Research ArticleCONDENSED MATTER PHYSICS

Super-heavy electron material as metallic refrigerant for adiabatic demagnetization cooling

See allHide authors and affiliations

Science Advances  09 Sep 2016:
Vol. 2, no. 9, e1600835
DOI: 10.1126/sciadv.1600835

Abstract

Low-temperature refrigeration is of crucial importance in fundamental research of condensed matter physics, because the investigations of fascinating quantum phenomena, such as superconductivity, superfluidity, and quantum criticality, often require refrigeration down to very low temperatures. Currently, cryogenic refrigerators with 3He gas are widely used for cooling below 1 K. However, usage of the gas has been increasingly difficult because of the current worldwide shortage. Therefore, it is important to consider alternative methods of refrigeration. We show that a new type of refrigerant, the super-heavy electron metal YbCo2Zn20, can be used for adiabatic demagnetization refrigeration, which does not require 3He gas. This method has a number of advantages, including much better metallic thermal conductivity compared to the conventional insulating refrigerants. We also demonstrate that the cooling performance is optimized in Yb1−xScxCo2Zn20 by partial Sc substitution, with x ~ 0.19. The substitution induces chemical pressure that drives the materials to a zero-field quantum critical point. This leads to an additional enhancement of the magnetocaloric effect in low fields and low temperatures, enabling final temperatures well below 100 mK. This performance has, up to now, been restricted to insulators. For nearly a century, the same principle of using local magnetic moments has been applied for adiabatic demagnetization cooling. This study opens new possibilities of using itinerant magnetic moments for cryogen-free refrigeration.

Keywords
  • Adiabatic demagnetization refrigeration
  • heavy fermion
  • quantum critical point

INTRODUCTION

There have been various reports on the recent 3He crisis due to the increasing imbalance of demand and supply (13), which affects a variety of applications, including medical, military, and scientific usages. Demand is rapidly expanded because of the increasing use from various applications, including neutron detectors in homeland security, whereas supply is limited because the gas is produced only through tritium decay in nuclear weapon stockpile and nuclear reactors (1, 2). The increasing demand is becoming unsustainable, leading to a steep rise in price by a factor of more than 10 from 2007 to 2009 (3). This crisis also affects the field of condensed matter physics, because 3He cryogenic refrigerators are most commonly used for cooling below 1 K. Therefore, finding alternative refrigeration techniques is an urgent issue in the field. One of the possible candidates to replace these cryogenic refrigerators is adiabatic demagnetization refrigeration (ADR) (4).

Current ADR for sub-kelvin cooling uses paramagnetic insulators as refrigerants, so-called paramagnetic salts with local magnetic moments (4, 5). At zero field, the moments are randomly oriented, whereas at high fields, the moments are aligned with reduced magnetic entropy (Fig. 1A). When the magnetic field is decreased from high to zero, disordered magnetic moments with increased entropy absorb heat from the lattice, leading to demagnetization cooling. The measure of effectiveness in cooling is magnetocaloric effect (MCE; ∂T/∂H|S), which quantifies the change of temperature caused by a change of the magnetic field in adiabatic conditions. Paramagnetic salts are widely used because of their large MCE in a temperature range below a few kelvin down to a few tens of millikelvin. However, their poor thermal conductance due to their insulating nature prevents an effective low-T heat transport. The same holds true for low-dimensional spin chains, tuned toward a quantum critical point (QCP) (see below) (6). Recently, an ADR down to ~0.2 K using the metallic compound YbPt2Sn has been demonstrated (7). In this compound, the magnetic moments of Yb ions, which are responsible for cooling, are fully localized and exhibit a very small mutual interaction. Thus, despite being metallic, it relies on the same principle of operation as that of paramagnetic salts. Furthermore, the coexisting conduction electrons lead to a small but finite Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, which causes magnetic ordering below 250 mK, preventing cooling to lower temperatures. Since the proposal by P. Debye (8) nearly a century ago, the same principle of using local magnetic moments for ADR has been applied. Below, we show a completely new approach to ADR, which is based on using itinerant magnetic moments in a super-heavy electron (super-HE) metal. Our example demonstrates cooling well below 0.1 K.

Fig. 1 Adiabatic demagnetization cooling by an HE refrigerant.

(A) Comparison of the demagnetization processes for conventional localized moment (left) and HE itinerant moment (right) refrigerants. For the latter (right), the entropy S is proportional to the number of thermally excited holes and electrons, D(EF)kBT, which corresponds roughly to the area depicted by orange color. Here, D is the density of states. These materials at zero field have a large and sharp peak with a width of ~kBTK in the density of states near EF. The density of states is being strongly suppressed by the application of fields exceeding μ0H = kBTK/gμB (upper right) (14, 2830), which could be used for adiabatic demagnetization cooling. Note that these are schematic sketches only, and the true D(E) in particular for H ≥0 will display a finer structure. (B) Crystal lattice structure of the super-HE refrigerant YbCo2Zn20 and the cage structure of Zn surrounding Yb. (C) Temperature dependence of electrical resistivity at zero field of Yb1−xScxCo2Zn20, with x = 0, 0.13, and 0.19.

An HE state is formed at temperatures below the Kondo temperature TK through an exchange interaction between f and conduction electrons (911). The HE state is easily destroyed by external parameters, such as magnetic field and pressure, owing to its low-energy scale kBTK. Because the electronic entropy of a metal is proportional to the electronic density of states at the Fermi energy D(EF) that is proportional to the effective electron mass m, the HE state has large entropy (at zero field). This large entropy is suppressed when the HE state is destroyed by applying a magnetic field μ0H of the order kBTK/gμB (Fig. 1A). Thus, with decreasing field from high values to zero, a very rapid increase of the entropy is expected. The latter will absorb heat from the lattice system, implying that HE materials can be used for ADR.

HE systems with effective doublet ground state contain a magnetic entropy of Rln(2) above TK. To optimally use all available magnetic entropy for ADR, the best parameters are the initial temperature Ti ~ TK, the initial field μ0Hi ~ kBTK/gμB = μ0HK, and the final field Hf = 0. Here, g is the g-factor of conduction electrons. For prototypical HE systems, such as YbRh2Si2, both TK ~ 25 K (12) and μ0HK ~ 10T (13) are too large for ADR applications that typically require Ti = 1 to 2 K and μ0Hi ≤ 8 T. On the other hand, for Kondo lattice metals with low TK ~ 1 K, typically, the exchange coupling between the moments leads to magnetic ordering, before the full moments are Kondo-screened upon cooling. For Ce-based HE systems, the RKKY interaction is typically of the order 10 K, whereas for Yb-based Kondo metals, it could be weaker. To retain a paramagnetic Kondo-screened state for an ADR refrigerant material with a low Kondo temperature of only 1 K, a very weak RKKY interaction energy is required. A careful scan of available results in the literature has revealed the most suitable system, YbCo2Zn20 (Fig. 1B) (14), which seems to fulfill these severe requirements.

Among all known paramagnetic HE metals, it has one of the lowest Kondo temperatures, TK = 1.5 K (14) (HE metals with this small or even smaller TK usually display a long range–ordered ground state). Reflecting the very low TK, the Kondo coherent state is formed only at very low temperatures, as evidenced by the maximum at 2.5 K in electrical resistivity shown in Fig. 1C. We ascribe the extraordinarily small TK and TRKKY to its crystal structure, which is the cubic CeCr2Al20 type shown in Fig. 1B, consisting of Zn cages surrounding the Yb atoms (14). These cages lead to a very weak hybridization between the Yb 4f and the Co 3d electrons. This effectively reduces the magnetic exchange interaction J, which enters both TK and TRKKY. The paramagnetic Fermi liquid (FL) ground state is evidenced by specific heat measurements, which show rapid increases of C/T with decreasing temperature below TK = 1.5 K and saturation at ~8 J/mol K2 (cf. Fig. 2), one of the largest values among heavy fermion materials (10, 11, 13). This super-HE state is very effectively suppressed by magnetic field, as indicated by the rapid decrease of the T2 coefficient in electrical resistivity (15). These properties set the system as the most promising candidate for a super-HE refrigerant. It should also be noted that YbCo2Zn20 exhibits a field-induced quadrupole ordering for the field only along the [111] direction (16). Because this ordering prevents the rapid decrease of the Sommerfeld coefficient, effective cooling is achieved only when the field is applied away from the [111] direction.

Fig. 2 Formation of the super-HE state in YbCo2Zn20 and the diverging effective mass in partially Sc-substituted material.

Electronic specific heat divided by temperature Cel/T for H//[100] plotted against temperature. The nuclear specific heat was subtracted. Cel/T at zero magnetic field for x = 0, 0.13, and 0.19, plotted with black open circles, red open squares, and blue solid triangles, respectively. The data for x = 0.19 at a magnetic field of 8 T along the [100] direction are indicated by blue solid diamonds. Inset: The calculated electronic entropy for x = 0.19 at zero field and 8 T under the assumption of constant Cel/T below the lowest measured temperatures. It is noted that this assumption leads to a slight underestimation of entropy at zero field. The gray lines with arrows indicate the demagnetization cooling process, starting from μ0H = 8 T and T = 1.5 K. The final temperature Tf is 0.075 K. The gray area indicates the amount of the heat ΔQc = 2.2 J/mol, which the material absorbs in the cooling process, whereas the area of rectangle a-b-c-d indicates the heat ΔQm = 5.6 J/mol, which is transferred from the material to the heat bath. These yield a high efficiency factor ΔQmQc of 40%.

Because the magnetic Grüneisen ratio ΓH = T−1(∂T/∂H)S (which is equal to the adiabatic MCE) diverges at any field-sensitive QCP (17), the usage of quantum critical (QC) materials for ADR was previously proposed (6). It has been reported previously, through measuring electrical resistivity under pressure, that YbCo2Zn20 can be tuned to a QCP around 1 GPa, beyond which long-range antiferromagnetic ordering is found (18). We therefore expect that the ADR performance of YbCo2Zn20 could be further enhanced when tuning this material toward its QCP. However, for any practical application, the need of applying a hydrostatic pressure leads to severe complications. Therefore, we investigated whether chemical pressure can be used to drive YbCo2Zn20 toward quantum criticality, and we found that partial substitution of Yb by the smaller Sc leads to an effective chemical pressure. Below, we investigate the possibility of using the super-HE system YbCo2Zn20 for ADR and study also the effect of quantum criticality, induced by chemical pressure, on the cooling performance.

RESULTS

Characterization

Using flux growth (see Materials and Methods), we grew single crystals of Yb1−xScxCo2Zn20, with x = 0, 0.13, and 0.19. The evolution of the lattice constant follows the Vegard’s law, indicating successful substitution of Sc, and the significant lattice contraction with substitution displays a clear chemical pressure effect (see the Supplementary Materials). The temperature dependence of the electrical resistivity of these materials is shown in Fig. 1C. The maximum around 2 K for x = 0 indicates the crossover to the coherent Kondo lattice state at low temperatures. In the Sc-substituted material with x = 0.13, the resistivity maximum is shifted to lower temperature, suggesting a suppression of TK by the effect of chemical pressure. Furthermore, the residual resistivity ρ(T→0) is substantially enhanced because of the structural disorder caused by chemical substitution. The resistivity maximum is completely suppressed by further Sc substitution of x = 0.19. Here, we note that this enhancement of electrical resistivity would benefit in ADR applications because it reduces eddy current heating in the demagnetization process. The unsubstituted material may not be used for application because of its very high purity, leading to the observation of the de Haas-van Alphen effect (16). We show here that thermal conductivity of Sc-substituted materials is still higher than that of paramagnetic salts. By using the Wiedemann-Franz law (with the resistivity value at the maximum, ~70 μohm·cm), we estimate the smallest electronic thermal conductivity at 50 mK to be 2 × 10−5 W/K cm, which is larger than a typical value of paramagnetic salts, 7 × 10−6 W/K cm (4, 19).

Figure 2 shows the electronic specific heat divided by temperature Cel/T of Yb1−xScxCo2Zn20. The data for x = 0 increase logarithmically as temperature is decreased and reach a large constant value of ~7 J/mol K2 at 0.2 K, indicating the formation of a super-HE state. Chemical pressure by Sc substitution suppresses FL behavior. The heat capacity coefficient for x = 0.13 and 0.19 displays further approximately logarithmic increase upon cooling below 0.2 K. This behavior is a signature of quantum criticality, as found in many other HE materials (20). At the lowest measured temperature of 0.05 K, Cel/T reaches a value as large as 8.5 J/mol K2. Similar to unsubstituted YbCo2Zn20 (16), Cel/T for x = 0.19 is reduced by almost two orders of magnitude by the application of a magnetic field of 8 T. Consequently, the large low-temperature electronic entropy found in zero field is almost completely shifted up to high temperatures by the magnetic field (Fig. 2, inset). This large entropy difference enables efficient ADR from 1.5 K at 8 T to 0.075 K at 0 T. We note that the difference of entropy is 90% of Rln(2) in this cooling process, indicating that cooling efficiency cannot be much better in any material, as long as the magnetic entropy of doublet ground states is used. In contrast to paramagnetic salts for which the specific heat follows the high-temperature tail of a Schottky anomaly, C/T ~ 1/T3, and thus rapidly decreases upon warming, Yb1−xScxCo2Zn20 with the milder ln(T) dependence displays huge C/T values up to a high temperature of ~2 K. As pointed out previously in the context of insulating QC magnets, this large value of specific heat at high temperatures leads to a significantly longer “hold” time compared to paramagnetic salts, which could be of advantage for ADR applications (6).

Quantum criticality of Sc-substituted materials

To compare the performance of the three different samples for ADR, we studied in detail the adiabatic MCE, which is equal to the magnetic Grüneisen parameter ΓH = T−1(∂T/∂H)S, using the alternating field method (21). For QC materials, this quantity is expected to display divergent behavior, which has been confirmed for several QC materials recently (17, 2224). We stress that the nature of a QCP is much more clearly characterized by ΓH compared to C/T. This is because ΓH is dominated by the QC contribution with the strongest T dependence, whereas C/T (for example, for antiferromagnetic quantum criticality in three dimensions) is dominated by the noncritical FL contribution, which in our case is extremely large (γ ~ 7 J/mol K2). Electrical resistivity is also not a good probe of quantum criticality in this study because the resistivity of the Sc-substituted materials has a large residual contribution (Fig. 1C).

Figure 3 shows temperature and magnetic field dependencies on ΓH for x = 0, 0.13, and 0.19. ΓH(T) for x = 0 increases strongly with decreasing temperature and passes through a maximum around 0.5 K. Because ΓH = −(dM/dT)/C (M, magnetization), the nonmonotonic behavior of ΓH(T) for x = 0 is probably related to the observed maximum in magnetic susceptibility at 0.3 K (15). As shown in Fig. 3B, the field dependence of ΓH for x = 0 at T = 80 mK changes sign at 0.5 T. This sign change is a typical signature of metamagnetic behavior and is consistent with the previously reported metamagnetism at 0.5 T (15), which is likely ascribed to itinerant moment magnetism, because 4f electrons are itinerant in this system. Upon partial substitution of Yb by Sc, the nonmonotonic behaviors in temperature and field dependencies weaken for x = 0.13 and completely disappear for x = 0.19. ΓH(T) at μ0H = 0.1 T for x = 0.19 diverges with T−1 ± 0.05 below ~0.6 K, and ΓH(H) above μ0H = 1 T follows 0.52/μ0H. The H−1 divergence instead of (HHQCP)−1 indicates that the system is tuned exactly to the QCP, with HQCP = 0. The T exponent and the obtained 1/H prefactor 0.52 ± 0.03 agree very well with the theoretical prediction for a zero-field QCP of an antiferromagnetic spin density wave (SDW), which yields ΓH(T,H) ~ H T−1 in the QC regime and ΓH(T,H) = 0.5/μ0H in the FL regime (17, 24). This theoretical prediction indicates the appearance of a maximum in ΓH(H), corresponding to a crossover field between the QC and the FL regimes, because ΓH(H) ~ H and ΓH(H) ~ 1/H in the former and the latter regimes, respectively. The positions of these maxima are shown by the dotted lines in Fig. 4. For x = 0.19, the deviation of ΓH(H) below μ0H = 1 T from the expected 1/H divergence is caused by the gradual crossover from the FL to the QC regimes with decreasing field. It is intriguing that the system exactly follows the theoretical prediction of an SDW QCP, although it contains significant dilution of magnetic Yb sites. Another likely possibility for causing QC behavior is a quantum Griffiths phase, which is expected in systems with large amounts of disorder. However, the observed critical behavior is incompatible with the Griffiths phase, because theory predicts for this case only a log(T) dependence in the Grüneisen parameter, which is much weaker than T−1 (25).

Fig. 3 Tuning to QCP by Sc doping in Yb1−xScxCo2Zn20.

(A) Magnetic Grüneisen ratio ΓH of Yb1−xScxCo2Zn20 with x = 0, 0.13, and 0.19 as a function of temperature for H//[100]. The divergence with a power law ~T−1 for x = 0.19 is indicated by the blue solid line. (B) ΓH at T = 80 mK as a function of magnetic field. The solid blue line is a fit to the data for x = 0.19 from μ0H = 1 to 8 T. Inset: Low-field region.

Fig. 4 Visualization of tuning to the QCP by Sc substitution.

Color-coded contour plot of the magnetic Grüneisen parameter ΓH of Yb1−xScxCo2Zn20 in H-T phase space. Magnetic field has been applied parallel to the [100] direction. Dotted lines indicate maximum positions in the field dependence of ΓH(H). These lines correspond to the crossover field to the FL regime at high fields (24). (See the main text for explanation.) For x = 0 and 0.13, the systems at low fields are influenced by quantum fluctuations of metamagnetism around 0.5 T (16), causing a finite-field extrapolation of the maximum position in ΓH(H) for T→0. For the critical concentration xc = 0.19, the line is extrapolated to zero, reflecting a zero-field QCP. ΓH obeys the expected QC behavior of SDW instability, namely, ΓH(T) ~ 1/T in the QC regime and ΓH(H) ~ 1/H in the FL regime (17, 24).

Detailed measurements of ΓH are summarized in the color-coded plot, shown in Fig. 4. There appears a small area of negative ΓH forx = 0 because of the sign change in its field dependence. This area of sign change is suppressed with Sc substitution, and monotonic field/temperature dependence is found for x = 0.19. The strongest enhancement toward the origin reflects the QCP at zero field for this composition. It is evident that the x = 0.19 sample is best suited for ADR toward the zero magnetic field, because the region of highest ΓH extends toward the origin for this material. We also note that further growth of ΓH(T) upon cooling to even lower temperatures is expected within the QC regime at H = 0.

Adiabatic demagnetization cooling by super-HEs

Using these data, we determined the cooling curves under adiabatic conditions (Fig. 5). All the curves display significant cooling effects. The curve for x = 0.19 shows a monotonic demagnetization cooling, whereas it is nonmonotonous for the unsubstituted material at low temperatures with a minimum around 0.6 T, which is caused by the metamagnetism (26). When starting from Ts = 2.5 K and μ0Hs = 8 T, the cooling effect for x = 0 is slightly better than that for x = 0.13 and 0.19, with a lower final temperature at zero field (Tf). Starting from Ti = 1 K, Tf becomes slightly lower with steeper slope at low temperature for x = 0.13 and 0.19 than that for x = 0. As shown in the inset of Fig. 2, a steeper slope of entropy, which is equal to C/T, leads to a lower final temperature. Because C/T is larger for x = 0.13 and 0.19 than for x = 0 below T ~ 100 mK, better cooling due to the QCP is realized at very low temperatures. We point out that C/T may diverge toward zero temperature at the QCP. Thus, cooling to arbitrary low temperatures may be possible because of the significant dS/dT = C/T, even in the limit of zero temperature. For x = 0.19, the final temperature reaches the lower limit of our experimental setup, which is 40 mK, when we set the initial temperature to a low value Ti = 1 K.

Fig. 5 Adiabatic demagnetization refrigeration of Yb1−xScxCo2Zn20.

Solid black, red, and blue curves represent the cooling curves for x = 0, 0.13, and 0.19, respectively. The curves are obtained by integrating the MCE (∂T/∂H|S) from 8 T to zero field. Magnetic field is applied parallel to the [100] direction. The points at low temperature below the lower limit of the thermometer calibration (40 mK) are obtained by extrapolating the calibration data. The out-of-calibration range is shaded by gray.

DISCUSSION

Here, we show that the HE metallic system Yb1−xScxCo2Zn20 can be used as a new type of refrigerant for ADR with itinerant magnetic moments. The steep increase of entropy due to the diverging C/T in the QC Sc-substituted material (Fig. 2) enables cooling down to temperatures even lower than the lowest temperature of our experimental setup, which is 40 mK. These results, along with the usage of most of the available magnetic entropy Rln(2) (Fig. 2, inset), indicate that the QC Sc-substituted YbCo2Zn20 is an ideal metallic refrigerant. HE metallic refrigerants have major advantages compared to insulating paramagnetic salts, such as an additional electronic thermal conductivity, stability in the air, and easy machining, because they are much less brittle compared to salts. In the conventional ADR systems, large amounts of Cu or Ag wires are spread uniformly in the paramagnetic salt pill to improve thermal conductance (4). One possible immediate application may be replacing the metallic wires with YbCo2Zn20, which would strongly improve the cooling performance. Thus, the HE refrigerants provide realistic possibilities of application and even replacement for conventional paramagnetic salts.

MATERIALS AND METHODS

Single crystals of Yb1−xScxCo2Zn20 were grown by the self-flux method (14, 27). The compositions of obtained crystals were determined by energy-dispersive x-ray analysis.

Electrical resistivity was measured using the standard four-probe technique. Single crystals of Yb1−xScxCo2Zn20 with a dimension of ~2 × 2 × 0.5 mm3 were used for specific heat and MCE experiments. The specific heat was measured by the standard quasi-adiabatic heat pulse method, using a dilution refrigerator.

The MCE was measured by using an alternating field technique adapted to a dilution refrigerator (21). We varied frequency (f) for the alternating field, depending on the temperature. The sample is in a quasi-adiabatic condition with a weak heat link to the bath, causing T-dependent relaxation time τ, which increases with decreasing T. To ensure the effective adiabatic condition for MCE measurements, it is necessary to maintain a condition, f ≫ 1/τ. We also point out that the relaxation time between the sample and thermometer (τ′) becomes longer with decreasing T, leading to the other necessary condition—f ≪ 1/τ′. Thus, one needs to maintain τ′ ≫ 1/f ≫ τ. This is easily checked by measuring the f dependence of the MCE signal. If the condition is maintained, the signal is f-independent. We obtained f-independent MCE signal for each measurement. Typically, we use 0.1 Hz at high temperatures above 1 K and 0.02 Hz at low temperatures below 0.2 K. This ensures perfect adiabatic conditions. Thus, our data of the MCE are identical to the magnetic Grüneisen parameter. We also changed the amplitude of the alternating field. Because the magnetic Grüneisen parameter becomes very sensitive to the field at low temperatures, we decreased the amplitude with decreasing temperature. We decrease it from 0.02 T at temperatures above 1 K to 0.004 T at temperatures below 0.2 K.

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/9/e1600835/DC1

fig. S1. Sc substitution dependence of lattice constant a.

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.

REFERENCES AND NOTES

Acknowledgments: Funding: Financial support for this work was provided by the German Science Foundation and the Grants-in-Aid for Scientific Research (no. 15K13521) from the Japan Society for the Promotion of Science. Part of this research was performed by P.C.C. and S.L.B. at the Ames Laboratory and supported by the Division of Materials Sciences and Engineering, Office of Basic Energy Science, U.S. Department of Energy. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under contract no. DE-AC02-07CH11358. Author contributions: P.G. conceived the project. Y.T. and P.G. planned and designed the experiments. Y.T. and B.P. performed the measurements. B.P., H.S.J., S.L.B., and P.C.C. synthesized and characterized the samples. Y.T. and P.G. discussed the results and prepared the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
View Abstract

Navigate This Article