermometer made from a commercially available RuO₂ chip resistor by removing the contacts and substrate. This thermometer was calibrated against a reference thermometer and attached to the pellet with GE 7031 varnish. Care was taken so as not to overheat the thermometer. Thus, the excitation was limited to less than 50 fW. Similarly a heater was prepared from a lower-resistance chip and also attached to the pellet. To limit thermal flux into the sample, superconducting wiring was utilized.

3. Results

3.1. Structure

Powder X-ray diffraction (PXRD) was performed at room temperature with a PANAlytical Empyrean diffractometer. The diffraction patterns were compared to the ICCD PDF-5+ database [42,43,44], and Rietveld refinements were conducted to determine phase purity. All samples contained only small fractions of foreign phases, of which the most common were identified as the corresponding rare-earth borates $(\mathrm{Yb},\mathrm{Gd}){\mathrm{BO}}_3$. Elevated concentrations of a foreign phase were found in Ba₃YbB₉O₁₈ at 1.2% wt. ${\mathrm{YbBO}}_3$ and in Ba₃GdB₉O₁₈ at 1.1% wt. ${\mathrm{GdBO}}_3$. A central peak at 44.5° can be attributed to the sample holder.

Both Ba₃YbB₉O₁₈ and Ba₃GdB₉O₁₈ form a hexagonal lattice of identical space group $P{6}_3/m$ with one rare-earth site forming layers in the $ab$ plane (Figure 1B,D). The rare-earth ions are coordinated in a hexagonal fashion, allowing for geometric frustration [36,37]. The layers differ by alternating orientations of the oxygen octahedra, further enhancing interlayer frustrations.

Ba₃YbB₃O₉ forms a hexagonal lattice of space group $P{6}_{3}cm$ [39]. In this compound, the ytterbium ions also form layers in the $ab$ plane; however, these layers contain two alternating Yb sites and are distorted (Figure 1A). The two sites share the orientation of the oxygen octahedra, and the layers differ by alternating orientations of the octahedra along the c axis. This arrangement allows for additional frustration between the two distinct Yb sites. A similar structure has been observed for Ba₃TbB₃O₉ [45].

Ba₃GdB₃O₉ exhibits major differences to the other compounds. It crystallizes in a trigonal lattice ($R\overline{3})$ with two distinct gadolinium sites forming alternating chains along the c axis (Figure 1C). These chains are separated from each other with nearest-neighbor chains offset along c. Similar structures have been known to harbor a manifold of magnetic phases in other substances [46,47]. There is little literature available on Ba₃GdB₃O₉ except for the structure, which we confirmed by PXRD [42]. The relevant parameters are listed in

Table 1. Structural data for the compounds, verified by powder XRD.

Compound	Lattice	Space Group	Unit Cell Parameters	Molar Volume
Ba3YbB9O18 [42]	Hexagonal	$P{6}_3/m$	$\mathrm{a}\text{:}7.1740\mathring{\mathrm{A}},\mathrm{c}\text{:}16.915\mathring{\mathrm{A}}$	$227.0{\mathrm{cm}}^3{\mathrm{mol}}^{-1}$
Ba3YbB3O9 [43]	Hexagonal	$P{6}_3/cm$	$\mathrm{a}\text{:}9.3830\mathring{\mathrm{A}},\mathrm{c}\text{:}17.441\mathring{\mathrm{A}}$	$133.5{\mathrm{cm}}^3{\mathrm{mol}}^{-1}$
Ba3GdB9O18 [42]	Hexagonal	$P{6}_3/m$	$\mathrm{a}\text{:}7.1934\mathring{\mathrm{A}},\mathrm{c}\text{:}17.206\mathring{\mathrm{A}}$	$232.2{\mathrm{cm}}^3{\mathrm{mol}}^{-1}$
Ba3GdB3O9 [44]	Trigonal	$R\overline{3}$	$\mathrm{a}\text{:}12.776\mathring{\mathrm{A}},\mathrm{c}\text{:}9.6063\mathring{\mathrm{A}}$	$141.8{\mathrm{cm}}^3{\mathrm{mol}}^{-1}$

. The distance between the (next) nearest neighbors governs the interaction between the rare-earth ions’ magnetic moments; thus, these parameters are listed in

Table 2. Distance between magnetic moments, verified by powder XRD. Parameters for the phonon contribution to the specific heat are listed as Einstein and Debye temperatures, as well as a mixing factor $\gamma$.

Compound	Nearest Neighbor (NN)	Next NN	ΘE	ΘD	γ
Ba3YbB9O18 [42]	$7.174\mathring{\mathrm{A}}$	$8.458\mathring{\mathrm{A}}$	$292\mathrm{K}$	$148\mathrm{K}$	$0.85$
Ba3YbB3O9 [43]	$5.417\mathring{\mathrm{A}}$	$8.721\mathring{\mathrm{A}}$	$89\mathrm{K}$	$364\mathrm{K}$	$0.04$
Ba3GdB9O18 [42]	$7.193\mathring{\mathrm{A}}$	$8.603\mathring{\mathrm{A}}$	$251\mathrm{K}$	$145\mathrm{K}$	$0.73$
Ba3GdB3O9 [44]	$4.776\mathring{\mathrm{A}}$	$7.710\mathring{\mathrm{A}}$	$68\mathrm{K}$	$343\mathrm{K}$	$0.05$

.

3.2. Magnetic Properties

To determine magnetic properties, pellets with a diameter of 3 mm and a height of 1 mm were prepared with 50% silver admixture. Figure 2 displays our data for the isothermal magnetization of the four different materials at temperatures between 0.4 and 10 K.

The magnetic moment of non-interacting dipoles can be described by the Brillouin function:

$$\mu \left(B\right)=gJ\left[{\displaystyle \frac{2J+1}{2J}}\coth \left({\displaystyle \frac{2J+1}{2J}}{\displaystyle \frac{g{\mu }_{\mathrm{B}}JB}{k_{\mathrm{B}}T}}\right)-{\displaystyle \frac{1}{2J}}\coth \left({\displaystyle \frac{g{\mu }_{\mathrm{B}}B}{2k_{\mathrm{B}}T}}\right)\right]+x_{0}B$$

(1)

with a total angular momentum of J and the Landé factor g. The function $\mu \left(B\right)$ (Equation (1)) was fitted to the data with fixed T and fixed $J=7/2$ and $g=2$ for the Gd compounds. A van Vleck contribution $x_{0}B$ was also added, although this did not become dominant for any of the materials.

The ytterbium samples (Figure 2A,B) can be approximated as free-ion systems, as the magnetic interaction energies are very small compared to thermal energy at the studied temperatures. Note that their magnetic interaction is much smaller compared to their Gd counterparts due to the smaller effective moment. For these samples, saturation was achieved in the $0.4\mathrm{K}$ and $2\mathrm{K}$ measurements. Fitting Equation (1) yields a g factor of $2.71$ for Ba₃YbB₃O₉ and $2.70$ for Ba₃YbB₉O₁₈. Significant anisotropy of g has been observed previously for similar Yb compounds [48] and for Ba₃YbB₃O₉ by Bag et al. [38] ($g_{ab}=2.19,g_c=3.38$), but the polycrystalline average observed is in good agreement with reports by Cho et al. [39] ($g_{\mathrm{PM}}=2.77\left(1\right)$).

Both gadolinium systems saturate close to the expected ${\mu }_{\mathrm{sat}}=7{\mu }_{\mathrm{B}}$ for high fields and low temperatures, indicating the expected $g=2$. Note that the single-ion approximation (black, Figure 2C,D) is inaccurate due to the interaction between ions for the data below 2 K. This manifests as a shift in the magnetic saturation to higher fields than expected in the free-ion model. This effect is enhanced in Ba₃GdB₃O₉ due to the higher magnetic moment density. Although the structural parameters are similar, the interaction in the gadolinium compounds is enhanced when compared to the ytterbium compounds, as the magnetic moments are significantly larger (${\mathrm{S}}_{\mathrm{eff}}=1/2$ vs. $\mathrm{S}=7/2$). An increase in interactions upon rare-earth replacement has been observed in the isostructural KBaYb(BO₃)₂ [19] and KBaGd(BO₃)₂ [20] as well as in the iso structural NaYbP₂O₇ [21] and NaGdP₂O₇ [24]. Therefore, a mean-field approximation was conducted by fitting Equation (2) to the $2\mathrm{K}$ and $0.4\mathrm{K}$ gadolinium datasets (pink, Figure 2C,D):

$$\mu ={\mu }_{\mathrm{sat}}{B}_J\left[{\displaystyle \frac{g{\mu }_{\mathrm{B}}J}{k_{\mathrm{B}}T}}\left(H+\alpha \mu \right)\right].$$

(2)

The mean-field model fits the data significantly better than the free-ion approximation; however, there are still significant deviations, likely due to crystal electric field (CEF) effects. For $2\mathrm{K}$, the mean-field model fits significantly better than for $0.4\mathrm{K}$. The mean-field parameter $\alpha$ appears to be temperature-dependent, as both samples exhibit a decrease in $\alpha$ for lower temperatures. This unexpected temperature dependence hints at influences that cannot be described well by the mean-field model—likely CEF effects and short-ranged magnetic correlations. For Ba₃GdB₉O₁₈, the onset of magnetic order could also play a role, even though this occurs at significantly lower temperatures.

The magnetic susceptibility $\chi$ of the samples was measured down to $0.4\mathrm{K}$. For high temperatures, an external field of $7\mathrm{T}$ was applied, and for low temperatures, a low field of $5\mathrm{mT}$ was utilized. The inverse susceptibility was fitted with a modified Curie–Weiss function (Equation (3)) to obtain effective moments and Weiss temperature ([49] Equation (18)):

$${\chi }^{-1}=(T-{\Theta }_{\mathrm{W}})\left[3k_{\mathrm{B}}{\mu }_0{N}_{\mathrm{A}}{\mu }_{\mathrm{B}}^2{\mu }_{\mathrm{eff}}^{-2}+{\chi }_0\right].$$

(3)

For high temperatures, Yb and Gd samples show a linear relation between ${\chi }^{-1}$ and T. For $T<100\mathrm{K}$, both Yb compounds deviate from the linear relation; as the CEF splits, the energy levels’ occupation shifts [50]. At low temperatures and low magnetic fields, the Yb samples exhibit Kramers behavior, thus returning to a linear ${\chi }^{-1}\left(T\right)$. To obtain ${\mu }_{\mathrm{eff}}$ and ${\Theta }_{\mathrm{W}}$ for the Yb samples, Equation (3) was fitted between $100\mathrm{K}$ and $300\mathrm{K}$, as shown in Figure 3. For Ba₃YbB₃O₉, the resulting effective moment of $4.73{\mu }_{\mathrm{B}}$ and Weiss temperature of ${\Theta }_{\mathrm{W}}=-108\mathrm{K}$ are close to reports by Gao et al. [36] (${\mu }_{\mathrm{eff}}=4.60{\mu }_{\mathrm{B}},{\Theta }_{\mathrm{W}}=-109\mathrm{K}$), as well as by Cho et al. [39] (${\mu }_{\mathrm{eff}}=4.89\left(1\right){\mu }_{\mathrm{B}},{\Theta }_{\mathrm{W}}=-105\left(1\right)\mathrm{K}$). For Ba₃YbB₉O₁₈, the effective moment of $4.75{\mu }_{\mathrm{B}}$ and Weiss temperature ${\Theta }_{\mathrm{W}}=-100\mathrm{K}$ are close to data reported by Khatua et al. [40] (${\mu }_{\mathrm{eff}}=4.73{\mu }_{\mathrm{B}},{\Theta }_{\mathrm{W}}=-90\mathrm{K}$), as well as by Cho et al. [39] (${\mu }_{\mathrm{eff}}=4.74\left(1\right){\mu }_{\mathrm{B}},{\Theta }_{\mathrm{W}}=-99\left(1\right)\mathrm{K}$). There is a small discrepancy compared to the observations by Liu et al. [41] (${\mu }_{\mathrm{eff}}=4.34{\mu }_{\mathrm{B}},{\Theta }_{\mathrm{W}}=-78.09\mathrm{K}$), but this dataset was collected at $0.2\mathrm{T}$, while our data was collected at a significantly higher field of $7\mathrm{T}$. The resulting effective moments of $4.73{\mu }_{\mathrm{B}}$ and $4.75{\mu }_{\mathrm{B}}$ are close to the theoretical ${\mu }_{\mathrm{eff}}=g_J\sqrt{J(J+1)}{\mu }_{\mathrm{B}}=4.54{\mu }_{\mathrm{B}}$ for a free Yb³⁺ ion with $J=7/2$ and $g_J=8/7$. The absolute values of the Weiss temperatures from the high-temperature fit of the two Yb materials cannot be interpreted as indicative of large coupling between the moments, because they result from the crystal field splitting, leaving only the low-lying doublet at low temperatures [36]. For Ba₃YbB₉O₁₈, the energy gap of the effective two-level system has been reported at $\Delta E/k_{\mathrm{B}}=210.61\mathrm{K}$ [41].

The Gd samples show a linear relation for ${\chi }^{-1}\left(T\right)$ down to approximately $25\mathrm{K}$, when polarization of the moments starts to become significant; therefore, fitting was performed between $100\mathrm{K}$ and $350\mathrm{K}$. The resulting effective moments are close to the theoretical ${\mu }_{\mathrm{eff}}=7.94{\mu }_{\mathrm{B}}$. For Ba₃GdB₃O₉, the effective moment of $7.7{\mu }_{\mathrm{B}}$ and Weiss temperature of ${\Theta }_{\mathrm{W}}=-2.0\mathrm{K}$ are comparable to reports on the similar frustrated oxide magnet KBaGd(BO₃)₂ by Sanders et al. [34] (${\mu }_{\mathrm{eff}}=7.70{\mu }_{\mathrm{B}},{\Theta }_{\mathrm{W}}=-1.64\mathrm{K}$), as well as by Jesche et al. [20] (${\mu }_{\mathrm{eff}}=7.91{\mu }_{\mathrm{B}},{\Theta }_{\mathrm{W}}=-1.4\mathrm{K}$). Notably, we observed a slightly negative ${\Theta }_{\mathrm{W}}$ even in the high-temperature fit. For Ba₃GdB₉O₁₈, the resulting effective moment of $7.3{\mu }_{\mathrm{B}}$ is slightly lower than the theoretical value for a free-ion model. Even though the nearest-neighbor distances are quite large in this compound, the reduction in ${\mu }_{\mathrm{eff}}$ could hint at interactions between the moments. The Weiss temperature of ${\Theta }_{\mathrm{W}}=-1.9\mathrm{K}$ is quite close to zero, indicating a weak antiferromagnetic exchange. These results also are in good agreement with other Gd³⁺ frustrated oxides [24]. All compounds exhibit negative ${\Theta }_{\mathrm{W}}$ in the high-temperature fits, suggesting dominant antiferromagnetic interaction.

For low temperatures and low fields, the ${\chi }^{-1}\left(T\right)$ behavior of all samples is linear (Figure 4). The Yb compounds now indicate fluctuating $\mathrm{S}=1/2$ Kramers doublets with reduced ${\mu }_{\mathrm{eff}}$ as compared to high temperatures.

For Ba₃YbB₃O₉, the effective moment of $2.52{\mu }_{\mathrm{B}}$ is very close to reports by Gao et al. [36] (${\mu }_{\mathrm{eff}}=2.61{\mu }_{\mathrm{B}},{\Theta }_{\mathrm{W}}=-2.3\mathrm{K}$), as well as by Cho et al. [39] (${\mu }_{\mathrm{eff}}=2.56\left(1\right){\mu }_{\mathrm{B}}$, ${\Theta }_{\mathrm{W}}=-0.24\left(1\right)\mathrm{K}$). The obtained Weiss temperature ${\Theta }_{\mathrm{W}}=-0.23\mathrm{K}$ is close to the results reported by Cho et al. [39], indicating very weak or highly frustrated interactions.

For Ba₃YbB₉O₁₈, the effective moment of $2.49{\mu }_{\mathrm{B}}$ and Weiss temperature ${\Theta }_{\mathrm{W}}=-0.15\mathrm{K}$ are very close to data reported by Khatua et al. [40] (${\mu }_{\mathrm{eff}}=2.32{\mu }_{\mathrm{B}},{\Theta }_{\mathrm{W}}=-0.12\mathrm{K}$). Reports by Cho et al. [39] (${\mu }_{\mathrm{eff}}=2.31\left(1\right){\mu }_{\mathrm{B}},{\Theta }_{\mathrm{W}}=-0.077\left(2\right)\mathrm{K}$) suggest a lower ${\Theta }_{\mathrm{W}}$ but are based on measurements to $2\mathrm{K}$.

Both Gd samples continue to exhibit a high effective magnetic moment of $7.55{\mu }_{\mathrm{B}}$ and $7.28{\mu }_{\mathrm{B}}$ at low temperatures. This is expected, as the full $S=7/2$ multiplet contributes to the magnetic moment in Gd³⁺ ions [20,24,34].

${\Theta }_{\mathrm{W}}$ for all samples is increased in this temperature range but remains negative. This suggests antiferromagnetic behavior with a low ordering temperature $T_{\mathrm{N}}$. Both Ba₃XB₉O₁₈ samples show higher ${\Theta }_{\mathrm{W}}$ than their respective Ba₃XB₃O₉ counterparts. The increase in ${\Theta }_{\mathrm{W}}$ corresponds to a lower magnetic moment density and, thus, reduced magnetic interactions. All compounds show no signs of long-range order down to the lowest observed temperatures.

3.3. Specific Heat

Specific heat was measured on the powder-pressed samples with silver admixture in the PPMS. The helium-3 data was scaled to the helium-4 data, and the contributions of silver as well as the phonons were subtracted. The phonon contribution was determined by fitting a combined Debye–Einstein model to the high-temperature data:

$$c_{\mathrm{ph}}=3\gamma RN{k}_{\mathrm{B}}{\left({\displaystyle \frac{{\Theta }_{\mathrm{E}}}{T}}\right)}^2{\displaystyle \frac{\exp \left(\frac{{\Theta }_{\mathrm{E}}}{T}\right)}{{\left(\exp \left(\frac{{\Theta }_{\mathrm{E}}}{T}\right)-1\right)}^2}}+9(1-\gamma )RN{k}_{\mathrm{B}}{\left({\displaystyle \frac{T}{{\Theta }_{\mathrm{D}}}}\right)}^3{\int }_0^{{\displaystyle \frac{{\Theta }_{\mathrm{D}}}{T}}}{\displaystyle \frac{x^3}{\exp x-1}}dx$$

(4)

with $x={\Theta }_{\mathrm{D}}/T$ and the mixing factor $0\le \gamma \le 1$. The resulting parameters are listed in Table 2. These fits were solely created to obtain a description of the low-temperature phonon contribution to specific heat. Since the temperature range for fitting is very limited ($T\le 30\mathrm{K}$), the obtained values of the Debye temperature ${\Theta }_{\mathrm{D}}$, the Einstein temperature ${\Theta }_{\mathrm{E}}$, and the mixing parameter $\gamma$ have no intrinsic physical significance and should not be interpreted beyond their role in providing a convenient empirical description of the data. For low temperatures, the microcalorimeter measurements were extended below $600\mathrm{mK}$ by means of ADR warmup analysis. In this process, a pellet is allowed to warm up to the starting temperature $T_0$ after a magnetic cooldown under parasitic heat load. From this warming curve $T\left(t\right)$, the specific heat $c_p$ can be calculated by

$$c_p={\displaystyle \frac{m_{\mathrm{mol}}}{m}}{\displaystyle \frac{dQ}{dT}}={\displaystyle \frac{m_{\mathrm{mol}}}{m}}{\displaystyle \frac{dQ}{dt}}{\left({\displaystyle \frac{dT}{dt}}\right)}^{-1}.$$

(5)

It has previously been shown that, for the PPMS puck-based setup, the assumption of constant ${\dot{Q}}_{\mathrm{parasitic}}\left(T\right)$ holds for $20\mathrm{mK}<T<1.5\mathrm{K}$ [20,24,30]. The specific heat from the ADR measurements was scaled to the zero field data measured by microcalorimetry to determine the heat load.

Magnetic specific heat $c_{\mathrm{m}}$ for a system of non-interacting magnetic dipoles in an external field can be described as follows ([51] Equation (9.15b)):

$$c_{\mathrm{m}}={\displaystyle \frac{x^{2}R}{4}}{\sinh }^{-2}\left({\displaystyle \frac{x}{2}}\right)-{\displaystyle \frac{x^2}{4}}{(2J+1)}^2{\sinh }^{-2}\left({\displaystyle \frac{x}{2}}(2J+1)\right)\mathrm{with}x={\displaystyle \frac{{\mu }_{\mathrm{B}}g{H}_{\mathrm{eff}}}{k_{\mathrm{B}}T}}.$$

(6)

The contribution of internal fields and antiferromagnetic exchange was considered through an effective field $H_{\mathrm{eff}}$ at the position of each magnetic ion. This effective field was determined through fitting the specific heat peak. The values of $H_{\mathrm{eff}}$ were mostly lower than the applied field, which may be attributed to the antiferromagnetic exchange in the compounds, while a slightly enhanced effective field was used for Ba₃GdB₉O₁₈ at an applied field of $1\mathrm{T}$. The specific heat data (Figure 5) was fitted with Equation (6). Ba₃GdB₃O₉ shows a clear maximum in specific heat that is shifted to higher temperatures with increased field. In zero field, a broad peak is also found, with no indication of a long-range ordered magnetic phase transition.

In a finite field, the molar specific heat of Ba₃GdB₉O₁₈ behaves similarly to that of Ba₃GdB₃O₉, as expected, also exhibiting a broad peak that is shifted to higher T with increasing field. The zero-field specific heat shows a clear phase transition anomaly at $108\mathrm{mK}$, followed by a broader peak at $368\mathrm{mK}$. The lambda-shaped peak could indicate antiferromagnetic order, given the negative sign of the Curie–Weiss temperature. The ordering occurs at a temperature close to but lower than $|{\Theta }_{\mathrm{W}}|=0.13\mathrm{K}$, which can be attributed to frustration in the magnetic lattice. There is also a small anomaly at elevated temperatures, which coincides with the magnetic ordering of the foreign phase GdBO₃. This has been previously reported at $1.72\mathrm{K}$ [52], which matches our observations of a contribution at $1.7\mathrm{K}$.

The origin of the absence of a magnetic transition in Ba₃GdB₃O₉ is unclear, as this material crystallizes in a different structure with two inequivalent Gd sites and chain motives. It can also not be excluded that an ordering would occur at temperatures lower than the minimal temperature of the ADR experiment.

From the measured specific heat data, magnetic entropy $S_{\mathrm{m}}$ was calculated using

$$S_{\mathrm{m}}=S_0+\int dT{\displaystyle \frac{c_{\mathrm{m}}}{T}}.$$

(7)

The offset constant $S_0$ is necessary, as $c_{\mathrm{m}}$ is only available starting from finite $T_{\min }$. This offset was determined from magnetization data at $2\mathrm{K}$ by

$$S_{\mathrm{m}}={\mu }_0{\int }_{B_0}^{B_1}dH{\displaystyle \frac{\partial M}{\partial T}}.$$

(8)

The resulting magnetic entropy shows a clear saturation effect at the expected value of $R\ln 8=17.3{\mathrm{Jmol}}^{-1}{\mathrm{K}}^{-1}$. Ba₃GdB₃O₉ saturates at slightly lower temperatures for a given field when compared to Ba₃GdB₉O₁₈. The contribution to $S_{\mathrm{m}}$ of the lambda-like peak in $c_{\mathrm{m}}$ for Ba₃GdB₉O₁₈ is small when compared with the broader peak at higher temperatures. This clearly indicates that the ordered moment comprises only a low fraction of the total Gd moment.

3.4. Actual Refrigeration Performance

To ascertain the actual refrigeration performance of the compounds, the $15\mathrm{mm}$ pellets were individually mounted to the PPMS puck-based setup described in Section 2. This setup was isothermally magnetized to an applied field of $5\mathrm{T}$ at a starting temperature of $2\mathrm{K}$. After magnetization, the remaining helium atmosphere was evacuated from the sample chamber, establishing near-adiabatic conditions. Subsequently, the field was fully ramped down at a rate of $5\mathrm{mT}/\min$. The sample temperature was recorded during the parasitic warming back to the sample chamber temperature (Figure 6). From the warmup curve $T\left(t\right)$, the refrigerant capacity q was calculated by integrating the known parasitic heat flow. This represents the amount of heat that the refrigerant can accept when warming to a specified temperature:

$$q\left(T\right)={\int }_{t\left(T_{\min }\right)}^{t\left(T\right)}{\displaystyle \frac{dQ}{dt}}d{t}^{\prime }.$$

(9)

The median $\dot{Q}$ of all ADR runs was $63\left(8\right)\mathrm{nW}$. Figure 6 displays the warmup data for all studied materials.

Table 3. Comparison of values for the effective magnetic moment and Curie–Weiss temperature (from Figure 4); minimal ADR temperature $T_{\mathrm{ADR}}$ for a full demagnetization from $2\mathrm{K},5\mathrm{T}$, and subsequent warmup back to $2\mathrm{K}$; refrigerant capacity q for a warmup from $T_{\mathrm{ADR}}$ back to $2\mathrm{K}$; and full entropy density, calculated from the magnetic ion density, for the four different studied materials, together with results from the literature for KBaYb(BO₃)₂ and KBaGd(BO₃)₂.

Compound	${\mathit{\mu }}_{\mathbf{eff}}$	ΘW	TADR	q	ΔSm
Ba3YbB9O18	$2.49{\mu }_{\mathrm{B}}$	$-0.15\mathrm{K}$	$37.0\mathrm{mK}$	$0.84{\mathrm{mJcm}}^{-3}$	$24.0\mathrm{mJ}{\mathrm{K}}^{-1}{\mathrm{cm}}^{-3}$
Ba3YbB3O9	$2.52{\mu }_{\mathrm{B}}$	$-0.23\mathrm{K}$	$40.0\mathrm{mK}$	$1.78{\mathrm{mJcm}}^{-3}$	$43.6\mathrm{mJ}{\mathrm{K}}^{-1}{\mathrm{cm}}^{-3}$
KBaYb(BO3)2 [19,25]	$2.28{\mu }_{\mathrm{B}}$	$-0.06\mathrm{K}$	$40\mathrm{mK}$	-	$57.95\mathrm{mJ}{\mathrm{K}}^{-1}{\mathrm{cm}}^{-3}$
Ba3GdB9O18	$7.59{\mu }_{\mathrm{B}}$	$-0.13\mathrm{K}$	$94.3\mathrm{mK}$	$29.4{\mathrm{mJcm}}^{-3}$	$66.8\mathrm{mJ}{\mathrm{K}}^{-1}{\mathrm{cm}}^{-3}$
Ba3GdB3O9	$7.55{\mu }_{\mathrm{B}}$	$-0.19\mathrm{K}$	$119.2\mathrm{mK}$	$56.0{\mathrm{mJcm}}^{-3}$	$104\mathrm{mJ}{\mathrm{K}}^{-1}{\mathrm{cm}}^{-3}$
KBaGd(BO3)2 [20,25]	$7.55{\mu }_{\mathrm{B}}$	$-0.55\mathrm{K}$	$122\mathrm{mK}$	-	$139\mathrm{mJ}{\mathrm{K}}^{-1}{\mathrm{cm}}^{-3}$

provides an overview of the relevant parameters as well as a comparison to other refrigerants. The Yb compounds achieved the lowest ADR temperatures $T_{\mathrm{ADR}}$, with Ba₃YbB₃O₉ attaining $40.0\mathrm{mK}$ and Ba₃YbB₉O₁₈ attaining $37.0\mathrm{mK}$. This is comparable to values reported for other frustrated oxides like KBaYb(BO₃)₂ (Tokiwa et al., $40\mathrm{mK}$ [19]) or NaYbP₂O₇ (Arjun et al., $45\mathrm{mK}$ [21]). There is no indication for long-range magnetic order in the Yb compounds down to the minimum temperatures attained during ADR, corroborating previous measurements [37,38,44]. This suggests that the obtained minimal temperatures are limited by the non-perfect adiabaticity of the setup, and not yet by the vanishing entropy in zero field.

Ba₃YbB₃O₉ warms 60% slower than Ba₃YbB₉O₁₈, as expected, since the former has a 71% higher magnetic moment density and 82% higher magnetic entropy density compared to the latter. This is also reflected in the 120% enhanced refrigerant capacity q.

As expected from the fact that the Gd materials—compared to their Yb counterparts—feature a stronger magnetic interaction and, thus, reduced entropy at higher temperatures, they exhibit higher $T_{\mathrm{ADR}}$, with Ba₃GdB₃O₉ attaining $119\mathrm{mK}$ and Ba₃GdB₉O₁₈ attaining $94.3\mathrm{mK}$. While the difference in minimal temperature $T_{\mathrm{ADR}}$ is quite small for the two Yb compounds, at $\Delta T_{\mathrm{ADR}},\mathrm{Yb}=3.0\mathrm{mK}$, the difference in minimal temperature is increased between the Gd materials, at $\Delta T_{\mathrm{ADR},\mathrm{Gd}}=24.7\mathrm{mK}$. Ba₃GdB₃O₉ warms 60% slower to 2 K than Ba₃GdB₉O₁₈, compatible with the 56% higher entropy density. This is also reflected in the 105% higher refrigerant capacity q. All substances reached significantly lower temperatures $T_{\mathrm{ADR}}$ during the ADR than their respective $|{\Theta }_{\mathrm{W}}|$, with all except Ba₃GdB₉O₁₈ showing no signatures of long-range order. This suggests a possible impact of geometric frustration. However, more detailed study of the magnetic properties would be required to draw a clearer conclusion.

To compare the refrigeration performance of various materials, the change in the volumetric magnetic entropy density $\Delta S=S_{B=0}-S_{B_i}$ was calculated for an initial field of $5\mathrm{T}$ (Figure 7A). Likewise, the refrigerant capacity q was calculated according to Equation (9) for a demagnetization from $B_i=5\mathrm{T}$ (Figure 7B). This indicates the amount of heat that the refrigerant can absorb during warmup to a specified temperature—in this case, $2\mathrm{K}$.

In terms of entropy change (Figure 7A), the Yb (blue, green) and Gd (orange, yellow) compounds differ in magnitude as well as temperature distribution. As expected, the Gd-based substances offer significantly higher entropy lift per unit volume, with Ba₃YbB₃O₉ outperforming Ba₃YbB₉O₁₈ due to its higher moment density. However, $\Delta S\left(T\right)$ shows a peak for these substances; this can be interpreted as representing the optimal operating conditions. Meanwhile, the Yb compounds offer a far broader plateau in $\Delta S\left(T\right)$, albeit at a significantly reduced magnitude. The tendency for Yb-based refrigerants to offer optimal performance over a wider temperature range has been observed for multiple mK refrigerants [25].

In comparison to the previously studied borates KBaYb(BO₃)₂ and KBaGd(BO₃)₂, the performance of the new refrigerants is limited [19,20,25]. KBaYb(BO₃)₂ (Figure 7A, dashed line) offers higher $\Delta S$ than Ba₃YbB₃O₉ and Ba₃YbB₉O₁₈ over the entire temperature range. Ba₃GdB₃O₉ and Ba₃GdB₉O₁₈ outperform KBaGd(BO₃)₂ (Figure 7A, solid line) for very low temperatures $(T<0.14\mathrm{K}$ and $T<0.17\mathrm{K})$ but offer only limited entropy lift at higher temperatures. This can be explained by the higher magnetic moment density in the diborates when compared to the triborates studied here. Overall, enhanced magnetic moment density combined with strong frustration seems to be preferable to more diluted frustrated systems for refrigeration purposes.

4. Conclusions

Magnetic measurements, specific heat, and ADR results for Ba₃YbB₃O₉, Ba₃YbB₉O₁₈, Ba₃GdB₃O₉, and Ba₃GdB₉O₁₈ are presented in this study. All of the materials exhibit slightly negative ${\Theta }_{\mathrm{W}}$, indicating weak antiferromagnetic interactions, which are stronger in case of the Gd materials due to their larger moment size. Magnetization of the ytterbium samples adhered closely to the single-ion model, while the gadolinium systems necessitated a mean-field approach. The susceptibility data reveals paramagnetic behavior to $0.4\mathrm{K}$, underpinned by specific heat data clearly exhibiting a Schottky-type anomaly.

Major MCE was observed in all substances during a demagnetization from $2\mathrm{K},5\mathrm{T}$. Both Ba₃GdB₉O₁₈ and Ba₃GdB₃O₉ achieved low ADR temperatures for gadolinium systems, while Ba₃YbB₉O₁₈ and Ba₃YbB₃O₉ were competitive with other rare-earth oxide quantum magnets with regard to temperature. All compounds offered significantly lower entropy densities and refrigerant capacities than the benchmark borates $\mathrm{KBa}(\mathrm{Gd},\mathrm{Yb}){\left({\mathrm{BO}}_3\right)}_2$. For Ba₃GdB₉O₁₈, the formation of long-range magnetic order was observed by means of ADR warmup analysis. The specific heat exhibited a clear lambda-shaped phase transition anomaly in zero field, followed by a broad maximum at higher temperatures. Data in $1\mathrm{T}$ are well described by a $S=7/2$ Schottky anomaly. Long-range order was not detected in Ba₃GdB₃O₉, even though this compound has a higher magnetic moment density. The absence of magnetic order in Ba₃GdB₃O₉ is likely caused by frustration due to the trigonal lattice. It also has to be considered that this material has two nonequivalent magnetic sites.

Both Ba₃YbB₉O₁₈ and Ba₃YbB₃O₉ attain low temperatures during demagnetization when compared to other frustrated Yb-based quantum magnets. While they are competitive with regard to temperature, they lack in terms of refrigerant capacity and entropy density when compared to incumbents like KBaYb(BO₃)₂.