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Frustrated magnet for adiabatic demagnetization cooling to milli-Kelvin temperatures

Abstract

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. This method uses almost non-interacting magnetic moments of paramagnetic salts where large distances suppress interactions between the moments. The large spatial separations are facilitated by water molecules, with a drawback of reduced stability of the material. Here, we show that the water-free frustrated magnet KBaYb(BO3)2 can be ideal for refrigeration, achieving at least 22 mK. Compared to conventional refrigerants, KBaYb(BO3)2 does not degrade even under high temperatures and ultra-high vacuum. Further, its magnetic frustration 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.

Introduction

Suppression of thermal fluctuations by lowering temperature gives access to intricate and potentially usable quantum effects. Major discoveries, such as the quantum Hall effect and superfluidity and superconductivity, have been made by explorations of matter close to absolute zero1,2. Recently, the development of quantum computers and sensors for dark matter detection rendered low-temperature refrigeration an important technological challenge3,4. One of the viable methods for reaching the milli-K (mK) temperature range is adiabatic demagnetization refrigeration (ADR) using paramagnetic salts5,6,7. Its main advantages over the currently dominant technique, 3He–4He dilution refrigeration, are the simple construction of a cooling device and its operation without the usage of expensive 3He. The recent crisis of 3He, which was caused by the increased demand due to the construction of neutron detectors for defense against nuclear terrorism, raised serious concerns about the strong dependence of the current technology on such a scarcely available and ever more expensive gas8,9,10. This triggered renewed interest in ADR, as well as interesting proposals of completely new types of ADR materials11,12,13,14,15,16,17,18,19.

The only advantage of 3He–4He dilution refrigeration is its capability of continuous cooling while conventional ADR is a single-shot technique. This makes the 3He–4He dilution refrigeration more commonly used than ADR. However, the situation may change thanks to recent developments of continuous ADR cooling20,21 and the availability of commercial continuous refrigerators based on ADR22. 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 \({\mathcal{J}}\). Because the interaction is weak, magnetic moments are easily aligned by the external magnetic field, causing a reduction of entropy (Fig. 1a, 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 (\(\sim\!{{{\Delta }}}_{0} \sim {\mathcal{J}}\)). Therefore, entropy, which is the driving force of ADR, decreases to zero below the temperature of \(T \sim {\mathcal{J}}\), thus putting a limit on the end temperature \({T}_{{\rm{f}}} \sim {\mathcal{J}}\) that can be reached via ADR with this material (Fig. 1b). As indicated by a black horizontal arrow in Fig. 1b, the 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 fields down to zero temperature, there are always weak but finite interactions in real materials.

Fig. 1: Adiabatic demagnetization refrigeration.
figure1

a Cooling process 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 frustration25,33 or a random-singlet state caused by structural disorder34. 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 Tm around Δ0, or \({\mathcal{J}}\), the latter remains disordered down to much lower temperatures where Δ0 is an energy level splitting at zero external fields because of the internal field caused by magnetic interaction, \({\mathcal{J}}\). For simplicity, a Schottky-type increase in entropy is assumed. Note, however, that the entropy increase may be much slower in a spin liquid because of its broad energy spectrum. Arrows with red and black dotted lines represent cooling processes for the conventional paramagnets (black) and frustrated magnets (red). c Crystal structure of KBaYb(BO3)2 with triangular layers of the Yb3+ ions. Black solid lines highlight the relative displacement of the adjacent triangular layers. d YbO6 octahedron surrounded by the BO3 triangles.

For many decades ever since Debye6 and Giauque7 independently proposed ADR, water-containing paramagnetic salts have been materials of choice for cooling in the mK range5,23,24. In these materials, the interactions are reduced by large distances between the magnetic ions, which are separated by water molecules. However, an abundance of water makes these salts prone to decomposition. They deliquesce in a 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. It is, therefore, desirable to find suitable ADR materials without water molecules.

One promising candidate for H2O-free refrigerant is KBaYb(BO3)2 with magnetic Yb3+ ions. At sufficiently low temperatures, only the lowest Kramers doublet of Yb3+ is occupied in this material25. In its crystal structure depicted in Fig. 1c, d, YbO6 octahedra with Yb3+ magnetic moments are separated spatially within the ab-plane by BO3 triangles. These negatively charged layers are interspersed along the c-axis by the positively charged K+ and Ba2+ ions that occupy the single crystallographic site. The shorter Yb–Yb distances of 5.41 Å within the ab plane connect magnetic Yb-ions into a triangular lattice, the feature that already caused interest in this26,27 and isostructural28,29,30 rare-earth borates from the perspective of frustrated magnetism25. The interplane Yb–Yb distances are only slightly longer (6.64 Å) and cause additional frustration because adjacent triangular layers are shifted relative to each other, as shown by solid lines in Fig. 1c. Such geometrical frustration may shift the magnetic ordering transition toward lower temperatures or even suppress the magnetic order completely. The distribution of K+ and Ba2+ results in uneven electric fields acting on the magnetic Yb3+ ions and may lead to random exchange couplings25 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 \(\sim {\mathcal{J}}\) (Fig. 1b). Moreover, the presence of soft modes in frustrated magnets amplifies the magnetocaloric effect and further lowers the end temperature of the cooling process17.

Here, we demonstrate the effective refrigeration with the frustrated magnet KBaYb(BO3)2 that shows excellent ADR performance on par with conventional water-containing paramagnetic salts and achieves an end temperature Tf of at least 22 mK, starting from a temperature of 2 K at a magnetic field of 5 T. The H2O-free refrigerant combines several properties that are beneficial for reaching lowest temperatures: the large separation between the magnetic ions, geometrical frustration of residual magnetic interactions, as well as their potential randomness.

Results

Magnetic properties

Magnetization measured down to 0.5 K shows the typical paramagnetic behavior (Fig. 2). Field-dependent magnetization is perfectly described by the Brillouin function augmented by the van Vleck term χ0H that originates from crystal-field excitations. Upon subtracting this van Vleck contribution χ0 from temperature-dependent susceptibility, one recovers linear behavior in 1/(χ − χ0) corresponding to the Curie–Weiss law. The fit returns the effective magnetic moment μeff = 2.28 μB and Curie–Weiss temperature θ = −60 ± 2 mK. We note that the reduced effective magnetic moment from free Yb3+ is due to crystalline electric field effect. We show in Supplementary Fig. 1 that at high temperatures μeff recovers the free-ion value for Yb3+. 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(BO3)2, making this material a quite challenging system to study in comparison with YbMgGaO4 and Yb3+ delafossites having Curie–Weiss temperatures of several Kelvin25. On the other hand, KBaYb(BO3)2 itself can be used to reach these extremely low temperatures, as we show below.

Fig. 2: Magnetic properties of KBaYb(BO3)2.
figure2

The field-dependent magnetization of KBaYb(BO3)2 at 0.5 K. Solid line is a fit to the data with gJBJ(x) + χ0H where g is the g-factor, \(J=\frac{1}{2}\) for the lowest Kramers doublet of Yb3+, and BJ is the Brillouin function for \(J=\frac{1}{2}\) with x = (gμBJμ0H) = (kBT). The second term is due to the van Vleck susceptibility χ0. The fit returns g = 2.54 and χ0/μ0 = 0.0111 μBT−1. The inset shows inverse magnetic susceptibility, 1/(χ − χ0) where χ = μ0H/M, measured at 0.1 T. Solid line is a linear fit in the 0.4 K ≤ T ≤ 2.0 K range, resulting in the effective magnetic moment μeff = 2.28 μB and Curie-Weiss temperature θ = − 60 ± 2 mK.

Specific heat

First hints for the ADR potential of KBaYb(BO3)2 are found in the magnetic specific heat data Cm(T) that were obtained via subtracting the lattice contribution given by the nonmagnetic reference compound KBaLu(BO3)2 (Fig. 3a). In zero field, specific heat increases toward lower temperatures and indicates the accumulation of entropy associated with the Kramers doublet of Yb3+. Magnetic field splits this doublet into the \({j}_{{\rm{z}}}=+\frac{1}{2}\) and \({j}_{{\rm{z}}}=-\frac{1}{2}\) 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 Fig. 2). Note that a similar anomaly has been reported for isostructural NaBaYb\({({{\rm{BO}}}_{3})}_{2}\)28.

Fig. 3: Estimation of cooling performance.
figure3

a Low-temperature magnetic heat capacity, Cm of KBaYb(BO3)2 at several external magnetic fields. Heat capacity of a non-magnetic reference KBaLu(BO3)2 was subtracted from the raw data as a phonon contribution. b Magnetic entropy Sm calculated by integrating Cm/T. In zero field, the entropy is vertically shifted to match Rln2 expected for the Kramers doublet. Two arrows show the adiabatic demagnetization refrigeration process, starting from T = 2 K at μ0H = 5 T.

By integrating Cm(T)/T, we calculate magnetic entropy (Fig. 3b) and observe that already at 0.5 T the full entropy of \(R{\mathrm{ln}}\,2\) associated with the lowest Kramers doublet of Yb3+ can be recovered above 80 mK. Higher fields shift the entropy toward higher temperatures, so that at 5 T only 10% of \(R{\mathrm{ln}}\,2\) is released below 2 K. Therefore, by adiabatic demagnetization of KBaYb(BO3)2 starting from 2 K and 5 T, one may expect to reach temperatures as low as 30 mK, as shown by arrows in Fig. 3b.

Refrigeration test under the near-adiabatic condition

We test this prediction in the actual cooling experiment that was performed with a 4.02 g pellet containing equal weights of KBaYb(BO3)2 and silver powder (10–50 μm particle size). Silver improves thermal conductivity within the pellet of insulating KBaYb(BO3)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 RuO2 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 the pressure of ~10−6 mbar) of the 3He–4He dilution refrigerator.

In order to minimize the heat flow into the sample from the bath, we used a feedback control of the bath temperature (Tb) to follow the sample temperature (Ts). Because the heat flow is \(\dot{Q}=\kappa \ {{\Delta }}T\) where ΔT = Tb − Ts and κ is the thermal conductance between the bath and the pellet, minimizing ΔT suppresses \(\dot{Q}\) significantly. With the bath temperature of 2 K, the pellet was slowly cooled to 2 K at μ0H = 5 T. Then we swept the magnetic field to zero with the 0.25 T min−1 rate while giving the feedback control to the bath. As shown in Fig. 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 (see Supplementary Fig. 3), we find the end temperature of about 16 mK. A small spike appearing around T = 80 mK (μ0H = 0.1 T) is probably due to the 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 Tb deviates from Ts below 0.4 K. This is because the cooling rate of Ts 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.

Fig. 4: Refrigeration test under nearly adiabatic conditions.
figure4

The solid black, solid red, and dotted gray lines are sample temperature (left axis), external magnetic field (right axis), and bath temperature (left axis), respectively. The sample pellet has weak thermal connection to the bath. The bath temperature is controlled to follow the sample temperature to minimize the heat flow. Below about 0.4 K, the bath cannot follow the sample temperature any more because the cooling rate is too fast to follow. The sample holder is in a vacuum chamber with a pressure of ~10−6 mbar. The end temperature is 16 mK. Note, however, that the thermometer for this measurement is calibrated down to 22 mK only. Temperature below this limit is obtained by extrapolating the calibration. (See Method and Supplementary Fig. 3).

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(BO3)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 KBaYb(BO3)2. Assuming \(\theta =-\frac{3}{2}{\mathcal{J}}\) in a nearest-neighbor triangular antiferromagnet and \({T}_{{\rm{N}}}/{\mathcal{J}}\simeq 0.2\) reported for Co-based frustrated materials25, we estimate \({\mathcal{J}}\simeq 40\) mK and TN 8 mK, which would be comparable with the end temperature of 16 mK achieved in our cooling experiment.

Actual refrigeration performance using PPMS

An immediate practical application of an ADR material would be its integration into a commercial physical property measurement system (PPMS) to achieve temperatures below 1.8 K without using 3He. 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 (Fig. 5). A resistive RuO2 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−1. 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 min 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, chromium potassium alum (CPA), and reaches 100 mK only31.

Fig. 5: Cooling performance in the commercial physical property measurement system.
figure5

The same sample as in Fig. 4 has been used. It is slowly cooled to T = 2K at μ0H = 5 T through the weak thermal link to the PPMS puck kept at 2 K. Magnetic field is swept from 5 to 0 T with a rate of 0.15 T min−1. The inset shows a photo of the setup with a two-step thermal insulation from the puck (bath). The brass stage is fixed to the puck with two plastic screws having Teflon spacers. The sample pellet is further insulated from the brass stage by inserting a plastic straw. To reduce thermal conductance, 50 μm-diameter 30 cm-long resistive wires (manganin) are used for temperature measurement by the same resistive thermometer as in the test shown in Fig. 4. This setup is shielded from radiation by a standard metallic cap of PPMS.

Discussion

We conclude that KBaYb(BO3)2 can be highly efficient in ADR while containing no water molecules and showing excellent stability upon both exposures to air and heating. Key parameters of the ADR materials are compared in Table 1. Figure 3 illustrates that even with the moderate field of 5 T almost all entropy \(R{\mathrm{ln}}\,2\) of the lowest Kramers doublet is used for refrigeration. The same is true for conventional paramagnetic salts5. Therefore, we list full entropies of the ground state as SGS. From the absolute values of SGS divided by volume, KBaYb(BO3)2 is clearly competitive with MAS and FAA as “high-temperature” paramagnetic salts that develop magnetic order below Tm 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 SGS/Vol. in CPA and especially CMN. It is in fact natural that the high density of the magnetic moments and low magnetic ordering temperature are unlikely to coexist. Nevertheless, KBaYb(BO3)2 combines these mutually exclusive properties by virtue of its magnetic frustration and structural randomness that both help to suppress Tm 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 YbPt2Sn and Yb3Ga5O12 that, however, develop magnetic short-range order below 0.25 K and long-range order below 54 mK, respectively, which limits their end temperatures12,13.

Table 1 Parameters of adiabatic demagnetization refrigeration materials.

In summary, we demonstrated the H2O-free refrigerant KBaYb(BO3)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 μ0H = 5 T at 2 K. The absence of water molecules in this refrigerant brings several additional advantages over conventional and commercially used paramagnetic salts. Delicate treatment of these materials is required to avoid degradation and ensure good thermal contact by incorporating wires to the refrigerant pill, which makes production very advanced21. With KBaYb(BO3)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(BO3)2 is free from all these problems. The chemical stability combined with the excellent cooling performance renders KBaYb(BO3)2 ideal for ADR applications.

Methods

Sample preparation and characterization

Powder samples of KBaYb(BO3)2 were prepared by a solid-state reaction of K2CO3, BaCO3, Yb2O3, and H3BO3 taken in stoichiometric amounts, with the 6% excess of H3BO3 and 2% excess of K2CO3 and BaCO3. The reactants were carefully ground, placed into alumina crucibles, and kept at 200 °C for 6 h to remove absorbed water. They were subsequently annealed at 700 °C for 24 h, re-ground, and re-annealed at 900 °C for another 24 h. All annealing steps were performed in the air. Sintered powders of KBaYb(BO3)2 were ground and pressed into small pellets with the mass of 18.4 and 2.69 mg used for measurements of magnetic susceptibility and specific heat, respectively.

X-ray powder diffraction pattern was recorded at room temperature using the Empyrean diffractometer from PANalytical (CuKα radiation, reflection mode). Rietveld refinement (Supplementary Fig. 4) confirmed the formation of KBaYb(BO3)2 with the lattice parameters a = 5.41120(2) Å, c = 17.5926(1) Å and atomic positions shown in Supplementary Table 1, in good agreement with the earlier report26. Several very weak peaks of Yb2O3 were detected, corresponding to 0.75(1) wt% of the impurity phase.

Considering that only 0.75(1) wt% of the Yb2O3 impurity is contained in the sample, and the magnetic ordering transition of Yb2O3 is at 2.3 K32, we infer that the anomaly should have a different origin. On the other hand, the small amount of entropy suggests that it is unrelated to KBaYb(BO3)2 itself. The most plausible explanation would be the presence of an amorphous or weakly crystalline impurity that is not seen in X-ray diffraction. Note also that a similar anomaly has been reported for the isostructural compound NaBaYb\({({{\rm{BO}}}_{3})}_{2}\)28.

Measurements and tests at low temperatures

Magnetic susceptibility was measured by an MPMS SQUID magnetometer from Quantum Design equipped with a 3He-refrigerator. Measurements of specific heat in the temperature range from 12 K down to 400 mK were performed by using a Quantum Design, PPMS, equipped with a 3He-refrigerator insert. Specific heat measurements of quasi-adiabatic heat pulse method at the lowest temperature down to 50 mK and the tests for ADR with this material have been performed, using a 3He–4He dilution refrigerator and PPMS.

Extrapolation of thermometer calibration

A ruthenium oxide thermometer is used for the tests of ADR with KBaYb(BO3)2. It is calibrated down to 22 mK. For temperatures below this lower limit, we have used an extrapolation of the calibration to lower temperatures. Supplementary Fig. 3 shows the extrapolation. Since T−1/4 for ln(R − R0) where R0 is the resistance at room temperature, is expected for variable-range hopping in semiconductors, we use a linear fit in Supplementary Fig. 3(left) to the low-temperature part of the data for extrapolation.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

We thank Andreas Honecker for fruitful discussions and for organizing a very useful and informative workshop on “New perspectives for low-temperature refrigeration with advanced magnetocaloric materials” in 2018. The work in Augsburg was supported by the German Research Foundation (DFG) via Project no. 107745057 (TRR80) and by the Federal Ministry for Education and Research through the Sofja Kovalevskaya Award of Alexander von Humboldt Foundation (AAT). The work in Tokai, Japan was supported by JSPS KAKENHI Grant nos. 20K22338 and 20KK0061.

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Y.T., A.A.T. and P.G. conceived and designed the study. Y.T. and S.B. performed the specific heat measurements, the tests using dilution refrigerator and PPMS. K.K. and A.J. performed the magnetic susceptibility measurements. K.K. and A.J. prepared and characterized the samples. Y.T., S.B., A.A.T., and P.G. discussed the results. Y.T. and A.A.T. prepared the paper.

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Correspondence to Yoshifumi Tokiwa.

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A utility model which covers the fundamental aspects of the publication has been filed in Germany by the University of Augsburg. It is registered under file number 20 2020 002 079. Inventors are Philipp Gegenwart, Alexander A. Tsirlin, and Sebastian Bachus. The remaining authors declare no competing interests.

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Tokiwa, Y., Bachus, S., Kavita, K. et al. Frustrated magnet for adiabatic demagnetization cooling to milli-Kelvin temperatures. Commun Mater 2, 42 (2021). https://doi.org/10.1038/s43246-021-00142-1

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