Quadrupolar ordering and exotic magnetocaloric effect in RB4 (R = Dy, Ho)

The interplay of charge, spin, orbital and lattice degrees of freedom has recently received great interest due to its potential to improve the magnetocaloric effect (MCE) for the purpose of magnetic cooling applications. Here, a new mechanism for a large entropy change with low magnetic fields in rare-earth tetraborides, especially for Ho1-xDyxB4 (x = 0.0, 0.5, and 1.0), is proposed. For x = 0.0, 0.5, and 1.0, the maximum entropy changes of the giant inverse MCE are found to be 22.7 J/kgK, 19.6 J/kgK, and 19.0 J/kgK with critical fields of 25 kOe, 40 kOe, and 50 kOe, respectively. For all compounds, systematic study on how the entropy changes as a function of the field and temperature is performed to investigate their correlation with consecutive double transitions, i.e., the magnetic dipolar order at T = TN and the quadrupolar order at T = TQ (TQ < TN). Based on Landau theory, it is found that this behaviour is attributed to the strong coupling between magnetic dipoles and quadrupoles in the presence of strong spin-orbit coupling and geometric frustration. Our work offers new insights into both academic and industrial interests in the discovery of giant MCE with various applications for magnetic cooling systems.

The magnetocaloric effect (MCE) is a thermodynamic property, in which heating or cooling occurs in magnetic materials when applying a magnetic field. For the conventional MCE, the cooling mechanism is based on the adiabatic demagnetization process. In contrast, the inverse situation can also occur, where the system is cooled via adiabatic magnetization. This is often termed as the inverse MCE. Refrigeration based on the conventional or inverse MCE is a solid-state cooling application, which is energy efficient, noise-free, and environmentally friendly. Thus, a large MCE is attractive as an alternative to conventional vapour refrigeration 1 . In particular, a large MCE in a low-temperature region is being actively studied for the purpose of gas liquefaction (hydrogen and helium), space technology, and diverse scientific research technologies. In principle, the effective magnetic cooling using MCE can be achieved with the materials, which show large magnetic entropy change. Thus, for the development of novel solid-state cooling, the design and discovery of new materials that exhibit a large magnetic entropy change are important.
Intuitively, a large magnetic entropy is expected in materials with a first order magnetic phase transition accompanied with a spontaneous magnetization jump. However, this gives rise to heat loss during the refrigeration cycle due to the hysteresis, irreversibility and the narrow working temperature range. As another promising candidate, a system with geometrical frustration may contain an enormous ground state degeneracy due to competing spin exchange interactions, so a large magnetic entropy change is expected when a magnetic field is applied [2][3][4] . In addition, more exotic scenarios with multipolar degrees of freedom and their influence on the entropy change have been proposed [5][6][7] . Because such multipolar degrees of freedom are expected to give rise to an anomalous MCE, it would be of great interest to find such systems, thereby allowing us to unveil the nature of the anomalous MCE.
Multipolar degrees of freedom and their importance have garnered significant attention in many correlated electronic systems, such as heavy fermions, frustrated magnets, multiferroics and superconductivity [8][9][10][11][12][13] . For alloys with heavy 4d, 5d transition metal ions or rare-earth ions, the spin and orbital degrees of freedoms are strongly entangled, and the system may contain a significant correlation between the spin-orbit coupled multipoles, resulting in their spontaneous ordering. When such multipolar degrees of freedom meet geometrical frustration, their interplay gives rise to multiple magnetic phase transitions, which significantly induces a large entropy change, even in the absence of a first order magnetic phase transition. Thus, exploring such multipolar degrees of freedom in a strongly spin-orbit coupled system is definitely required when investigating the potential candidate materials with a large MCE. Not only regarding technological applications, this also deepens our understanding of complicated spin systems in the presence of various competing exchange interactions and their exotic multipolar order, which is, in principle, very challenging to detect. Thus, it is often termed as the 'hidden order' [14][15][16][17] .
Motivated by the above, we propose a new candidate of material rare-earth tetraborides for a large inverse MCE, i.e., positive magnetic entropy change, and elucidate their theoretical origin, pursuing possible controllability. A rare-earth tetraboride with a chemical formula RB 4 (R = rare-earth elements) is a system in which strong spin-orbit coupling and geometrical frustration coexist. In the presence of spin-orbit coupling and a crystalline electric field, the spin states in rare earth ions with valence 3+ split into several doublets or singlets in terms of the total angular momentum J basis. It is known that, depending on rare-earth ions, four low lying energy states are quite well separated from the other excited states, forming pseudo-quartets 18,19 . Furthermore, the lattice structure formed by R ions in the c plane is the Shastry-Sutherland lattice (SSL), which is a geometrically frustrated system, exhibiting double magnetic transitions, magnetic dipole ordering at T = T N1 and quadrupolar ordering at T = T N2 20 . Since the 1970s, the physical properties and magnetic structure of RB 4 compounds have been studied [21][22][23][24][25] and, recently, the detailed ground state of RB 4 (R = Dy, and Ho) was re-investigated by resonant X-ray scattering and X-ray and neutron diffraction 18,[26][27][28][29] . The two successive magnetic transitions of these compounds are related to the collinear antiferromagnetic transition at T = T N1 along the c-axis, the quadrupolar ordering at T = T N2 and the strong quadrupolar fluctuation between them. In addition, a structural transition from tetragonal to monoclinic is also observed at the quadrupolar ordering temperature, which indicates strong quadrupole-lattice coupling.
The field-induce entropy change of high-purity single crystals of RB 4 (R = Dy and Ho) is investigated by examining the temperature-and field-dependence of magnetization with an applied field along the c-axis and the ab plane. It is quite remarkable that these materials exhibit a large entropy change in low fields, i.e., a maximum magnetic entropy change near the quadrupolar ordering of +19.6 J/kg·K, +19.0 J/kg·K, and +22.7 J/kg·K at the critical fields of 50 kOe, 40 kOe, and 25 kOe for DyB 4 , Dy 0.5 Ho 0.5 B 4 , and HoB 4 , respectively. The exotic entropy change increases with an increasing magnetic field below the critical field but decreases above the critical field. The critical field is the regime where the non-collinear magnetic ground state breaks down, thus indicating that the entropy change is clearly correlated with the quadrupolar ordering. Figure 1 shows the temperature-dependence of the magnetization divided by the applied magnetic field, parallel and perpendicular to the c-axis, namely, M(T)/H with H = 10 kOe, for a single crystal of HoB 4 . There are two successive magnetic transitions at T N2 = 5.7 K and T N1 = 7 K for both applied magnetic fields parallel and perpendicular to the c-axis. Thus, this can be split into three distinct phases, I (T > T N1 ), II (T N1 > T > T N2 ), and III (T N2 > T) in the low-field region with a decreasing temperature. In phase I, the paramagnetic phase follows the Curies-Weiss law, M(T)/H = C/(T − θ), where C = N 0 μ eff 2 /3k B , N 0 is Avogadro's number, k B is the Boltzmann constant, and the effective magnetic moment, μ eff , is determined to be 10.4 μ B , where μ B is the Bohr magneton, and the Weiss temperature, θ, is −12.7 K and −11.6 K for the magnetic field parallel and perpendicular to the c-axis, respectively. The μ eff values are close to the theoretical value of Hund's rule for the ground state of the isolated Ho 3+ ions (μ eff = 10.6 μ B ) 28,30 . In phase II, it is known that the incommensurate magnetic order is dominant. However, the commensurate magnetic order evolves with a decreasing temperature, which also accompanies the elastic softening. In phase III, it is also reported that the quadrupolar ordering and magnetic dipole ordering coexist and the lattice distortion is also stabilized as a consequence of the strong quadrupole-strain interactions 28 . www.nature.com/scientificreports www.nature.com/scientificreports/ Figure 2(a,b) show the isothermal magnetization curves at various temperatures in the range of 2 K ≤ T ≤ 50 K with a field applied along the c-axis and perpendicular to the c-axis for HoB 4 , respectively. The isothermal data, along the c-axis, at T = 2 K show two meta-magnetic transitions at H ≈ 20 kOe and 35 kOe. The magnetic moment of the Ho 3+ ion is 6.6 μ B at H = 50 kOe, which is smaller than the maximum moment of the Ho 3+ ion (10.6 μ B ). This indicates that the canted antiferromagnetic moments by the coupling between the quadrupolar moments and the magnetic dipole moments undergo a field-induced phase transition with an increasing field 28,31 . Typical paramagnetic behaviour is observed at T = 50 K. On the other hand, the isothermal data, perpendicular to the c-axis, show similar but weaker meta-magnetic transition near H ≈ 20 kOe as compared to other configuration.

Results and Discussion
The magnetic entropy change, ΔS M , can be estimated from the Maxwell equation in the approximated form where M i+1 and M i are the experimentally measured values at temperatures T i+1 and T i , respectively, in the magnetic field interval, Δ = .
H 0 25 kOe i . The temperature dependence of the magnetic entropy change of HoB 4 is calculated using isothermal magnetization data (Fig. 2) with a magnetic field applied along the c-axis and in the ab plane. These are plotted in Fig. 3(a,b), respectively. For the field applied along the c-axis, a large positive entropy change is observed below T = T N1 with a maximum value of 22.7 J/kg·K at T = T N2 with the field change from 0 Oe to 25 Figure 4 shows the temperature-dependence of magnetization divided by the applied magnetic field, parallel and perpendicular to the c-axis, namely, M(T)/H with H = 10 kOe, for a single crystal of DyB 4 . Similar to the HoB 4 case, there are two successive magnetic transitions at T N2 = 13.0 K and T N1 = 20.5 K for an applied magnetic field parallel and perpendicular to the c-axis, respectively 32 . It is determined that the origins of these two transitions are quite similar to those in HoB 4 and are responsible for the magnetic order and quadrupolar order. However, the types of their orderings in DyB 4 are distinct from the ones in HoB 4 . At T = T N1 , collinear antiferromagnetic ordering is developed along the c-axis, while quadrupolar ordering is developed at T = T N2 , which accompanies the structural distortion and magnetic order in both the ab-plane and the c-axis 29 .
The isothermal magnetization curves at various temperatures in the range of 2 K ≤ T ≤ 50 K with a field applied along the c-axis and perpendicular to the c-axis for DyB 4 are plotted in Fig. 5(a,b), respectively. The magnetizations along the c-axis at T = 2 K and 5 K undergo field-induced transitions near H ≈ 45 kOe and the magnetic moment of a Dy 3+ ion is observed to be 3.8 μ B at H = 50 kOe 33 . Because the theoretical value of the Dy 3+ ion moment is 10.6 μ B , the transition is likely to be a meta-magnetic transition in the orbital ordered state, corresponding to the first meta-magnetic transition at H = 25 kOe in HoB 4 . The isothermal curves are found to follow paramagnetic behaviour at T = 20 K and 50 K. On the other hand, the isothermal data, perpendicular to the c-axis, show typical antiferromagnetic features without meta-magnetic transitions.
The temperature dependences of the magnetic entropy change for DyB 4 is calculated using the isothermal magnetization data for a magnetic field applied along the c-axis and in the ab plane (Fig. 5) and are plotted in Fig. 6(a,b). For a field applied along the c-axis, a large positive entropy change, ΔS (50 kOe) M , is observed below T = T N2 with a maximum value of 19.6 J/kg·K at T = T N2 , as shown in Fig. 6(a)  www.nature.com/scientificreports www.nature.com/scientificreports/ It is quite interesting that the maximum entropy change (ΔS M ) occurs at T = T N2 with the magnetic field of the meta-magnetic transition for both HoB 4 and DyB 4 . Because the meta-magnetic transition is a field-induced spin reorientation, which is strongly coupled with the quadrupole moment, a steep increase in ΔS M near T = T N2 with a maximum ΔS M at T = T N2 along the c-axis should be associated with the orbital ordering degeneracy release. In particular, the entropy change of HoB 4 at T N2 = 5.7 K increases as the field increases, reaching its maximum at the field change of 25 kOe and then decreasing with a further increase in the field change. Similarly, the entropy change of DyB 4 is maximized at T N2 = 13.0 K and the critical field H = 45 kOe, which is expected to be suppressed with a further increase of the field. This field dependence of the entropy change is quite unusual compared to the conventional behaviour, which originates purely from magnetic moments (unprecedented behaviour to our knowledge). Thus, the large magnetocaloric effect of both HoB 4 and DyB 4 near T = T N2 is a consequence of the magnetic moment reorientation, strongly coupled with quadrupolar ordering in the presence of strong spin-orbit coupling.
To understand such behaviour of a large inverse MCE, it is important to note that the system is geometrically frustrated and metallic. Geometrical frustration and thus competing exchange interactions are crucial for inducing two successive magnetic transitions 34,35 . In addition, the magnetic moment and quadrupole moment are strongly coupled and mediated via itinerant electrons in this metallic system. This leads to a sudden change in the magnetization near the critical field, especially at transition temperature T N2 , where the quadrupole moment is being developed. Within Landau theory, one can qualitatively understand this phenomena taking three order parameters into account, namely, the antiferromagnetic order (M s ), the ferroquadrupolar order (Q) and the uniform magnetization (M u ).  To observe the correlation of the field dependence of the entropy change with the spin-orbit interaction strength, a Dy 0.5 Ho 0.5 B 4 single crystalline specimen is synthesized. Figure 7(a) shows the temperature dependence of the magnetization with a magnetic field of H = 10 kOe parallel and perpendicular to the c-axis. Two successive transitions at T N1 = 13.8 K and T N2 = 9.5 K are observed, which are in-between those of DyB 4 and HoB 4 , as expected. The entropy change for the field applied parallel to the c-axis is calculated from the isothermal magnetization data (see Supplementary Data) and is plotted in Fig. 7 The data for three compounds (DyB 4 , Dy 0.5 Ho 0.5 B 4 and HoB 4 ) are summarized in Table 1. It is believed that the large positive entropy change near T = T N2 is not due to a simple antiferromagnetic transition at T = T N1 , but instead due to the strong correlation between the quadrupole ordering and magnetic moment. There is no significant variation in the maximum ΔS M values of the three compounds, indicating that the accumulated magnetic degeneracies due to a quadrupolar interaction with a magnetic dipole moment are similar for the three compounds. The critical field, i.e., the magnetic field at which the maximum entropy change is found, decreases gradually to be 50 kOe for DyB 4 , 40 kOe for Dy 0.5 Ho 0.5 B 4 and 25 kOe for HoB 4 . Because these fields are close to those for the field-induced meta-magnetic transitions, the critical field should be correlated with the coupling www.nature.com/scientificreports www.nature.com/scientificreports/ strength between the magnetic and quadrupole moments. Thus, with a further increase in the magnetic field above the critical field, the effects of quadrupolar ordering on the magnetic entropy change, ΔS M , would decrease, contrary to the conventional MCE.
For comparison, materials, which exhibit inverse MCE, are also listed in Table 1    Such an exotic inverse MCE is unique, originating from an interplay of a strong spin-orbit coupling and geometric frustration. Strong spin-orbit coupling and crystal field splitting lead to the formation of a pseudo-quartet for low lying states and allows for multipolar degrees of freedom, including both magnetic dipoles (linear in total angular momentum J) and quadrupoles (quadratic in J). In rare-earth tetraborides, such multipolar degrees of freedom lie in a geometrically frustrated Shastry-Sutherland lattice and interact with one another via itinerant electrons. This induces consecutive double phase transitions in the presence of strong coupling between magnetic dipoles and quadrupoles. Thus, the magnetization change as a function of temperature is expected to be enhanced at T = T N2 with the critical magnetic field of the meta-magnetic transition, resulting in a large positive entropy change and abnormal field dependence. This new mechanism opens a potential pathway to understanding the physical origin of a large inverse MCE in rare earth tetraborides. In addition, it enables us to enhance the controllability of the MCE with several parameters and applications for other candidate materials with strong spin-orbit coupling.

Method
The single crystals of RB 4 (R = Dy and Ho) are prepared by a high-temperature metal flux method using the Al flux 29,30 . A stoichiometric mixture of rare earth metals (≥99.9%, China Rare Metal Material Co., LTD.) and boron pieces (99.9%, RND Korea) are placed in an alumina crucible (99.8%, Samhwa Ceramic Company) together with the Al (99.999%, RND Korea) flux at a mass ratio of RB 4 : Al = 1: 50. The mixture is placed in a heated tube furnace with an MoSi 2 heating element. The furnace is heated at a rate of 300 °C per hour to 1650 °C under a high-purity argon atmosphere after dehydration and cooled slowly at a rate of 4.8 °C per hour to 650 °C. The single crystals are separated from the flux by dissolving the excess Al in NaOH.
The crystal structures are characterized using x-ray diffraction measurements (XRD; Rigaku D/MAX-2500 with a Cu target) at room temperature. The XRD data are collected on pulverized single crystals of DyB 4 and HoB 4 . The XRD patterns show a single phase of DyB 4 and HoB 4 without any observable impurity peaks. The Bragg peak positions are in good agreement with the tetragonal symmetry of the ThB 4 -type structure and space group P4/mbm (#127) 30,32 . The lattice parameters are determined from LeBail refinements using FULLPROF software. The refined lattice parameters are a = 7.099(5) Å and c = 4.015(9) Å for DyB 4 and a = 7.083(8) Å and c = 4.005(1) Å for HoB 4 . The temperature-and field-dependent magnetizations are measured using a superconducting quantum interference device magnetometer (SQUID; Quantum Design MPMS XL).