Frustrated magnet for adiabatic demagnetization cooling to milli-Kelvin temperatures

Generation of very low temperatures has been crucially important for applications and fundamental research, as low-temperature quantum coherence enables operation of quantum computers and formation of exotic quantum states, such as superfluidity and superconductivity. One of the major techniques to reach milli-Kelvin temperatures is adiabatic demagnetization refrigeration (ADR). This method uses almost non-interacting magnetic moments of paramagnetic salts where large distances suppress interactions between the magnetic ions. The large spatial separations are facilitated by water molecules, with a drawback of reduced stability of the material. Here, we show that an H$_2$O-free frustrated magnet KBaYb(BO$_3$)$_2$ can be ideal refrigerant for ADR, achieving at least 22\,mK upon demagnetization under adiabatic conditions. Compared to conventional refrigerants, KBaYb(BO$_3)_2$ does not degrade even under high temperatures and ultra-high vacuum conditions. Further, its frustrated magnetic network and structural randomness enable cooling to temperatures several times lower than the energy scale of magnetic interactions, which is the main limiting factor for the base temperature of conventional refrigerants.

ing, while conventional ADR is a single-shot technique. This makes the 3 He-4 He dilution refrigeration more commonly used than ADR. However, the situation may change thanks to recent developments of continuous ADR cooling [20,21] and availability of commercial continuous refrigerators based on ADR [22]. Therefore, ADR has the potential of becoming the main cooling technology already in near future, at least in the mK temperature range.
ADR uses magnetic moments of almost-ideal paramagnets with very weak magnetic interaction J . Because the interaction is weak, magnetic moments are easily aligned by external magnetic field, causing reduction of entropy (Figure 1(a,b)). Even at zero external magnetic field H = 0, magnetic moments of such almost-ideal paramagnets experience some small internal magnetic field produced by adjacent magnetic moments through magnetic interactions. This causes a tiny Zeeman splitting and magnetic order at the same energy scale (∼ ∆ 0 ∼ J ). Therefore, entropy, which is the driving force of ADR, decreases to zero below the temperature of T ∼ J , thus putting a limit on the end temperature T f ∼ J that can be reached via ADR with this material (Figure 1(b)). As indicated by a black horizontal arrow in Figure 1(b), entropy difference between H=0 and H =0 is the key for ADR. While a perfect paramagnet with zero magnetic interactions would be an ideal refrigerant, having maximum entropy at zero field down to zero temperature, there are always weak but finite interactions in real materials.
In commonly used paramagnetic salts, these interactions are reduced by large distances between the magnetic ions, which are separated by water molecules. However, abundance of water makes these salts prone to decomposition. They deliquesce in humid atmosphere and dehydrate in vacuum or upon even mild heating. Therefore, for repeated use without degradation, stable water-free materials with very weak magnetic interactions are desirable. Furthermore, ADR would be certainly beneficial for applications in ultra-high-vacuum (UHV) apparatus, especially in scanning tunneling microscopy and angle-resolved photoemission spectroscopy where the necessity of chamber baking at high temperature and high vacuum for reaching UHV makes the use of current ADR materials essentially impossible.
For many decades ever since P. Debye [6] and W.F. Giauque [7] independently proposed ADR, water-containing paramagnetic salts have been materials of choice for cooling in the mK range [5,23,24]. Here, we demonstrate the effective refrigeration with an H 2 O-free frustrated magnet KBaYb(BO 3 ) 2 that shows excellent ADR performance on par with conventional water-containing paramagnetic salts, and achieves an end temperature T f of at least 22 mK, starting from a temperature of 2 K at a magnetic field of 5 T. In this compound, Yb 3+ ions are responsible for magnetic properties and thus for the ADR. At sufficiently low temperatures, only the lowest Kramers doublet of Yb 3+ is occupied [25].
The crystal structure of KBaYb(BO 3 ) 2 depicted in Figs. 1(c,d) reveals isolated YbO 6 octahedra surrounded by six BO 3 triangles. These negatively charged units are interspersed by the positively charged K + and Ba 2+ ions that occupy the single crystallographic site. The shorter Yb-Yb distances of 5.41Å within the ab plane connect magnetic Yb-ions into a triangular lattice, the feature that already caused interest in this [26,27] and isostructural [28][29][30] rare-earth borates from the perspective of frustrated magnetism [25]. The interplane Yb-Yb distances are only slightly longer (6.64Å) and cause additional frustration, because adjacent triangular layers are shifted relative to each other, as shown by solid lines in Fig. 1 connecting Yb-ions from the neighboring layers. Such geometrical frustration may shift the magnetic ordering transition toward lower temperatures or even suppress the magnetic order completely. The distribution of K + and Ba 2+ results in uneven electric fields acting on the magnetic Yb 3+ ions and may lead to random exchange couplings [25] that are also detrimental for the magnetic ordering. Sizable remaining entropy due to the suppression of the magnetic long-range order facilitates the cooling even below ∼ J (Figure 1(b)). Moreover, the presence of soft modes in frustrated magnets amplifies magnetocaloric effect and further lowers the end temperature of the cooling process [17]. The enhanced ADR performance was indeed observed in frustrated magnets, but temperatures in the mK-range could not be reached because of the relatively strong magnetic interactions [13,31]. KBaYb(BO 3 ) 2 is a novel type of an H 2 O-free refrigerant combining several properties that are beneficial for reaching lowest temperatures: large separation between the magnetic ions, geometrical frustration of residual magnetic interactions, as well as their potential randomness.

II. RESULTS
Magnetization measured down to 0.5 K shows the typical paramagnetic behavior (Figure 2). Field-dependent magnetization is perfectly described by the Brillouin function augmented by the van Vleck term χ 0 H that originates from crystal-field excitations. Upon subtracting this van Vleck contribution χ 0 from temperature-dependent susceptibility, one recovers linear behavior in 1/(χ − χ 0 ) corresponding to the Curie-Weiss law. The fit returns the effective magnetic moment µ eff = 2.28 µ B and Curie-Weiss temperature θ = −60±2 mK.
We note that the reduced effective magnetic moment from free Yb 3+ is due to crystalline electric field effect. We show in Supplementary information that at high temperatures µ eff recovers the free-ion value for Yb 3+ . The small value of θ indicates that temperatures on the order of 10 mK would be needed to explore cooperative magnetism and frustrated magnetic behavior of KBaYb(BO 3 ) 2 , making this material a quite challenging system to study in comparison with YbMgGaO 4 and Yb 3+ delafossites having Curie-Weiss temperatures of several Kelvin [25]. On the other hand, KBaYb(BO 3 ) 2 itself can be used to reach these extremely low temperatures, as we show below.
First hints for the ADR potential of KBaYb(BO 3 ) 2 are found in the magnetic specific heat data C m (T ) that were obtained via subtracting the lattice contribution given by the non-magnetic reference compound KBaLu(BO 3 ) 2 ( Figure 3a). In zero field, specific heat increases toward lower temperatures and indicates the accumulation of entropy associated with the Kramers doublet of Yb 3+ . Magnetic field splits this doublet into the j z = + 1 2 and j z = − 1 2 levels and causes a Schottky anomaly that shifts toward higher temperatures upon increasing the field. A weak anomaly is seen in the zero-field data around 400 mK, but its nature is probably extrinsic, as it contains only a tiny amount of the magnetic entropy (See Supplementary Information). Note that a similar anomaly has been reported for isostructural NaBaYb(BO 3 ) 2 [28].
By integrating C m (T )/T , we calculate magnetic entropy ( Figure 3b) and observe that already at 0.5 T the full entropy of R ln 2 associated with the lowest Kramers doublet of Yb 3+ can be recovered above 80 mK. Higher fields shift the entropy toward higher temperatures, so that at 5 T only 10% of R ln 2 is released below 2 K. Therefore, by adiabatic demagnetization of KBaYb(BO 3 ) 2 starting from 2 K and 5 T, one may expect to reach temperatures as low as 30 mK, as shown by arrows in Figure 3 We test this prediction in the actual cooling experiment that was performed with a 4.02g pellet containing equal weights of KBaYb(BO 3 ) 2 and silver powder (10 to 50 µm particle size). Silver improves thermal conductivity within the pellet of insulating KBaYb(BO 3 ) 2 . To improve thermal conductance between the particles, we sintered the pressed pellet at 600 • C.
The pellet was mounted on a sapphire-plate sample holder, which is held on a plastic frame by four 50 µm thin fishing wires. A resistive RuO 2 thermometer is attached to the pellet with glue. To suppress the thermal conduction from the heat bath, thin superconducting filaments are used as wire leads on the resistive thermometer for temperature measurements. This setup was attached to the cold bath in a vacuum chamber (with pressure of ∼ 10 −6 mbar) of the 3 He-4 He dilution refrigerator.
In order to minimize the heat flow into the sample from the bath, we used a feedback control of the bath temperature (T b ) to follow the sample temperature (T s ). Because the heat flow isQ = κ ∆T where ∆T = T b − T s and κ is the thermal conductance between the bath and the pellet, minimizing ∆T suppressesQ significantly. With the bath temperature of 2 K, the pellet was slowly cooled to 2 K at µ 0 H = 5 T. Then we swept magnetic field to zero with the 0.25 T/min rate while giving the feedback control to the bath. As shown in Figure 4, the pellet is quickly cooled down to less than 22 mK, which is the lowest limit of the thermometer calibration. By extrapolating this calibration to lower temperatures, we find the end temperature of about 16 mK. A small spike appearing around T = 80 mK (µ 0 H = 0.1 T) is probably due to magnetic flux pinning of the superconducting magnet.
After reaching the lowest temperature, the sample warms up slowly due to the heat flow from the surroundings (the bath and radiation). Note that T b deviates from T s below 0.4 K. This is because the cooling rate of T s is too fast for the bath to follow. The large temperature difference indicates a weak thermal conductance between the sample and the bath. Because of the difference there is finite heat flow into the sample. Therefore, under ideal adiabatic conditions the end temperature would be even lower than the extrapolated 16 mK.
We note that there is no visible anomaly in sample temperatures other than the spike at 80 mK. A phase transition of second (or first) order would cause a kink (or a local plateau).
The absence of such an anomaly confirms the absence of magnetic ordering in KBaYb(BO 3 ) 2 at least down to 22 mK and most likely down to about 16 mK, the temperature several times lower than |θ| = 60 mK. This behavior is in line with the geometrical frustration expected in An immediate practical application of an ADR material would be its integration into commercial PPMS to achieve temperatures below 1.8 K without using 3 He. To this end, we constructed a stage where the sample pellet is placed on a tall plastic straw and thermally isolated from the heat bath ( Figure 5). A resistive RuO 2 thermometer is glued on the pellet and connected with thin resistive manganin wires to suppress heat flow. Then the setup is shielded by a metallic cap to reduce thermal radiation from the surroundings. Using the "high-vacuum mode" of the PPMS (pressure below 10 −4 mbar), we set the bath temperature to 2 K at a magnetic field of 5 T. Then the pellet temperature slowly approaches 2 K by a weak thermal link through the manganin wires and plastic straw. After reaching 2 K, the field is swept to zero with a rate of 0.15 T/min. The pellet reaches temperatures below 40 mK, with a spike in the sample temperature around 0.1 K due to flux pinning of the superconducting magnet. The pellet warms up back to 2 K in about 50 minutes by the weak heat conduction. This warming rate is slow enough for data collection of transport properties, such as electrical or Hall resistivity. Although the difference in setup has to be taken into account, the performance of our stage is better than the commercial ADR option for PPMS, which utilizes the paramagnetic salt, CPA, and reaches 100 mK only [32].

III. DISCUSSION
We conclude that KBaYb(BO 3 ) 2 can be highly efficient in ADR while containing no water molecules and showing excellent stability upon both exposure to air and heating. Key parameters of the ADR materials are compared in Table I. Figure 3 illustrates that even with the moderate field of 5 T almost all entropy R ln 2 of the lowest Kramers doublet is used for refrigeration. The same is true for conventional paramagnetic salts [5]. Therefore, we list full entropies of the ground state as S GS . From the absolute values of S GS , KBaYb(BO 3 ) 2 is clearly competitive with MAS and FAA as "high-temperature" paramagnetic salts that develop magnetic order below T m of 170 and 30 mK, respectively, and thus cannot cool below these temperatures. The lower-temperature ADR comes at the cost of the higher dilution of the magnetic ions and, therefore, lower S GS in CPA and especially CMN. It is in fact natural that high density of the magnetic moments and low magnetic ordering temperature are unlikely to coexist. Nevertheless, KBaYb(BO 3 ) 2 combines these mutually exclusive properties by virtue of its magnetic frustration and structural randomness that both help to suppress T m well below the energy scale of magnetic interactions given by |θ| = 60 mK. The material also shows an outstanding density of magnetic ions, 6.7 nm −3 , which is much higher than in any of the paramagnetic salts and would allow the design of more compact ADR apparatus. Even higher density has been reported for YbPt 2 Sn and Yb 3 Ga 5 O 12 that, however, develop magnetic short-range order below 0.25 K and long-range order below 54 mK, respectively, which limits their end temperatures [12,13].
In summary, we demonstrated the H 2 O-free refrigerant KBaYb(BO 3 ) 2 with the high entropy/volume ratio and excellent ADR performance of achieving at least 22 mK and possibly temperatures below 16 mK upon demagnetization from µ 0 H = 5 T at 2 K. The absence of water molecules in this new refrigerant brings several additional advantages over conventional and commercially used paramagnetic salts. Delicate treatment of these materials is required to avoid degradation and ensure a good thermal contact by incorporating wires to the refrigerant pill, which makes production very advanced [21]. With KBaYb(BO 3 ) 2 , improving thermal contact is much easier because rough treatments are possible. In the present study, for example, we ground the sample into fine powder and sintered the pressed pellet of a mixture with silver powder at high temperatures. Because of their low stability, water-containing paramagnetic salts have to be sealed air-tight and cannot be heated.
The proposed material KBaYb(BO 3 ) 2 is free from all these problems. The chemical stability combined with the excellent cooling performance render KBaYb(BO 3 ) 2 ideal for ADR applications.

IV. METHODS
Powder samples of KBaYb(BO 3 ) 2 were prepared by a solid-state reaction of K 2 CO 3 , BaCO 3 , Yb 2 O 3 , and H 3 BO 3 taken in stoichiometric amounts, with the 6% excess of H 3 BO 3 and 2% excess of K 2 CO 3 and BaCO 3 due to their marginal loss upon heating. The reactants were carefully ground, placed into alumina crucibles, and kept at 200 • C for 6 hours to remove absorbed water. They were subsequently annealed at 700 • C for 24 hours, re-ground, and re-annealed at 900 • C for another 24 hours. All annealing steps were performed in air.
We confirmed the desired crystal structure by X-ray diffraction with a small amount of 0   1. (a) Adiabatic demagnetization processes of a conventional paramagnet (left) and a frustrated magnet with impeded magnetic order (right). Green ovals represent singlet pairings in a short-range correlated but long-range disordered state, such as a spin liquid caused by magnetic frustration [25,33] or a random-singlet state caused by structural disorder [34]. (b) Entropy curves at zero and a finite magnetic field as a function of temperature for conventional paramagnets (black) and a system with suppressed magnetic order (red). While the former orders below some temperature T m around ∆ 0 or J , the latter remains disordered down to much lower temperatures.
For simplicity, a Schottky-type increase in the entropy is assumed. Note however that the entropy increase may be much slower in a spin liquid because of its broad energy spectrum. ∆ 0 is an energy level splitting at zero external field because of the internal field caused by magnetic interactions.
Arrows with red and black dotted lines represent cooling processes for the conventional paramagnets (black) and frustrated magnets (red).

II. MAGNETIC SUSCEPTIBILITY
Inverse magnetic susceptibility of KBaYb(BO 3 ) 2 , measured at 1 T, as a function of temperature is shown in Fig. S2. The obtained effective magnetic moment from fitting the data at high temperatures, µ eff =4.56µ B , is in good agreement with 4.54 µ B for free Yb 3+ ion.

III. CONTRIBUTION OF THE IMPURITY PHASE
A weak anomaly is visible in our zero-field specific heat data around 0.4 K. Here, we estimate magnetic entropy associated with this anomaly. We assume a smooth curve that connects the data points below and above the anomaly (see the gray line in Fig. S3a). The difference between the experimental data and this curve shows a λ-type anomaly in ∆C/T typical of a second-order phase transition (Fig. S3b) lower limit, an extrapolation has to be used, as shown in Figure S4. Since ln(R−R 0 ) ∼ T −1/4 , with the resistance at room temperature R 0 , is expected for variable-range hopping in semiconductors, we use a linear fit in Fig. 4(left) for the extrapolation. This corresponds to the final temperature not exceeding 16 mK.