Journal Pre-proof Magnetism and superconductivity of rare earth borides S. Gabani, K. Flachbart, K. Siemensmeyer, T. Mori PII:
To appear in:
Journal of Alloys and Compounds
Received Date: 12 July 2019 Revised Date:
6 October 2019
Accepted Date: 26 November 2019
Please cite this article as: S. Gabani, K. Flachbart, K. Siemensmeyer, T. Mori, Magnetism and superconductivity of rare earth borides, Journal of Alloys and Compounds (2019), doi: https:// doi.org/10.1016/j.jallcom.2019.153201. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Magnetism and superconductivity of rare earth borides S. Gabani1, K. Flachbart1*, K. Siemensmeyer2, T. Mori3, 4* 1
Institute of Experimental Physics, Slovak Academy of Sciences, Watsonova 47, 04001 Kosice, Slovakia 2 Helmholtz Zentrum Berlin, Hahn-Meitner Platz 1, 14109 Berlin, Germany 3 National Institute for Materials Science (NIMS), International Center for Materials Nanoarchitectonics (WPIMANA) and Center for Functional Sensor & Actuator (CFSN), Namiki 1-1, Tsukuba 305-0044, Japan 4 University of Tsukuba, Graduate School of Pure and Applied Sciences, 1-1-1 Tennoudai, Tsukuba 305-8671, Japan *[email protected]
, [email protected]
Abstract Rare earth (RE) borides have yielded systems exhibiting interesting and varied magnetic behaviors. The electron deficient boron atomic frameworks are a good combination with rare earth atoms, which are relatively localized and contribute outer shell electrons to stabilize the various structures formed. At the same time, the variations in the boron network structure and the arrangement of the rare earth atoms occupying the voids, have resulted in a variety of interesting geometric magnetic behavior, from frustration derived from the Shastry-Sutherland lattice in REB4, to 1D, 2D, 3D behavior in the boron icosahedral compounds. The rare earth borides have also yielded systems with particularly fascinating physical properties, e.g. heavy fermions, Kondo insulators, controversial topological insulators, magnetic polaron-induced ferromagnetism, magnetic quadrupole ordering, etc. as described in detail in this review. On the other hand, it is peculiar that in the presence of rich phonon modes, which mainly originate from the light atomic weight boron framework, superconductivity occurs in RE borides only at rather low temperature (∼ 0.4 K) in LuB12 and the origins of this behavior are also discussed in detail. Introduction Rare earth borides, compounds of boron and rare earth elements, form a wide class of inorganic compounds exhibiting a rich spectrum of physical properties. Especially interesting are their electronic and magnetic physical properties, which are determined by the valence of rare earth atoms, as well as by their unfilled 4f-shells. Therefore, within this class of materials one can find nonmagnetic metals (as e.g. LuB4 and LaB6), a superconductor (LuB12), a dense Kondo system (CeB6), a ferromagnetic semimetal (EuB6), intermediate valence systems often considered also as Kondo insulators which may harbor topologically protected surface states (SmB6, YbB12), and there are many rare earth borides (diborides, tetraborides, hexaborides, dodecaborides, and even insulating higher borides, etc.) which order magnetically [1, 2, 3, 4, 5, 6]. In the following sections, some significant properties of rare earth borides will be discussed in detail together with presentation of latest results in this area. 1. Superconducting properties of rare earth borides MgB2 is possibly the most well-known superconducting boride and much research and literature have been devoted to it (see e.g. the following review articles [7, 8]). In this review, we focus on the rare earth borides. For a better understanding of the electronic properties of RE 1
borides, it is often that the properties of nonmagnetic lanthanum (without 4f-electrons) and lutetium (with a fully occupied 4f-shell) borides are investigated. The best known among these compounds is lanthanum hexaboride LaB6, often used as a reference material for hexaborides, which has a high electrical conductivity, mainly due to the light effective mass of conduction electrons. The conduction band of LaB6 band has a 5d character and its Fermi surface consists of weakly deformed almost spherical ellipsoids centered at the X point of the Brillouin zone [9, 10]. LaB6 belongs also to very good thermoelectronic emitters, finding utilization in technical and industrial applications because of its low work function, high melting point and chemical stability. The crystal structure of LaB6 with B6 octahedra in the corners of a simple cubic lattice leads to a rich variety of phonon modes [11, 12, 13, 14]. Based on the rich phonon modes, one of the reasons for the extensive investigation of LaB6 was the expectation of superconductivity in this compound between 27 and 61 K , which should originate from the interaction of conduction electrons with high frequency phonon modes of the boron sublattice. Experimental results, however, have shown that the transition temperature is probably very low (superconductivity in this compound has not been reliably observed down to 0.1 K ), and so the question of superconductivity in LaB6 is still open. Nevertheless, it should be noted that after superconducting MgB2 with a critical temperature of Tc ≈ 39 K, yttrium hexaborides YB6 (with the same crystal structure as LaB6) exhibits the second highest transition temperature Tc ≈ 7.4 K among borides . In the last few years the properties of YB6 have been investigated rather intensely, above all its electronic band structure , optical spectra , and the low temperature and superconducting properties. Very recent studies on the superconducting properties of this compound can be found e.g. in [20, 21, 22]. Up to now, no superconductivity has been observed in rare earth tetraborides or diborides. 1.1. Superconductivity of LuB12 and its solid solutions As is the case of LaB6 for hexaborides, LuB12 is often used as a reference material for the other rare earth dodecaborides. Due to the relatively low transition temperature  there was however, until recently quite little known about the superconducting properties of this material. Later investigations have shown that LuB12 is a good metallic conductor with a ρ(T) ~ T5 dependence of resistivity below about 30 K and a superconducting transition temperature Tc ≈ 0.4 K . The residual resistivity ratio ρ (300 K) / ρ (4.2 K) ≈ 70 and the residual resistivity ρ0 of LuB12 above Tc yield a low-temperature electron mean free path of l ≈ 200 nm. Band structure calculations and experimental Fermi surface investigations of LuB12 [23, 24] have shown that there are two conduction bands intersecting the Fermi level. The upper band corresponds to a simply sphere-like connected Fermi surface, the other band shows similarities to noble metal Fermi surfaces. The superconducting properties of LuB12 were for the first time investigated in detail in . The obtained normal state C(T) heat capacity data can be well described using the relation C(T) = γ T + β T3, where the first term represents the electronic contribution, and the second one, the phonon contribution to specific heat. From these data it was determined that the Sommerfeld coefficient is γ ≈ 4 mJ.mole-1.K-2 and the Debye temperature TD ≈ 1100 K (derived from β). In the superconducting state, the C(T) dependence has a discontinuity at Tc ≈ 0.4 K, clearly indicating a second order phase transition. The obtained ratio ∆C(TC) / γ Tc ≈ 1.3 of the specific heat discontinuity relative to the normal-state electronic specific heat at the transition temperature points to weak-coupling BCS-type superconductivity in LuB12. 2
The corresponding phonon spectra of LuB12 with modes at ∼14, ∼24 and ∼38 meV which were in  obtained by point-contact spectroscopy are in agreement with phonon modes obtained from neutron scattering experiments [26, 27]. Further investigations of LuB12 have shown that soft vibrations of Lu3+ ions have only a small contribution to the electron-phonon coupling, which leads to low Tc in this compound . Another factor which may contribute to a low Tc is the development of an electron instability due to the formation of dynamic charge stripes in LuB12  (for more details see text below). A comparison of BCS-type superconductors LuNB12 (Tc ≈ 0.42 K) and ZrNB12 (Tc ≈ 6 K) based on heat capacity C(T) and magnetization measurements carried out on high quality single crystals with various boron isotopes (with N = 10, 11, and with natural composition of 10B and 11 B) was made in . It was shown that ZrB12 is a type-II superconductor that has three Einstein type vibration modes of Zr4+-ions with energies θE1(ZrNB12) ≈ 200 K and θE2,3(ZrNB12) ≈ 450 K, whereas only one quasi-local mode of Lu3+-ions with θE1(LuNB12) ≈ 160–170 K could be detected. This difference of vibration spectra has been found to be probably the main reason for the difference between Tc of ZrB12 and LuB12. For a better understanding of the processes in LuB12 it should be noted that rare earth dodecaborides (including LuB12) represent one of the simplest model objects with a face-centered cubic (fcc, NaCl-type) crystal lattice built by B12 cuboctahedra in which the RE atoms are centered in large B24 cages formed by six neighboring B12 units (see Fig. 1 (a) and (b)). Strong covalent bonds between boron atoms (both within B12 units and between them) form a rigid boron framework (Fig. 1 (b)), which changes only insignificantly within the REB12 family (see e.g. ).
Fig. 1. (a) - Unit cell of LuB12 with Lu (red spheres) and B12 clusters (B is shown by green spheres), (b) - two large B24 polyhedra with central Lu atoms and a smaller B12 cubooctahedron between them, (c) - schematic view of the rattling vibrations of Lu3+ ions in a double-well potentials induced by freezing of RE ions in random (glass) positions inside the B24 cages with decreasing temperature (see also text) . Reprinted with permission from Reference . The large difference between the size of B24 cages (r(B24) ≈ 1.1 – 1.15 Å ) and the radii of RE ions (0.8 – 0.97 Å) leads then to formation of loosely bound states of the heavy RE ions in rigid B24 cages. This results in low frequency (14 - 18 meV)  dispersion-less (Einstein-like, rattling) vibrations in the phonon spectrum of dodecaborides. Moreover, as local structural distortions and boron vacancies can be expected to be present in the cubic REB12 lattice, it has been shown  that these crystal structures tend to form a cage-glass phase at low temperatures (through an order-disorder phase transition), in LuB12 at T* ∼ 60 K. This corresponds to freezing of RE ions in random (glass) positions inside B24 cages (see Fig. 1 (c)). Later X-ray diffraction studies  discovered a significant tetragonal distortion of the atomic structure of LuB12 in the vicinity of this phase transition. Additionally, a new structure analysis of LuB12 single crystals performed at various temperatures made it possible to observe in detail this symmetry lowering. It develops along a singular direction of the dodecaboride lattice and leads also to an asymmetry of the electronic density distribution within the unit cell. This distribution correlates with the anisotropy of magnetoresistance, it becomes more pronounced with decreasing temperature, and forms a filamentary structure of conduction channels “charge stripes” almost along the  axis [29, 35, 36] (see Fig. 2 below).
Fig. 2. (a) - anisotropy of LuB12 magnetoresistance in polar coordinates: ∆ρ/ρ0 = [ρ(ϕ, B) - ρ(ϕ0, B)]/ρ(ϕ0, B), ϕ0 = 270°, B || [-1 -1 0], (b) - anisotropic electronic density distribution reconstructed from the X-ray analysis . Reprinted with permission from Reference . While LuB12 appears to display a clear s-wave behavior of superconductivity, there have been considerable debates as regards the nature of superconductivity in ZrB12. For example, it has been suggested that ZrB12 is either a single-gap s-wave [37, 38], a two-gap s-wave , or a dwave  superconductor, with its Fermi surface composed of one open and two closed sheets [41, 42]. Therefore, it is interesting to study the superconducting properties of substitutional Lu1xZrxB12 solid solutions. Very recent investigations  of the superconducting phase diagram of LuB12 as well as of Lu1-xZrxB12 solid solutions with x ≤ 0.45 based on susceptibility measurements down to 50 mK show a linear increase of critical temperature Tc and a quadratic increase of critical magnetic field Hc with increasing Zr concentration (see Fig. 3 and 4).
Fig. 3. Phase diagrams of Lu1-xZrxB12 solid solutions obtained from ac-susceptibility measurements down to 50 mK . Reprinted with permission from Reference .
Fig. 4. Variation of Tc and Hc as a function of Zr concentration (0 ≤ x ≤ 0.45) in Lu1-xZrxB12 solid solutions . Reprinted with permission from Reference . 5
On the other hand, a strong suppression (see the Fig. 5 below) of superconductivity was observed in LuxZr1−xB12 compounds with a high Zr concentration . This strong Tc suppression can be attributed to pair breaking arising in the vicinity of nanosized Lu clusters which seem to exhibit a magnetic moment (even if Lu ions are non-magnetic). Further investigations of the superconducting and magnetic properties of four samples of LuxZr1−xB12 (x = 0.04, 0.07, 0.17, and 0.8) using muon spin rotation (µSR) and magnetometry measurements  revealed a strong magnetic signal in both the µSR and magnetometry data only in one sample (x = 0.07), likely caused by the formation of static moments of size ≈ 1 µB due to a clustering effect of Lu3+ ions. In all other samples only a small magnetic signal in the µSR data was observed. This sample dependence was attributed to a clustering effect in which the distribution of Lu3+ may affect the formation of static moments. Remarkably, these authors found also that the increase in Tc with decreasing x (in LuxZr1−xB12 compounds with for x ≤ 0.17) is accompanied by the formation of nodes in the superconducting gap, which provides a potentially new example of an s- to s+dwave crossover in a superconductor. Thus, it will be interesting to investigate in detail the crossover from superconductivity in LuB12 to this in ZrB12 in a wide range of concentrations.
Fig. 5. Suppression of superconductivity in LuxZr1−xB12 compounds as a function of Lu concentration . Reprinted with permission from Reference . 2. Magnetic properties of rare earth borides In rare-earth metallic compounds, the principal interaction which couples the magnetic moments of RE ions is the indirect (RKKY) type exchange. The long range and oscillatory character of this coupling in the presence of other interactions, e.g. the crystal electric field (CEF) anisotropy, frequently leads to competing interionic interactions. The role of the CEF is usually fundamental, however, sometimes even weaker interactions like magnetoelastic coupling, dipoledipole or quadrupolar interactions may also be important. The interplay between these competing interactions can lead to incommensurate amplitude-modulated magnetic structures, complex magnetic phase diagrams and magnetic phase transitions driven by temperature or by applied magnetic field. The frustrated magnetic systems (e.g. in tetraborides and dodecaborides) seem to 6
be especially interesting among RE borides, where due to the specific crystal structure and competitive exchange interactions, magnetic forces act on the moment, which cannot be simultaneously satisfied. This suppresses magnetic ordering and leads to a degeneration of the ground state, i.e. to the formation of several, different magnetic states with the same energy. As RE borides form various crystal structures, they form many suitable systems for magnetism research. The magnetic properties of rare earth borides attracted the attention of physicists already by the mid 1980s. Several early overviews of the magnetic properties of RE borides can be found in [1, 2, 46], a later one in . Here we will describe recent results on the magnetic properties of RE diborides, tetraborides, hexaborides, dodecaborides, and other boron-rich borides. 2.1. RE diborides The magnetism of rare earth diborides (REB2), which have a layered structure, has up to now, not been as extensively studied as the tetraborides, for example. Ferromagnetic (FM) transitions were observed for TbB2 (Tc = 151 K), DyB2 (Tc = 55 K), HoB2 (Tc = 15 K) and ErB2 (Tc = 16 K) , and YbB2 (TN = 5.6 K) was reported to undergo an antiferromagnetic (AFM) transition . Some additional data on REB2 compounds (e.g. heat capacity, thermal expansion coefficients) can be found in [48, 49]. Recently, thulium diboride was synthesized in the FM phase , and in  the mechanical properties and magnetic phase stability of REB2 compounds were studied. In order to discover and understand the magnetism in these materials in  the magnetic and electronic properties (as magnetic moments, band structures, densities of states and chemical bonding) were investigated theoretically. These calculations came to the conclusion that REB2 (R = Pr, Gd, Tb, Dy, Ho, Er, and Tm) compounds are FM, which is in agreement with experimental results. Other REB2 compounds (RE = Nd, Eu, and Yb) are antiferromagnets of AFM-I type (lower layer uu, upper layer dd), while Sm and Pm compounds adopt the type AFM-III configuration (lower layer ud, upper layer du). Their results also confirm that the strong covalent B–B bonding and the RE–B bonding play a critical role in the incompressibility and hardness of REB2. On the other hand, the spin–orbital interaction and the on-site Coulomb potential of 4f orbitals of RE elements play an important role to reach the correct magnetic ground state in this series of materials. Nevertheless, practically all investigations on rare earth diborides so far have been performed on polycrystalline or sintered samples. It would be therefore desirable, to study in detail the magnetic, as well as other properties of RE diborides on single crystals.
REB2, hexagonal structure, P6/mmm Ground state
TC , TN (K)
La Ce 59 3+ Pr 60 Nd3+ 61 Pm3+ 62 Sm3+ 58
FM, metal AFM-type I, metal AFM-type III, metal AFM-type III, metal
    7
Eu Gd3+ 65 Tb3+ 66 Dy3+ 67 Ho3+ 68 3+ Er 69 Tm3+ 70 Yb 71 Lu 64
AFM-type I, metal FM, metal FM, metal FM, metal FM, metal FM, metal FM, metal AFM-type I, metal
151 55 15 16 5.6
  ,  ,  ,  ,  ,  
Tab. 1. Ground state (FM – ferromagnetic, AFM – antiferromagnetic) and critical temperatures of rare earth diborides (for more details see the text). 2.2. RE tetraborides Rare earth tetraborides (REB4) crystallize in a tetragonal lattice  (see Fig. 6 (a)). As one of the three valence electrons of RE3+ ions goes into the conduction band, these compounds are good metals and the Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction between magnetic ions is playing an important role. In the above mentioned tetragonal lattice the RE ions lie in sheets perpendicular to the c-axis and within thee (a-b) planes mapped onto the frustrated Shastry-Sutherland lattice, which consists of squares and equilateral triangles. Between these RE sheets there are planes of boron atoms grouped into B6 octahedra and boron-dimer pairs. The magnetism of tetraborides is strongly influenced by crystal field effects (CFE) which are responsible for their strongly anisotropic magnetic properties. Earlier results on polycrystalline  and single crystalline samples  have shown that LaB4 and LuB4 are diamagnetic, CeB4 and YbB4 show paramagnetic ground states, PrB4 has a ferromagnetic ground state, and that most of RE tetraborides (NdB4, SmB4, GdB4, TbB4, DyB4, HoB4, ErB4, TmB4) order antiferromagnetically with Néel temperatures (TN) being in the range between ∼ 7 and ∼ 44 K. Probably the most investigated among RE tetraborides is TmB4 which exhibits a rich magnetic phase diagram (see Fig. 7) [54, 55, 56, 57, 58, 59].
Fig. 6. (a) Crystal structure of REB4. The RE and B layers are labeled. (b) The RE sublattice (perpendicular to c-axis) showing the frustrated Shastry-Sutherland lattice . Reprinted with permission from Reference .
Fig. 7. Magnetic phase diagram of TmB4 as derived from magnetization data for B || (001) (full symbols) . Reprinted with permission from Reference .
Fig. 8. The magnetic structure of Ising spins in TmB4 in the ab plane . For clarity only the domains along a direction are drawn. Red and blue circles indicate the up or down magnetic moments, the different surface colors indicate the periodicity of various stripe structures. Labels (i)–(iii) refer to the phase identification in Fig. 7 (above). In (i) the position of the Tm3+ ions in the ab plane is expanded with the Shastry-Sutherland bonds marked in red. Note the practically identical length of exchange paths J1 and J2. Reprinted with permission from Reference . Crystal field effects (CFE) at Tm3+ sites lift the degeneracy of the J = 6 multiplet. Consequently the ground state is a doublet MJ = ± 6 which induces a strong Ising-like magnetic anisotropy with magnetic moments of Tm ions oriented along the c-axis below its Néel temperature TN = 11.7 K. In the ordered antiferromagnetic state the magnetization M for magnetic fields B || c reaches saturation MS at about 4 T accompanied by magnetization plateaus at 1/2 MS and 1/8 MS. On the other hand, for fields B ⊥ c the saturation of M is reached only at much higher fields (above 30 T). Results of recent investigations of TmB4 can be found e.g. in [59, 60, 61, 62, 63]. Transport measurements under hydrostatic pressure up to 26.5 kbar carried out in a piston cylinder pressure cell between 1.7 and 14 K and in magnetic fields up to 6 T  exhibit shifts of ordering temperatures TN as well as shifts of boundaries between different magnetic phases, and the observed pressure dependences can be described by the relation dln(TN)/dp = +(0.16 ÷ 0.18) % / kbar. In  it was shown that the precise value of the magnetization at the fractional plateau and the magnitude of magnetization hysteresis strongly depends on the field history. These results lend support to the model proposed in  regarding the origin of the fractional plateau. Hysteretic magnetoresistance results studied in  suggest that complex structures arise at magnetic domain walls that strongly affect the transport properties, and that a combination of high-resolution neutron scattering, microscopic experiments and theoretical modeling is required to determine the magnetic structure and the origin of the observed unconventional anomalous Hall effect in TmB4. Using high-resolution specific heat and magnetization measurements it was 10
shown  that the 1/8 fractional plateau phase in TmB4 can be viewed as an example of emergence of an Archimedean lattice. These authors suggest that the 1/8 fractional plateau is smoothly connected to the antiferromagnetic phase on ramping the field from H = 0 and thus would not be a distinct thermodynamic ground state (but rather a metastable variant) of TmB4. In , low temperature angle-dependent magnetotransport in single crystals of TmB4 was investigated. The observed results were attributed to the anisotropic Fermi surface topology of TmB4. Thus, despite intensive investigations of TmB4, there still remain many open questions, concerning above all, the microscopic origin of the fractional magnetization plateaux. The strong magnetic anisotropy of TmB4 makes this compound suitable for the investigation of the rotating magnetocaloric effect (R-MCE). R-MCE represents a new and interesting magneto-thermodynamic phenomenon where the temperature variation in magnetic material is caused by rotating this material in a constant field instead of moving it in and out of a magnet, or by actively changing the magnetic field. In , R-MCE of TmB4 was investigated based on the detailed temperature dependence of heat capacity in various magnetic fields of a single crystalline sample for crystal axes orientations c || B and c ⊥ B. The results exhibit a marked cooling area above TN, which was confirmed by direct R-MCE measurements. Additionally, in  a comparison of R-MCE determined using independent measurements of heat capacity and magnetization was performed. The comparison of these two approaches has shown that estimates of the R-MCE based on magnetization measurements (which usually present a simpler and faster way to obtain the necessary data) provide similar results to those obtained from detailed temperature dependencies of heat capacity. However, to take the advantage of magnetization measurements it is necessary to make a right approximation concerning heat capacity data. Theoretical approaches elucidating the magnetism in TmB4 usually include localized Ising spins on the SSL, itinerant electrons, and the interaction between them and with the external magnetic field, i.e. the juxtaposition of 2D magnetism and 3D electrical conduction. Recent specific articles can be found e.g. in [66, 67, 68, 69, 70]. ErB4 has a similar magnetic structure as TmB4. It orders antiferromagnetically below TN = 15.4 K with magnetic moments oriented along the c-axis . The magnetization (M) along the c-axis reaches saturation at magnetic field of B ≈ 4 T, perpendicular to the c-axis the saturation field is at about ∼ 14 T. Both M(B) dependencies exhibit a plateau at half-saturated magnetization . However, no fractional plateaus, as in TmB4, were observed in ErB4. Investigations of DyB4 [73, 74, 75] have shown that this tetraboride orders antiferromagnetically at TN = 20 K, then there is a second transition at a lower temperature, TQ = 12.5 K. At this second transition, the X-ray scattering experiments found quadrupole ordering and a structural transition from tetragonal to possibly monoclinic structures, which is indicative of a quadruple-lattice coupling. This quadrupolar ordering is mainly due to the ordering of the Dy ion quadrupole moment (asymmetric charge distribution). In  it was shown that in the antiferromagnetic phase at TQ < T < TN the in-plane components of the magnetic phase become zero with magnetic moments pointing only along the c-axis. However, the quadrupolar ordering changes this qualitatively, and the magnetic moments of Dy ions aligned along the c axis get tilted by ∼ 30°. Different to DyB4, the compound TbB4 shows successive antiferromagnetic transitions at TN1 = 44 K and TN2 = 24 K , but the magnetic moments of Tb ions are oriented perpendicularly to the c-axis [71, 77]. The magnetic moments point towards the equivalent  direction between TN1 and TN2. Below TN2 the moments slightly tilt from this direction and the 11
quadrupolar interaction is suggested to be the origin of this tilt . Subsequently, authors  reported studies on multiple phase transitions in TbB4 by means of high-field magnetization and magnetostriction measurements up to 54 T. They found that a number of magnetization plateaus appear when the magnetic field is oriented along the c axis, i.e. perpendicular to the magnetic easy plane. The analysis of magnetization indicates a significant role of magnetic frustration and quadrupole interaction on the magnetic properties of TbB4. Later experimental investigations of heat capacity and lattice parameters  identified that below TN1 there is probably a transition of TbB4 from the tetragonal structure into the orthorhombic one. NdB4 has also been found to exhibit successive phase transitions at T0 = 17.2 K, TN1 = 7.0 K and TN2 = 4.8 K  with magnetic phases established as follows: paramagnetic phase I (T > T0), phase II (TN1 < T < T0), phase III (TN2 < T < TN1), and phase IV (T < TN2). In  the observed neutron powder diffraction patterns in phase II could be successfully explained by a coplanar structure with static magnetic moments in the tetragonal ab-plane. Later, polarized neutron scattering studies on NdB4 single crystals have been carried out  which confirm a non-collinear structure of the in-plane moments (see Fig. 9 below).
Fig. 9. The magnetic structure of NdB4, phase II (between T0 = 17.2 K, TN1 = 7.0 K) . Reprinted with permission from Reference . Moreover, they found that in phases III and IV there is an incommensurate modulation of the magnetic moment along the c-axis. Further detailed investigations  have refined the magnetic phase diagram of NdB4 and have shown that it exhibits also a fractional magnetization plateau at M/Msat ≈ 1/5 before saturating in field of 33 kOe. The reconstructed magnetic phase diagram for both H || c and H ⊥ c is shown in Fig. 10.
Fig. 10. Magnetic phase diagrams of NdB4 for (a) H || c and (b) H ⊥ c . For both field directions, phase I corresponds to a non-collinear antiferromagnetic order, while phases II and III are two different incommensurate structures . Phases IV and V for H || c correspond to the 1/5-magnetisation plateau and the fully polarized state respectively. Reprinted with permission from Reference . HoB4 is another tetraboride that displays multistep magnetic transitions and non-collinear magnetism (similar to NdB4). The 4f moments from the 4f10 electron configuration of Ho+3 ions become antiferromagnetically ordered at TN1 = 7.1 K, followed by a first-order transition at TN2 = 5.7 K, for a field parallel to the c-axis [84, 85]. 11B pulsed nuclear magnetic resonance (NMR) measurements performed on a single crystal of HoB4  below TN1 and TN2 have shown that the magnetic properties of this compound originate from the magnetic frustration and from quadrupole moment disorder effects. A recent study of HoB4  has been carried out using single crystal neutron diffraction complemented by magnetization measurements. In zero field magnetic phase transitions at TN1 = 7.1 K to an incommensurate state and at TN2 = 5.7 K to a noncollinear commensurate antiferromagnetic structure were observed. Between 18 and 24 kOe a ferrimagnetic up-up-down structure was observed. This field range is marked by the previously reported M/Msat = 1/3 magnetization plateau. The region between 21 and 33 kOe is characterized by antiferromagnetic neutron scattering reflections, the maximum of which coincides with the appearance of a narrow magnetization plateau with M/Msat ≈ 3/5. A further increase of the magnetic field results in a polarized state above 33 kOe, while the incommensurate reflections are clearly present in all fields up to 59 kOe. Based on the results obtained a H-T phase diagram of HoB4 for the H || c containing magnetic phases which overlap and show significant history dependence was proposed (see Fig. 11).
Fig. 11. Magnetic phase diagram of HoB4 constructed from neutron diffraction data (squares, filled circles, triangles, and diamonds) and magnetization (empty circles) measurements . All field dependent measurements were made by ramping the field up. IT denotes the intermediate temperature phase between TN1 and TN2. For further details see . Reprinted with permission from Reference . GdB4 possesses also a non-collinear magnetic structure [88, 89] with TN = 42 K, but a magnetization curve with no plateaus. The latter fact seems to be a consequence of GdB4 having small magnetic anisotropy compared to other tetraborides (as it has a half-filled 4f shell and no significant orbital moment). In  the Fermi surface (FS) properties of GdB4 were investigated. The FS is characterized by a rich variety of extreme cross-sections. The appearance of these extreme cross-sections is most likely connected with the hybridization of 5d-, 4f-states of Gd and 2p-states of B and due to the splitting 5d-states of Gd into the isotropic lower-energy band and distorted upper energy band . Corresponding calculations of the electronic structure and exchange interactions in GdB4 can be found in . There is up to now rather little information about the tetraborides of Ce, Sm and Yb. As regards CeB4, its susceptibility shows an almost temperature-independent Pauli paramagnetic behavior . Band structure calculations in  have shown that the valence state of Ce is rather close to trivalent (f 1). Heat capacity measurements of this tetraboride  revealed an anomalously large electronic contribution. In SmB4 two magnetic phase transition temperatures at TN = 25 K and Tm ≈ 7 K were observed by magnetization measurements for in magnetic field perpendicular to the c-axis . The magnetic transition at TN = 25 K is caused by antiferromagnetic order with Sm3+ moments that lie in the ab-plain. However, this antiferromagnetic transition temperature was not observed in magnetic field along the c-axis. The transition at Tm ≈ 7 K was observed for both the parallel and perpendicular field direction. For YbB4 it was found  that it has a paramagnetic ground state. An interesting subject among RE tetraborides are the magnetic properties of solid solutions as e.g. RE1−xYxB4 (RE = Tb and Dy), where magnetic ions are replaced by nonmagnetic ones. As the concentration of Y increased, TN was observed to systematically decrease, and an exotic weak ferromagnetic transition was observed . The weak ferromagnetism 14
occurred in the magnetic easy plane for Tb1−xYxB4, and along the magnetic easy axis for Dy1−xYxB4. It was also shown that the emergence of weak ferromagnetism is not due to an individual atomic effect but rather to a collective correlation effect. REB4, tetragonal structure, P4/mbm Ground state 57
TC , TN (K)
References ,  , , , [93-95] ,  , [80-83]
La Ce3+ 59 3+ Pr 60 Nd3+ 61 Pm 62 Sm 63 Eu 64 Gd3+
DIA, metal PARA, metal FM, metal AFM, , metal
24, 25 4.8, 7, 17
AFM, , metal
AFM, , metal
AFM, , metal
AFM,, metal AFM, , metal AFM, , metal PARA DIA, metal
Ho Er3+ 69 Tm3+ 70 Yb2+/3+ 71 Lu3+ 68
, , [88-92] , , , [7724, 44 79] 20, TQ = 12.7 , [73-76] 5.7, 7.1 13, 15.4 9.7, 11.7
, , [83-87] , , ,  [53-65] , , ,  , 
Tab. 2. Ground state (DIA – diamagnetic, PARA – paramagnetic, FM – ferromagnetic, AFM – antiferromagnetic), easy axis (a-axis // , c-axis // , ab-plane // ) and critical temperatures (TQ – critical temperature for quadrupolar phase transition) of rare earth tetraborides. 2.3. RE hexaborides Rare earth hexaborides form a very interesting series of compounds and in the last years a lot of progress has been made in their study. In this series one can find dense Kondo system behavior and electric quadrupolar ordering in CeB6 (see e.g. [99, 100], low-carrier-density magnetism with a narrow semiconducting gap in EuB6 (see e.g. , a nonmagnetic narrow gap in YbB6 (see e.g. ), and mixed valence and Kondo insulating properties in SmB6, which is considered as a material containing topologically protected surface states (see e.g. ). Other REB6 compounds (PrB6, NdB6, GdB6, TbB6 and DyB6) are mostly antiferromagnetics [1, 2, 46]. Here the most recent results achieved studying these compounds will be discussed in more detail. CeB6 is a heavy-fermion system showing an electron mass enhancement of the order of 100  which arises due to hybridization of localized f-electrons of Ce ions with itinerant conduction electrons. The magnetism of CeB6 is strongly influenced by the competition of Kondo screening and the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. While the Kondo 15
screening is associated with quenching the local (4f1) moments that favors paramagnetic behavior, the RKKY interaction is promoting magnetic order among these moments mediated by conduction electrons. The magnetic phase diagram of this compound is quite complex: basic antiferromagnetism with a commensurate structure, which sets in below TN = 2.3 K, is preceded by a hidden order state at TQ = 3.2 K, the so called antiferroquadrupolar phase (AFQ), which has been explained by ordering of quadrupole moments [105, 106]. Quadrupolar ordering is orbital in nature and arises due to distortion of the electronic charge density of unpaired electrons in their 4f orbitals (in CeB6 the Ce ions are in 4f1 configuration). This ordering has for long been elusive to neutron diffraction experiments and was first directly visualized by X-ray scattering . Recently, it was shown that this elusive phase manifests itself also as a Fermi surface instability which was observed based on a 3D tomographic sampling of the electronic structure by angleresolved photoemission spectroscopy (ARPES) and the comparison with inelastic neutron scattering data . In this case, the hidden order is expected to be mediated by itinerant electrons. In further experiments  it was found that the antiferro-quadrupolar (AFQ) ordering in CeB6 is strongly influenced by the crystal-field splitting and controlled by spin-orbit coupling. In PrB6 the crystalline electric field (CEF) splits the multiplet of the Pr3+ ions into a number of sublevels with a ground state that corresponds to a triplet. The first excited level has an energy of Е = 27 meV . At low temperatures this system undergoes two successive magnetic phase transitions, at ТN1 ≈ 7 K a transition to an incommensurate antiferromagnetic order occurs, while at ТN2 ≈ 4 K a commensurate antiferromagnetic phase , which is characterized by the wave vector Q = (0.25, 0.25, 0.5) , is formed. It was also shown that in both the incommensurate and commensurate magnetic ordered phases the antiferroquadrupolar (AFQ) interaction plays an important role [112, 113, 114]. Moreover, in low magnetic field (B < 0.1 T) at ∼ 20 K a magnetisation anomaly (i.e. a spontaneous magnetisation with a low magnetic moment) occurs. In  it was assumed that displacements of Pr ions due to the presence of defects in the crystal structure can lead to structural inhomogeneities and to the formation of small domains with short-range magnetic correlations, which can lead to such an anomaly. This was confirmed by inelastic magnetic neutron scattering on single crystalline Pr11B6  which shows that below 24 K domains arise where the degeneracy of the ground state of Pr ions is partially lifted because of the deformation of their local surrounding. The energy dispersion of these excitations is similar to the dispersion of magnons in the ordered state, which allows to claim that this spatially inhomogeneous phase contains antiferromagnetic correlations between local moments. Neutron diffraction measurements made on NdB6, which orders antiferromagnetically with a Neel temperature of 8.6 K, indicate a simple antiferromagnetic doubling of the unit cell (type I antiferromagnetism) with ordering vector q = (0, 0, ½)  and a magnetic moment of 1.74 Bohr magnetons per neodymium ion. In  dHvA measurements have been reported in both the AFM and PM regimes of NdB6, which appear to be separated by a second order upper critical field for antiferromagnetic ordering. The Fermi surface was observed to change radically, but predictably, across this transition, providing an example of a system where the effect of a one-dimensional magnetic periodic potential on doubling the unit cell can be tuned by varying only the applied magnetic field. The Fermi surface within the paramagnetic phase resembled that observed in non-magnetic hexaborides such as LaB6. NdB6, thus providing an essentially localized f-electron example for understanding similar field induced antiferromagnetism-toparamagnetism transitions in more complex 4f-electron systems where the 4f-charge-degrees of 16
freedom become involved. Further studies of thermal and transport properties focused on the anisotropic behavior depending on applied magnetic field direction  have shown anisotropy in NdB6 for H parallel to . In this direction the competition between the crystalline electric field effect and the ferro-quadrupolar interaction plays an important role. This competition might be important for the appearance of successive metamagnetic phase transitions for H parallel to . SmB6 is a material that has been studied for a long time. It was identified as the first homogeneous mixed valence system (the oxidation state of Sm-ions is intermediate between Sm2+ (4f6) and Sm3+ (4f55d1) configurations with an average valence of ∼ 2.56) in the 1960s (see e.g. [3, 120]) and as the first Kondo insulator (exhibiting a narrow gap of Eg ≈ 10 - 20 meV at the Fermi level. It originates from hybridization between localized (4f) and itinerant (5d) states [121, 122, 123]. Moreover, in  an enhancement of the electronic part of heat capacity below 2 K, as in heavy fermion systems, was observed and attributed to the formation of a coherent state within the metal-like in-gap states of this compound at lowest temperatures. Recently, SmB6 has come back into focus following theoretical predictions that it may display topologically protected surface states below its Kondo temperature due to the interplay of spin-orbit coupling with strong electron-electron interactions . Based on residual electrical conductivity ∼ 5 K it was suggested that this occurs due to correlated, topologically protected surface states with a linear dispersion formed within the energy gap, and in many recent experiments performed on this material a pronounced surface-dominated transport has been observed at lowest temperatures (see, e.g., [126, 127]). Also other types of measurements were performed to look for signatures of topological characteristics in SmB6, among them e.g. angle-resolved photoelectron spectroscopy (ARPES), which should be able to identify whether the surface states are topological or not. However, in case of SmB6 there is up to now no unanimity: ARPES results are interpreted differently in different publications (see e.g. [128, 129]. This means, that although some experimental results (as e.g. magneto-transport data) may represent a smoking-gun-proof for the topological Kondo insulator behavior in this system, the debate on the true meaning of data from surface-sensitive probes (as e.g. ARPES) is still open. Regarding the magnetic properties of SmB6, according to  in the high-temperature region the observed data follow a Curie-Weisslike dependence indicating local-moment behavior coming from magnetic (4f5) states of Sm3+ ions. The expected non-magnetic ground state of this Kondo insulator, which develops with lowering temperature, is formed by non-magnetic 4f6 and 4f55d1 configurations. However, below 10 K an additional increase of magnetization is observed. According to  it can be accounted for by a small amount of bare Sm3+ ions, magnetic 4f55d1 states or by magnetic rare-earth impurities. However, very recent investigations  indicate that the dependence of magnetization at lowest temperatures may be related not only to localized magnetic moments, but also to the Pauli paramagnetism of surface electrons. Additionally, it should be noted that measurements on SmB6 under hydrostatic pressure  show that the bulk insulating state of this compound vanishes at pressure ≈ 10 GPa and that at this pressure homogeneous long range magnetic (antiferromagnetic) order appears. With divalent Eu2+ ions in the REB6 structure EuB6 is a ferromagnetic semimetal . The transition into the ferromagnetic state, however, shows an intriguingly complex behavior with two consecutive transitions at about Tc1 ≈ 15.3 K and Tc2 ≈ 12.6 K . The paramagnetic to ferromagnetic transition is accompanied by a drastic reduction of resistance in zero magnetic field as well as by a colossal magnetoresistance (CMR) effect in finite magnetic fields, which is the strongest around Tc1. It has been proposed that the large negative magnetoresistance (MR) in 17
EuB6 at Tc1 is related to a percolation-type transition resulting from the overlap of ordered magnetic clusters / magnetic polarons (MP), which causes delocalization of the charge carriers (holes) . MP begin to form at temperatures T* ∼ 35 - 40 K  and upon cooling or increasing magnetic field grow in size and/or number until they percolate (form links) at T ≤ Tc1. These polaronic clusters finally merge at T ≤ Tc2, where bulk ferromagnetic order sets in - a scenario in accordance with transport investigations . In µSR experiments, two distinct and spatially separate regions associated with different magnetic behavior have been observed upon cooling through transitions at Tc1 and Tc2, providing evidence for magnetic phase separation that can be interpreted in terms of gradual clustering of magnetic polarons . Later on in , using combined scanning tunneling microscopy and locally resolved magnetic stray field measurements, the existence of local inhomogeneities in the electronic density of states due to the formation of magnetic polarons was visualized. Moreover, micro-Hall magnetometry measurements of the total stray field revealed evidence for magnetic clusters also in finite magnetic fields. In carbon doped EuB6, specifically in EuB5.99C0.01 , an anomalously large negative magnetoresistance was observed and attributed to the effect of fluctuations in carbon concentration. It was shown that in the bulk ferromagnetic (FM) state carbon-rich regions give rise to helimagnetic domains, which are responsible for an additional scattering term in the electrical resistivity. Above temperatures of the FM ordering (Tc = 4.3 K) the carbon-rich regions act as “spacers” preventing magnetic polarons (MP) to link, to form FM clusters, and eventually to percolate. Such spacers, being in fact volumes incompatible with the existence of MP and FM state may be responsible for the decrease of the percolation temperature and for the observed magnetoresistivity increase in this doped EuB6. GdB6 is known to have two successive magnetic transitions occurring at TN ≈ 16 K and TN2 ≈ 9 K, both being antiferromagnetic (AFM) phases [141, 142]. It should be noted that TN ≈ 16 K is rather small compared to magnetic transitions temperatures of other / similar heavy rare earth hexaborides (TN = 19.5 K and 25.6 K for TbB6 and DyB6, respectively ). The transition at TN ≈ 16 K is a first order phase transition which is accompanied by coherent displacements of Gd3+ ions . To explain the observed phase transition and AF structure magnetoelastic phenomena (due to atomic displacement waves) were taken into account [145, 146]. These displacements are connected with the fact that the ionic radius of Gd is considerably smaller than the lattice constant of GdB6, and in the high-temperature phase (between TN and TN2) the displacement structure has a multiaxial character. In , based on electron spin resonance measurements, an estimate of the displacement of Gd3+ ions (∆r ≈ 0.4 Å) from the centrosymmetric position was made. With further temperature decrease this structure becomes slightly suppressed, and at TN2 ≈ 9 K, which is again a first order phase transition, a more complex displacement scheme than in the higher-temperature AF phase appears [146, 148]. These structural features determine in many aspects also some physical characteristics of this compound above TN . Effects of atomic displacement, as in GdB6, can be expected also in TbB6, and in this regard a first-order transition at TN can be expected. Indeed, experimental investigations of heat capacity , magnetic properties , elastic measurements , and powder neutron diffraction  provide clear indications of a first-order magnetic transition in TbB6 at TN ≈ 19.5 K (this temperature is sample dependent). Powder neutron-diffraction patterns of the magnetic structure can be indexed considering 〈 ¼ ¼ ½〉 magnetic wave vectors , as in GdB6. But unlike GdB6, only a single AF phase is stable down to the lowest temperatures in TbB6. Recent investigations of this compound by means of magnetic susceptibility, magnetostriction, 18
and X-ray diffraction  point to a tetragonal symmetry in the antiferromagnetic state. In this phase the X-ray diffraction investigation shows the emergence of charge / displacement reflections related to wave vectors of 〈 ½ 0 0〉 and of 〈 ½ ½ 0〉 types. The 〈½ 0 0〉 type reflections result from the formation of static atomic displacement waves which can be described as a compromise between the RKKY exchange coupling and a single-ion magnetoelastic energy. However, further X-ray investigations will be needed to clarify the origin of the 〈½ ½ 0〉 type reflections in TbB6. This study, moreover, confirmed the influence of displacements in magnetic RE hexaborides. Regarding DyB6, in [150, 154] successive phase transitions at 32 K (TQ) and at 26 K (TN) in heat capacity measurements were observed. Also the appearance of multistep meta-magnetic transitions in magnetization curves at T = 4.2 K was found with a strong magnetic anisotropy . Neutron diffraction experiments unveiled the occurrence of a rhombohedral lattice distortion below 32 K  and a large softening of the elastic constant towards 32 K with decreasing temperature . Based on these results ferroquadrupolar (FQ) order was proposed to be realized below TQ = 32 K. Below TN a single-q AFM structure with qM = (¼ ¼ ½) develops [158, 159], while also a charge-modulation superlattice characterized by q2 = (½ ½ 0) is observed. However, the physical properties of this compound have not been analyzed in detail. In , mean-field calculations were performed for a two-sublattice model where the crystalline electric field (CEF) effect, and antiferromagnetic and ferroquadrupolar interactions are taken into account, focusing on the nature of the FQ ordered state. The heavy rare-earth compound HoB6, similar as DyB6, exhibits huge elastic constant softening above its ordering temperature TQ = 6.1 K. Below this temperature ferro-quadrupolar order appears . Antiferromagnetic (AFM) ordering develops at TN = 5.6 K. It was shown that with increasing magnetic field the elastic constant softening becomes suppressed and the point indicating the ferro-quadrupolar transition (TQ) shifts to higher temperatures . Neutron scattering experiment indicate, moreover, that at TQ the crystal structure of HoB6 changes from cubic to trigonal . On the other hand, the existence and stability of the neighbouring magnetic hexaboride, ErB6, appears to be a controversial topic in literature due to problems in reproducing previous preparation procedures . The possible occurrence of a correlated topological insulator (TI) state in YbB6  renewed the interest to this non-magnetic narrow-gap semiconductor (with an energy gap of ∆Eg ≈ 0.1 eV) . However, up to now no clear evidence of the surface component in low temperature electrical resistivity  and no parabolic dispersion of the two-dimensional metallic states from high-resolution ARPES studies  was found to support the topologically protected character of the surface states. Recent high pressure investigations of YbB6  have shown that the nonmagnetic Yb2+ (4f14) ions gradually change to magnetic Yb3+ (4f13) ones above 18 GPa concomitantly with the increase of electrical resistivity. The authors assume that YbB6 may become a topological Kondo insulator at high pressures above 35 GPa.
REB6, sc structure CsCl - type, Pm3m Ground state
TC , TN (K)
AFM, heavy metal
References , ,
2.3, TQ = 3.2 , , , [104-108]
7.0, 4.0 19
, , , , [109-
Nd3+ 61 Pm 62 Sm2+/3+ 63 Eu2+ 64 Gd3+ 65
AFM-type I, metal
PARA, heavy semiconductor FM, heavy semimetal AFM, metal
, , [120-123] 12.6, 15.3 , [133-139] 9, 16 , , , [141-149] , , , , [15019.5 153] , , , , [15426, TQ = 32 160] 5.6, TQ = 6.1 , , 
Er Tm 70 Yb2+ 71 Lu
, , , [117-119]
DIA, heavy semiconductor
, , [164-168]
Tab. 3. Ground state (DIA – diamagnetic, PARA – paramagnetic, FM – ferromagnetic, AFM – antiferromagnetic) and critical temperatures (TQ – critical temperature for quadrupolar phase transition) of rare earth hexaborides. 2.4. RE dodecaborides Rare earth dodecaborides crystallize in the NaCl type cubic structure (UB12) which may be presented as combination of two interpenetrating simple cubic sublattices, one sublattice based on metal ions, the other consisting of boron cage units (B12). Within this lattice the rare earth ions are accommodated in the centers of cubooctahedra formed by 24 boron atoms. Taking into account the strong covalent bonds between boron atoms, the crystal structure of RE dodecaborides can be described also as a rigid boron network with loosely-bound RE atoms embedded in cavities. One of the requirements to form dodecaborides is therefore the correlation between the dimension of cubooctahedra cavity and RE ion radius. According to this, RE dodecaborides are formed by RE atoms with a smaller ion size as Tb, Dy, Ho, Er, Tm, Yb and Lu. Primary studies of TbB12, DyB12, HoB12, ErB12 and TmB12 have shown that these compounds are antiferromagnetics with TN of ∼ 22.0 K, ∼ 16.5 K, ∼ 7.5 K, ∼ 6.7 K and ∼ 3.4 K, respectively [1, 46, 169]. Here recent results showing more details about the magnetic structure of RE dodecaborides are presented. Neutron diffraction measurements  have shown that the magnetic structure of TbB12 below Néel temperature is a sinusoidally modulated one. This structure can be described as a modulation of magnetic moments (stacked parallel in (111) sheets, but with an antiparallel orientation of the neighbouring sheets) which propagates along three main crystallographic directions. From further measurements of the heat capacity (i.e. from the magnitude of heat capacity, the discontinuity at TN and from the shape of heat capacity in the region of TN)  it follows that the magnetic order of this compound exhibits characteristics of amplitude-modulated 20
antiferromagnetic structures coming from the interplay between the crystal field effect and the indirect RKKY interaction. The ordered state of TbB12, moreover, reveals two first order phase transitions at 18.2 K and 14.6 K, most likely due to magnetic structure changes. A similar (amplitude-modulated antiferromagnetic) structure was also found for the magnetic ground state of DyB12, HoB12, ErB12 and TmB12 . However, the first excited CEF state of HoB12, ErB12 and TmB12 appears to be located so high that practically only the ground state is thermally populated in the vicinity and below the AF transition. And indeed, in these compounds (as well as in DyB12) no further phase transition as a function of temperature, except at TN, are observed. HoB12 was investigated in more detail. Magnetization measurements up to 9 T  have shown up three magnetic phases in the B vs. T phase diagram, depending on the orientation of applied field. The heat capacity in zero field exhibits a very steep increase at TN = 7.4 K, but its maximum is reached only at a lower temperature. In applied magnetic field up to 8 T additional λ-like anomalies were observed confirming the phase boundaries between the various phases determined from magnetization measurements. Powder neutron diffraction in zero magnetic field revealed an antiferromagnetic structure below TN with basic reflections indexed as (1/2±δ, 1/2±δ, 1/2±δ), where δ = 0.035 (Fig. 12), pointing to an incommensurate magnetic structure along thee crystallographic orientations. The analysis of all results has shown that an amplitude-modulated incommensurate structure represents the magnetic order of HoB12. Further neutron scattering investigations on a single crystalline sample  confirmed the amplitude-modulated incommensurate magnetic structure of HoB12 at 2 K (Fig. 13). However, already well above TN neutron scattering experiments show strong diffuse scattering indicating strong correlations between Ho - moments along the  direction (Fig. 14). The behavior of this component resembles low dimensional magnets which are known to show long range order only at T = 0. In HoB12 correlations perpendicular to the  direction get relevant only close to TN, they diverge at TN. Thus, a complex ordering process can be observed in this magnetic dodecaboride, where the frustration of the fcc-lattice is lifted in steps.
Fig. 12. Incommensurate amplitude-modulated (1/2±δ, 1/2±δ, 1/2±δ) magnetic structure on the fcc lattice of HoB12 along the (111) axis . Reprinted with permission from Reference .
(111) 0 -2
q (kk0) Fig. 13. Reciprocal space map of neutron scattering reflections observed for HoB12 at 2 K in zero field . The points show the observed magnetic reflections. Air scattering streaks seen at the strong reflections are of instrumental origin. Reprinted with permission from Reference .
2 1 0 -1 -2 -3 -2
Fig. 14. Diffuse neutron scattering on HoB12 above TN . Reprinted with permission from Reference . The influence of substitution of magnetic Ho atoms through diamagnetic Lu ones (in Ho1Lu B x x 12 solid solutions), as a result of which both chemical pressure and magnetic dilution take place with increasing content of Lu, was investigated in . The obtained TN vs. Luconcentration (x) phase diagram has shown indications for the existence of a critical point close to xc ≈ 0.9 which separates the region of magnetic order (starting with HoB12 for x = 0) and the region without ordering (ending with superconducting LuB12 for x = 1). On Ho0.8Lu0.2B12, moreover, a precise angular dependent magnetoresistance (∆ρ/ρ) analysis has shown that its angular (polar) H-ϕ-T magnetic phase diagrams have the form of a „Maltese cross” (Fig. 15 22
below)  reflecting at the various temperatures the interplay between charge carriers and the magnetic subsystem as a function of crystallographic orientation. In the analysis it was suggested that the main positive linear field contribution to (∆ρ/ρ ∼ H) may be attributed to itinerant electron scattering on spin density waves (on the 5d component of the magnetic structure), whereas the negative quadratic term (−∆ρ/ρ ∼ H2) is due to scattering on local 4f-5d spin fluctuations of Ho3+ ions. It is argued the observed distribution is the consequence of strong renormalization of the indirect exchange interaction (of RKKY type) due to the presence of dynamic charge stripes in the matrix of this AF metal.
Fig. 15. Polar H-ϕ (H increases from the central point from 0 to 80 kOe) magnetic phase diagram of Ho0.8Lu0.2B12 at temperatures 2.1 K (a) and 4.2 K (b). The phase boundaries are derived from magnetoresistance data . Reprinted with permission from Reference . YbB12 is (as is SmB6) an intermediate valence compound (with a mixture of Yb2+ (4f14) and Yb3+ (4f13) states and an average valence of ∼ 2.92) which exhibits properties of a Kondo insulator. It is (again as SmB6) also considered to be a suitable candidate for topological Kondo insulators. As the Yb2+ ions are non-magnetic and the Yb3+ ions magnetic, the non-magnetic ground state of YbB12 at low temperatures is formed mostly by the configuration of non-magnetic 4f135d1 states and a few 4f14 states. The insulating state of YbB12 is characterized by a clear 23
energy gap with a size of ∼ 40 meV, which was detected e.g. by optical conductivity measurements . Moreover, a finite density of states (DOS) at the Fermi level (EF) has been observed at low temperatures using time-resolved photoemission spectroscopy and considered to originate from metallic topological surface states [164, 177]. Later on ARPES investigations on a clean surface of YbB12 (001) were reported . It was shown that the in-gap states are metallic and come from the sample surface. While these surface states do not hybridize with the Yb 4f states lying just below the Fermi level (EF) at room temperature, strong hybridization occurs at low temperatures. The reconstructed metallic surface states are continuously dispersed across the bulk band-gap. This low-temperature behavior is in agreement with the expected behavior for surface states in topological Kondo insulators. Further experiments of heat capacity in magnetic field up to 60 T  have shown that under the influence of a high magnetic field the Kondo insulator YbB12 transforms into a Kondo metal (Fig. 16), maintaining its high Kondo temperature (TK ≈ 200 K). This result proves that the energy gap collapse does not lead to the breakdown of the Kondo bound state. A steep increase of the magnetization at the insulator to metal transition, moreover, manifests a sharp density of states at the Fermi level formed via the Kondo effect.
Fig. 16. The C (heat capacity)/T (temperature) plot in the magnetic field (B) vs. Temperature plot showing Kondo insulator to Kondo metal transition in YbB12 under the influence of magnetic field . Reprinted with permission from Reference .
REB12, fcc structure NaCl - type, Fm3m-Oh5 Ground state
La Ce 59 Pr 60 Nd 61 Pm 62 Sm 63 Eu 64 Gd 58
metal (under pressure)
AFM, metal (sintered)
14.6, 18.2, 22
AFM, metal AFM, metal PARA, heavy semiconductor
Er3+ 69 Tm3+ 70
, , , ,  , , , , , , ,  , , , , , , , , ,  , , [23-25], 
Tab. 4. Ground state (DIA – diamagnetic, PARA – paramagnetic, AFM – antiferromagnetic) and critical temperatures of rare earth dodecaborides. 2.5. Several ternary and quartenary RE metallic borides REMB4 (M = metals: transition metals, Al) The REMB4 (M = metals: transition metals, Al) compounds predominantly form layered structures with an analogy to the AlB2-type structure. This is because in approximation, the composition is 2 metal atoms versus 4 boron atoms, namely approximately 1 metal versus 2 boron atoms, i.e. similar to AlB2. Since RE and M atoms are different, instead of homogeneous hexagon [B]  rings sandwiching the metal atoms, the compound forms [B]  and [B]  rings to accommodate the different sized RE and M atoms. There are two different patterns of arranging the  and  rings, the YCrB4-type or α-type structure (space group Pbam), first discovered by Kuzma  and the ThMoB4-type or β-type structure (space group Cmmm), first discovered by Rogl . The first magnetic transition in these compounds was discovered in αTmAlB4. An antiferromagnetic transition was observed at 5.8 K . Later it was indicated that an additional transition occurred in-plane at TN1 ~ 40 K with short-range order or low dimensional character . Regarding the magnetization curves, strong anisotropic behavior was observed with the c-axis indicated to be the easy axis. As shown below in Fig. 17, successive sharp metamagnetic transitions were also observed at Hc1 = 11 kOe and Hc2 = 24.5 kOe with field 25
aligned along the c-axis . This multi plateau-like behavior of the magnetization has similarities to what has been observed in some of the REB4 compounds, but the detailed behavior has not yet been fully elucidated for α-TmAlB4.
Fig. 17. Magnetization of α-TmAlB4 . Adapted from Reference . Another interesting feature of the magnetism of TmAlB4 was that multiple magnetic anomalies in α-TmAlB4 were found below the Neel temperature TN2 = 5.8 K [182,183]. Intrinsic tiling variations in the boron layer (building defects), namely, β -type structural fragments inserted into the α-type structure were indicated to be the origin of this behavior (Fig. 18) . α-type defects were also found to be inserted in the β-type structure compounds , and these defects were indicated to be ubiquitous, but typically only possible to be structurally detected by TEM or a very high level of single crystal XRD analysis. Finally, utilizing unusual flux crystal growth conditions, defect-free single crystals of α-TmAlB4 were successfully grown . The multiple magnetic anomalies and “missing entropy” problem were found to disappear in the defect-free crystals, indicating their large effect on the physical properties (Fig. 19). The defects were also measured to reduce thermal conductivity significantly . In regards to thermal conductivity of such layered compounds, an interesting insight was obtained when the cross plane thermal conductivity of AlB2 was found to be larger than that along the graphitic layers inplane . This illustrates the differences in bonding between carbon based network structures and electron deficient boron networks which are stabilized by electrons from metal atoms in the voids, leading to strong bonding between the boron and metal atoms. An important point regarding building defects is that, as mentioned, they are difficult to detect by conventional methods and can often be unperceived. Therefore, such building defects may possibly be the origin of anomalous unsolved physical behavior in other non-boride layered systems also [185,186]. As for other magnetic behavior, spin glass-like behavior was observed for α-HoAlB4 and α-ErAlB4, although the origin of the particular disorder for these rare earths is still under investigation . And there exists the related compound RE2MB6, which also has metal:boron 26
ratio of 1:2 and possessing a layered structure with [B], [B], and [B] rings, instead of just [B] and [B] rings for REMB4, since RE2MB6 is closer to REB2. Tm2AlB6 was found to have a magnetic field induced state with extremely high field stability .
Fig. 18 Schematic view of the building defects in αTmAlB4 . Reprinted with permission from Reference .
Fig. 19 Specific heat of α-TmAlB4 . Adapted from Reference .
Rare earth borocarbides
A recent review by Halet and coauthors has comprehensively reviewed the synthesis, structure, and physical properties of the various rare earth borocarbides so they will not be given in detail here . Just to simply summarize, amongst the rare earth borocarbides, the REB2C2 compounds appear to possess the most interesting properties. The crystal structure of REB2C2 has boron and carbon mixed 2D layers sandwiching metal atom layers, similar to the overall scheme for the previous subsection’s REMB4 compounds. For RE = La, Ce, Pr, Nd, Sm, Y, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, REB2C2 is tetragonal (space group P4/mbm) with  and  [BC] rings forming the 2D layer, while ScB2C2 takes a different structure because of the small size of Sc and is orthorhombic (Pbam) with  and  [BC] rings. Amongst the various rare earth borocarbides YB2C2 and LuB2C2 are the only compounds to exhibit superconductivity with transition temperatures of Tc = 3.6 K and 2.4 K, respectively [190,191]. A rich variety of magnetism has been reported for magnetic rare earth element compounds of REB2C2, REB2C, and RE2B3C2 27
. TbB2C2, DyB2C2, HoB2C2 are antiferromagnetic, but were also found to possess antiferromagnetic quadrupole ordering also . The detailed magnetic behavior is complex, originating from a competition between conventional antiferromagnetic coupling from the RKKY interaction, and antiferromagnetic quadrupole coupling . 2.7. Rare earth transition metal borides In regards to rare earth transition metal borides, superconductivity has been observed for several RETr2B2C (Tr = transition metal) compounds. The RETr2B2C compounds are tetragonal (space group I4/mmm), with the structure having Tr2B2 layers separated by REC layers which has been described as a “filled” version of the ThCr2Si2-type structure . Further variants of this structure are also known; (REC)nTr2B2, where Tr2B2 layers are separated by n REC layers. The observed superconductivity transition temperatures have been relatively high, with Tc = 23 K, 16.5 K, 15 K, and 12 K for YPd2B2C, LuNi2B2C, YNi2B2C, and (LaN)3Ni2B2, respectively . The most striking behavior has been reported in RETr2B2C (R = Dy, Ho, Er, Tm), where both superconductivity and magnetism have manifested. There is an intricate interplay between magnetic ordering and superconductivity in these systems, and from investigation of this understanding has deepened regarding vortex formation, for example . These compounds have been previously reviewed in detail [197,198]. Amongst the rare earth transition metal borides, the Nd2Fe14B-type compounds are a prominent system because of its magnetism. These compounds have been widely applied as permanent magnets. A Nd15Fe77B8 compound were first synthesized in 1983, and found to possess magnetic remanence, coercivity, and energy product of 1.23 T, 880 kA/m, 90 kJ/m3, respectively, which values were significantly larger than anything previously known . Subsequent development of the Nd2Fe14B-type magnets has further improved the magnetic properties and reduced costs, leading to wide ranging applications. The boron content of Nd2Fe14B is relatively low compared to other compounds discussed in this review, but boron was the critical component to realize a new crystal structure (tetragonal with space group of P42/mnm) from which the excellent magnetic properties are derived. 2.8. RE boron icosahedral borides; Insulating borides The rare earth boride compounds reviewed up to now have all been systems, which are good metals in the case of trivalent rare earth borides. As a result, the magnetic interactions in these systems are typically supplied by the conduction electron-mediated RKKY interaction. It has been noted that for all known rare earth borides with [B]/[RE] >12, the basic boron building block of the crystal structure is the B12 icosahedron, and all known compounds are insulators/semiconductors [4,6]. The thermoelectric properties of these compounds have been found to be of interest [4,6,200]. In particular relation to magnetism, the Sm phase of the REB66 compound was found to have an enhanced power factor, suggested to be possibly related to the indicated mixed valency of Sm . Enhancing the thermoelectric properties through magnetism is a hot trend in the field [202-204]. Considering a compound like REB50, it is rare earth dilute, i.e. possessing significant average RE-RE spacing, and as an insulator, lacking the RKKY interaction, so magnetic behavior is not expected at moderate temperatures. Despite this, relatively strong magnetic interaction is observed, and it has been proposed that the B12 icosahedron is mediating the magnetic interaction [205-208]. 28
In this review, we will focus on the variety of magnetism which has been observed in the RE boron icosahedral borides. 2.8.1. REB50 and REB44Si2: 1D dimer-like transition REB50 and REB44Si2 are basically of the same crystal structure-type , with Si assumedly partially replacing several boron sites in the case of REB44Si2. The magnetic properties were found to be of note, when the first magnetic transition ever observed was discovered for TbB50 at TN = 17.5 K . REB50 (RE = Tb, Dy, Ho, Er) were also found to have similar antiferromagnetic transitions . The introduction of Si expands the lattice constants of REB44Si2 compared to REB50 [211,212], and using the Curie-Weiss temperature θ as a measure of the magnetic interaction, the values for REB44Si2 are generally smaller than REB50 as would be expected. However, θ of TbB44Si2 is significantly larger compared to TbB50, despite the larger atomic separations, and this has not been fully explained yet. The f-electron dependence of θ does not agree with that of dipole-dipole coupling or RKKY interaction. µefffreeion (µB/R atom) 9.72 10.65 10.62 9.58 7.94
TbB50 DyB50 HoB50 ErB50 GdB44Si2
µeff (µB/R atom) 10.3 11.9 11.5 9.74 7.87
-15.3 -13.7 -13.7 -5.4 -7.2
17.5 6.2 7.5 4.6 7
30.0 18.0 16.0 8.0 -
    -
TN (K) HC (kG)
Tab. 5. Magnetic parameters of REB50 (RE = Tb, Dy, Ho, Er) and REB44Si2 (RE = Gd, Tb, Dy, Ho, Er). Regarding the nature of the antiferromagnetic transition in REB50 and REB44Si2, the peak in the magnetic susceptibilities are broad and likewise the specific heat , indicating the transition is of short range order nature. This was also supported by neutron scattering measurements . Furthermore, a magnetic dilution experiment of TbB50 revealed that Hc which is also a measure of the magnetic interaction, was invariant against the dilution, indicating that the antiferromagnetic transition is dimer-like . And from ESR experiments on GdB44Si2 , it was finally indicated that the antiferromagnetic transition occurs along the c-axis 1D RE bond alternating chain. This coupling is along the direction where the RE atoms are adjacent to
B12 icosahedra, which first indicated to us that the boron icosahedra might be mediating the magnetic interaction. The interesting features of the magnetism in REB50 and REB44Si2 are the unexpectedly strong coupling for this relatively magnetically dilute insulators, and also the 1D dimer-like behavior which is a result of the rare earth atom coordination in the boron cluster matrix. 2.8.2. REB15.5CN, REB22C2N, and REB28.5C4: 2D spin glass Spin glass behavior was discovered in the series of homologous compounds REB15.5CN, REB22C2N, and REB28.5C4 [218,219]. Taking REB22C2N as representative, the structure is rhombohedral (space group R-3m) and for example for the Ho phase, a = b = 5.614 Å, c = 44.625 Å. The compound has a layered structure along the c-axis with B12 icosahedral and C-B-C chain layers residing in between B6 octahedral and rare earth atomic layers (Fig. 20 left). REB15.5CN
Fig. 20 Structure of RE-B-C(N) homologous phases (left) and configuration of only the RE atoms in the RE-B-C(N) homologous phases (right) [218,220,221]. Adapted from Reference . The origin of the spin glass behavior was surmised to be from disorder from the partial occupancies of the rare earth sites, and also geometrical frustration from the triangle 2D RE layers (Fig. 20 right). As an important point, the shortest RE-RE separation is actually not in the triangle layer (5.62 Å in the case of e.g. HoB22C2N), but between the triangle layers which are stacked in A-B stacking (3.54 Å in the case of e.g. HoB22C2N). However, because frustration is indicated, the strongest magnetic interaction is indicated to occur in the RE triangle with much larger spacing. As was the case for REB50 and REB44Si2 discussed above, the interaction is strong adjacent to the B12 icosahedra which are situated above the RE triangle. In contrast, the coupling is indicated to be weak in the much closer RE triangle adjacent layers which is coupled by the B6 octahedra. µeff (µB/R atom)
Tab. 6. Magnetic parameters of RE-B-C-N (RE = Dy, Ho, Er). Regarding the nature of the spin glass behavior in RB15.5CN, RB22C2N, and RB28.5C4 with HoB22C2N as representative, it was indicated as follows, that these are 2D spin glass systems . Strong frequency dependence of the ac-susceptibility was found, but could not be analyzed satisfactorily by the dynamical scaling theory of a three dimensional spin glass. A more detailed investigation of the behavior of relaxation times by Cole-Cole analysis (Fig. 21) showed that the data could be described well in terms of a generalized Arrhenius law (Fig. 22): ln(τ/τ0) ∝ T-(1+φν), τ0 = 5.3x10-6 s, 1+φν = 2.5
Fig. 21. Distribution function of relaxation times g(τ) of HoB22C2N . Adapted from Reference 224. 31
Fig. 22. Temperature dependence of τc . Adapted from Reference 224. It was demonstrated that this was not a super paramagnetic system (which also exhibits large frequency dependence) but a new 2-dimensional spin glass system . 2.8.3. REB18Si5: 3D LRO GdB18Si5 is rhombohedral (space group R-3m), with lattice constants a = b = 10.07 Å, c = 16.45 Å, and the only RE for this compound which exhibited a magnetic transition above 2 K. An antiferromagnetic transition was observed at TN = 3.2K (Fig. 23) and it was indicated that the spins are ordered in the a-b plane . A λ-type peak is observed in the magnetic specific heat at 3.2 K (Fig. 24), indicating long range order antiferromagnetic transition occurring in this system .
Fig. 23. Susceptibility of GdB18Si5 . Adapted from Reference 225. Fig. 24. Specific heat and magnetic entropy of GdB18Si5 . Adapted from Reference 225. 32
Fig. 25. In-plane susceptibility of GdB18Si5 . Adapted from Reference 226. An interesting dependence in the magnetization of GdB18Si5 at low magnetic fields was observed when the field was varied in-plane, with a reorientation of the spin, a spin flip, appearing to occur at fields below 300 G (Fig. 25). The origin of this behavior is likely due to the round spin of the gadolinium ions, but the actual observation of a spin flip at such low magnetic fields is interesting . Adapted from Reference 226. 3.6.4 REB66: Short range order (possible spin glass) The crystal structure of REB66 is cubic with space group of Fm3c. It is the most magnetically dilute rare earth boride, with a partial occupancy of approximately 50% on the rare earth site. Despite these facts, interesting magnetic behavior has been observed at measurable temperatures. Short range antiferromagnetic ordering was first observed in TbB66 and GdB66 at 0.34 K and 0.20 K, respectively . Neutron scattering measurements on 11B enriched samples showed a lack of long range ordering behavior, and the possibility of it being a spin glass was raised . Spin glass-like behavior was further reported for HoB66 with a characteristic temperature of 0.98 K . It is likely that GdB66, TbB66, and HoB66 are all spin glasses, which is not unreasonable considering the extremely large disorder in REB66 from the complex crystal structures and low partial occupancies of some sites, especially the rare earth atoms.
Summary As reviewed in this paper, the rare earth borides have yielded systems exhibiting a very rich tapestry of magnetism and superconductivity. The electron deficient boron frameworks, i.e. 2D atomic sheets and atomic cluster networks, are a good combination with rare earth atoms which are relatively localized and unobtrusively contribute outer shell electrons to stabilize the various structures formed. At the same time, the variations in the boron network structure and the 33
arrangement of the rare earth atoms occupying the voids, have resulted in a variety of interesting geometric magnetic behavior, from frustration derived from the Shastry-Sutherland lattice in REB4, to 1D, 2D, 3D behavior in the boron icosahedral compounds. Particularly fascinating physical properties of the RE borides, such as heavy fermions, Kondo insulators, controversial topological insulators, magnetic polaron-induced ferromagnetism, magnetic quadrupole ordering, etc. were reviewed. The origins of the unexpectedly low superconducting transition temperature in LuB12 are also discussed in detail. Another attractive feature of these compounds for research is that at least for the metallic rare earth borides, it is generally possible to grow relatively large single crystals of high quality, and this has spurred elucidation of physical properties. As described in this paper, there are still mysteries to solve, and the research on rare earth borides is expected to continue with interest. Acknowledgments This work was supported by the project VEGA 2-0032-16, by the Slovak Research and Development Agency under the contract no. APVV 17-0020 and by the project DAAD - SAS. Liquid nitrogen for experiments was sponsored by U.S. Steel Kosice. TM acknowledges support from JSPS KAKENHI JP16H06441, JP17H02749 and CREST JPMJCR15Q6, JPMJCR19Q4. References  J. Etourneau, Critical survey of rare-earth borides: occurrence, crystal chemistry and physical properties, J. Less-Common Met. 110 (1985) 267.  J. Etourneau and P. Hagenmuller, Structure and physical features of the rare-earth borides, Phil. Mag. B 52 (1985) 589.  P. Wachter, Intermediate valence and heavy fermions, in Handbook on the physics and chemistry of rare earths, edited by K.A. Gschneidner, L. Eyring, and S. Hüfner, Vol. 19, NorthHolland, Amsterdam, 1994, pp. 177.  T. Mori, Higher borides, in Handbook on the physics and chemistry of rare-earths, edited by K.A. Gschneidner, Jr., J.-C. Bunzli, and V. Pecharsky, Vol. 38, North-Holland, Amsterdam, 2008, pp. 105.  T. Mori, The rare earth elements: Fundamentals and application, edited by D. Atwood John Wiley & Sons Ltd., Chichester, 2012, pp. 263.  T. Mori, Thermoelectric and magnetic properties of rare earth borides: Boron cluster and layered compounds, J. Solid State Chem. 275 (2019) 70.  C. Buzea and T. Yamashita, Review of the superconducting properties of MgB2, Supercond. Sci. Technol. 14 (2001) R115-R146.  W. Li, J. Kang, S. Fu, Y. Hu, P. Hu, M. Zhu, Y. Li, Rare earth doping effects on superconducting properties of MgB2: A review, J. Rare Earths 37 (2019) 124.
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Comprehensive review of magnetism and superconductivity of rare earth borides Borides heavy fermions, Kondo insulators, topological insulators, etc. Borides frustrated magnetism, polaron-induced ferro-, quadrupole ordering, etc.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: