Antiferroquadrupolar Research Articles

Symmetry Analysis of Zero-Field Antiferroquadrupole Order in CeB6: Extremely Low-Frequency 11B-NQR Study

The zero-field state in the antiferroquadrupole (AFQ) ordered phase of CeB6 has been investigated by a 11B-nuclear quadrupole resonance (NQR) measurement that is sensitive to changes in local charge distribution. In order to achieve this experiment, we have improved our low-frequency NQR techniques, so that the 11B-NQR spectrum has been successfully observed at an NQR frequency νQ ∼ 560 kHz, exceptionally low frequency among direct NQR measurements on strongly correlated electron systems. The temperature dependence of the NQR spectrum reveals that νQ rapidly decreases just below AFQ ordering temperature TQ = 3.3 K, while the spectral shape does not show any sign of site splitting below TQ. The latter is incompatible with the theoretical prediction based on the Oxy-type AFQ order, suggesting that the ordered state in the zero-field region differs from that in the nonzero-field region. Some new possible interpretations to explain the obtained results are proposed.

Possible New State in CeB6 Uncovered by Extremely Low-Frequency Nuclear Quadrupole Resonance Experiments

Extremely low-frequency nuclear quadrupole resonance analyses of CeB6, a prototypical antiferroquadrupole (AFQ) material, indicate the possible existence of a new state in the zero-field AFQ ordered phase.

First-Principles Study of the RKKY Interaction and the Quadrupole Order in the Pr 1-2-20 Systems PrT2Al20 (T = Ti, V)

Electronic states and quadrupole orders in the Pr 1-2-20 systems PrT2Al20 (T=Ti, V) are investigated on the basis of the first-principles calculations. The effective 196 orbital model is derived to reproduce the first-principles electronic structures of LaT2Al20 (T=Ti, V) without contribution from the Pr 4f electrons which are considered to be well localized and is employed to calculate the Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions between quadrupole and octupole moments of the Pr ions. Within the random phase approximation for the RKKY Hamiltonian, the most divergent susceptibility is found to be the quadrupole one for the wave vector Q = (0,0,0) in the case of PrTi2Al20 while that for Q = (pi/a,0,pi/a) in the case of PrV2Al20 as consistent with experimental observations in the both cases which exhibit the ferro-quadrupole (FQ) and the antiferro-quadrupole (AFQ) orders, respectively. We also discuss the ordered states using the mean-field approximation and find that, in the case of PrTi2Al20, the 1st-order phase transition to the O20 FQ order with a tiny discontinuity takes place as predicted by the Landau theory. In the case of PrV2Al20, the system exhibits two distinct O22 AFQ orders, AFQ-I and AFQ-II, and shows subsequent two phase transitions, the 2nd-order one from normal to AFQ-I and the 1st-order one from AFQ-I to AFQ-II, that may be responsible for the double transitions observed by specific heat measurements.

Crystal-field excitations and quadrupolar fluctuations of 4f-electron systems studied by polarized light scattering

We present Raman-scattering results for three materials, CeB6, TbInO3, and YbRu2Ge2, to illustrate the essential aspects of crystal-field (CF) excitations and quadrupolar fluctuations of 4f-electron systems. For CF excitations, we illustrate how the 4f orbits are split by spin-orbit coupling and CF potential by presenting spectra for inter- and intra-multiplet excitations over a large energy range. We discuss identification of the CF ground state and establishment of low-energy CF level scheme from the symmetry and energy of measured CF excitations. In addition, we demonstrate that the CF linewidth is a sensitive probe of electron correlation by virtue of self-energy effect. For quadrupolar fluctuations, we discuss both ferroquadrupolar (FQ) and antiferroquadrupolar (AFQ) cases. Long-wavelength quadrupolar fluctuations of the same symmetry as the FQ order parameter persists well above the transition temperature, from which the strength of electronic intersite quadrupolar interaction can be evaluated. The tendency towards AFQ ordering induces ferromagnetic correlation between neighboring 4f-ion sites, leading to long-wavelength magnetic fluctuations.

Heavy Fermion Metal CeB6 in SubTHz and THz Range: The Electron Spin Resonance and Neutron Scattering Studies

Cerium hexaboride attracts attention of solid-state community for almost 50 years. Nowadays, this material is considered as a unique example of a system with orbital ordering resulting in formation of low-temperature magnetic phase with quadrupolar order, the so-called antiferroquadrupole phase. In crystal electric field 4f 1 state of Ce3+ splits into Γ7 doublet and Γ8 quartet, the latter is believed to be the ground state, as long as just Γ8 allows quadrupolar effects. An additional tool for studying rich physics of CeB6 appeared since the discovery in this material the ESR, which is specific to antiferroquadrupole (AFQ) phase and is missing in the paramagnetic phase. In the present review, we have brought together electron spin resonance and inelastic neutron scattering data, which allowed obtaining most detailed structure of the magnetic resonance modes in the AFQ phase in a wide-frequency range up to 0.6 THz. It is shown that apart paramagnetic resonance main mode with homogeneous precession of the magnetization, there are antiferromagnetic resonance modes as well, some of them existing in the AFQ phase. Analysis of the g factors demonstrates that available theories based on Γ8 ground state are facing difficulties with accounting experimental data, thus providing a challenge for developing of a theory correctly describing dynamic magnetic properties of this interesting material.

Structural Phase Transition in Antiferromagnet CeCoSi Compared to Isostructural LaCoSi and PrCoSi

Synchrotron X-ray powder diffraction experiments were carried out at T = 300 K under pressures on a tetragonal metallic compound CeCoSi exhibiting a possible antiferroquadrupole (AFQ) ordering at l...

Magnetoelectric Effects and Charge-Imbalanced Solenoids: Antiferro Quadrupole Orders in a Diamond Structure

We study magnetoelectric (magneto-current) effects in a diamond structure under antiferro quadrupole (AFQ) orders. The AFQ orders break the spacial inversion symmetry and cause the current-induced magnetization. The current-induced magnetization strongly depends on the types of the order parameters and the direction of the current. This gives a way to the experimental identification of AFQ order parameters. We also discuss the current-induced magnetization under the AFQ orders in the diamond structure can be intuitively understood in terms of charge-imbalanced solenoids.

Revealing the Nature of Antiferroquadrupolar Ordering in Cerium Hexaboride: CeB_{6}.

The cerium hexaboride (CeB_{6}) f-electron compound displays a rich array of low-temperature magnetic phenomena, including a "magnetically hidden" order, identified as multipolar in origin via advanced x-ray scattering. From first-principles electronic-structure results, we find that the antiferroquadrupolar (AFQ) ordering in CeB_{6} arises from crystal-field splitting and yields a band structure in agreement with experiments. With interactions of p electrons between Ce and B_{6} being small, the electronic state of CeB_{6} is suitably described as Ce(4f^{1})^{3+}(e^{-})(B_{6})^{2-}. The AFQ state of orbital spins is caused by an exchange interaction induced through spin-orbit interaction, which also splits the J=5/2 state into a Γ_{8} ground state and a Γ_{7} excited state. Within the smallest antiferromagnetic (AFM) (111) configuration, an orbital-ordered AFQ state appears during charge self-consistency, and it supports the appearance of a "hidden" order. Hydrostatic pressure (either applied or chemically induced) stabilizes the AFM (AFQ) states over a ferromagnetic one, as observed at low temperatures.

Appearance of the octupole ordered phase IV in CexLa1−xB6

We investigated the physical properties of ${\mathrm{Ce}}_{x}{\mathrm{La}}_{1\ensuremath{-}x}{\mathrm{B}}_{6}$ at $x\ensuremath{\sim}0.8$, below which the ${T}_{\ensuremath{\beta}}$-type antiferro-octupole (AFO) ordered phase IV appears as a result of the larger suppression rate of ${T}_{\mathrm{Q}}$ than ${T}_{\mathrm{N}}$ by La doping. The most important result is that while the peak of the specific heat at ${T}_{\mathrm{Q}}$ is rapidly suppressed and broadened by La doping, that at ${T}_{\mathrm{IV}}$ is sharp and large. This indicates that although the ${T}_{\ensuremath{\beta}}$-AFO order in the phase IV is robust against the local lattice distortion induced by La doping, the ${O}_{xy}$-type antiferroquadrupole (AFQ) ordered phase II is very weak. The ${T}_{xyz}$-AFO interaction is robust against La doping from the observation of the pronounced enhancement of ${T}_{\mathrm{Q}}$ even in a small $x$ region. Based on these La-doping effect of the multipole interactions, we carried out the mean-field calculation for the four-sublattice model to reproduce the magnetic phase diagrams of ${\mathrm{Ce}}_{x}{\mathrm{La}}_{1\ensuremath{-}x}{\mathrm{B}}_{6}$. Based on the calculated results, we propose that the small splitting of the quartet is induced by La doping in phase I to explain the magnetic phase diagram for $x<0.65$. We could obtain the calculated results roughly consistent with the experimental results, although there appear new problems. We classified the mechanisms of the four different types of the competition among the four interactions with roughly the same magnitude, which induce the interesting and complicated properties in ${\mathrm{Ce}}_{x}{\mathrm{La}}_{1\ensuremath{-}x}{\mathrm{B}}_{6}$.

La Substitution and Pressure Studies on CeCoSi: A Possible Antiferroquadrupolar Ordering Induced by Pressure

The La-substitution and pressure effects on antiferromagnet CeCoSi with a tetragonal crystal structure were investigated in order to clarify the origin of the pressure induced ordered phase (PIOP) with an enormously high transition temperature of T0 ∼ 38 K. By substituting Ce with La, the PIOP shifts to high pressures, and the maximum value of T0 is suppressed, indicating that the PIOP originates from a Ce–Ce interaction. At T0, the magnetic susceptibility exhibits not a cusp, but an increase with decreasing temperature, as is frequently observed in antiferroquadrupolar (AFQ) ordering. Based on these results, we propose that a possible origin of the PIOP is an AFQ ordered phase of Ce-4f electron. Due to the lack of local inversion symmetry at the Ce sites, the AFQ order could be accompanied by odd parity multipolar moments. Since there is no clear evidence of successive transitions below T0, the remaining Kramers degeneracy could be lifted by the Kondo coupling.

Doping-induced redistribution of magnetic spectral weight in the substituted hexaborides Ce1−xLaxB6 and Ce1−xNdxB6

We investigate the doping-induced changes in the electronic structure of CeB$_6$ on a series of substituted Ce$_{1-x}R_x$B$_6$ samples ($R$ = La, Nd) using diffuse neutron scattering. We observe a redistribution of magnetic spectral weight across the Brillouin zone, which we associate with the changes in the Fermi-surface nesting properties related to the modified charge carrier concentration. In particular, a strong diffuse peak at the corner of the Brillouin zone ($R$ point), which coincides with the propagation vector of the elusive antiferroquadrupolar (AFQ) order in CeB$_6$, is rapidly suppressed by both La and Nd doping, like the AFQ order itself. The corresponding spectral weight is transferred to the $X(00\frac{1}{2})$ point, ultimately stabilizing a long-range AFM order at this wave vector at the Nd-rich side of the phase diagram. At an intermediate Nd concentration, a broad diffuse peak with multiple local maxima of intensity is observed around the $X$ point, evidencing itinerant frustration that gives rise to multiple ordered phases for which Ce$_{1-x}$Nd$_x$B$_6$ is known. On the La-rich side of the phase diagram, however, dilution of the magnetic moments prevents the formation of a similar $(00\frac{1}{2})$-type order despite the presence of nesting. Our results demonstrate how diffuse neutron scattering can be used to probe the nesting vectors in complex f-electron systems directly, without reference to the single-particle band structure, and emphasize the role of Fermi surface geometry in stabilizing magnetic order in rare-earth hexaborides.

Electron nematic effect induced by magnetic field in antiferroquadrupole phase of CeB6

Spatial anisotropy generated spontaneously in the translationally invariant metallic phase, i.e. electron nematic effect, addresses a great challenge for both experimentalists and theoreticians. An interesting option for the realization of the electron nematic phase is provided by the system with orbital ordering, as long as both orbitally ordered states and electron nematic phases possess broken spatial symmetry. Here we report the detailed study of the angular dependences of the magnetoresistance in the orbitally ordered antiferroquadrupole (AFQ) phase of CeB6. Our data allowed revealing the electron nematic effect, which develops when magnetic field exceeds a critical value of 0.3–0.5T. As a result, new transition inside the AFQ phase corresponding to the change of the symmetry of magnetic scattering on spin fluctuations in CeB6 is discovered.

Influence of Pt substitution on magnetic properties of multipolar ordering compounds Ce(Pd,Pt)3S4

We have studied the magnetic properties of the multipolar ordering compounds Ce(Pd1−xPtx)3S4 with 0.00 ≤x≤0.53 by means of magnetic susceptibility and magnetization measurements. In CePd3S4, a simultaneous phase transition of the antiferro quadrupolar (AFQ) ordering and ferro magnetic (FM) ordering has been observed at 6.3K. It has been suggested that the primary order parameter of CePd3S4 is the quadrupole moments, and it has not been understood why the FM ordering occurs at very high temperature which is almost the same magnetic transition temperature of GdPd3S4. GdPd3S4 shows an antiferromagnetic (AFM) transition at 5.8K. With increasing Pt substitution in CePd3S4, the FM transition temperature TC(x) is rapidly suppressed to 2.4K for x≃0.3 and approaches asymptotically to 1.9K(x=0.53). The results of magnetization curve suggest that the ordered state below TC(x) remains FM and AFQ ordered state for the whole range of x. For x≥0.29, TC(x) reaches at around 2K, a new AFM transition was observed at TN(x)≃7K. We determined the T−x phase diagram, and discuss the phase transitions at TC(x) and TN(x). The results suggest the possibility of the presence of the correlation between the magnetic interaction and the quadrupole interaction, and the correlation is not understood based on the previous multipolar model.

Antiferroquadrupolar Ordering in Quadrupolar Kondo Lattice of Non-Kramers System PrTa2Al20

Single crystals of PrTa2Al20 have been investigated by means of single-crystal structural analysis and measurements of magnetization, specific heat, and electrical resistivity. The crystalline-electric-field level scheme of the Pr ions has a nonmagnetic Γ3 doublet ground state and a Γ5 magnetic excited state with an energy separation of 53 K. The 4f-electron contribution to the electrical resistivity shows −log T magnetic Kondo scattering above 50 K and a downward curvature characteristic of a quadrupolar Kondo lattice formation below 20 K. A phase transition appears at 0.65 K in zero field and shifts to higher temperatures in applied fields, indicating that this transition is antiferroquadrupolar (AFQ) in nature. The largely enhanced Sommerfeld coefficient ∼1.5 J/(mol K2) in the AFQ state may indicate the formation of heavy quasiparticles. The hierarchically arranged sequence of the magnetic Kondo regime, quadrupolar Kondo lattice regime, and AFQ ordered state in the wide temperature range 0.2–300 K demo...

Effect of Zn-site substitution with Ga on non-Fermi liquid behavior in PrIr2Zn20

PrIr2Zn20 exhibits an antiferroquadrupolar (AFQ) order at TQ = 0.11K, above which temperature the electrical resistivity ρ(T) shows an upward curvature and the magnetic specific heat divided by temperature Cm/T follows − lnT dependence. The non-Fermi Liquid (NFL) behaviors have suggested formation of a quadrupole Kondo lattice. In the present work, we have studied the effect of Ga substitution for Zn on the NFL behavior by the measurements of the specific heat C, the magnetic susceptibility χ, and ρ of PrIr2Zn20-xGax (0 ≤ x ≤ 0.25). With increasing x, the characteristic temperature T0 defined as the temperature where the magnetic entropy Sm reaches (3/4)Rln2 is increased by a factor of 3.5. Similarly, another characteristic temperature TR below which ρ(T) starts decreasing with the upward curvature increases with x by a factor of 1.2. The increments of both T0 and TR may be attributed to the possible split of the Γ3 doublet by symmetry lowering of the Pr sites. Otherwise, the quadrupole Kondo lattice would be stabilized by the enhanced c-f hybridization due to the increment of the 4p electronic density and/or the chemical pressure effect.

Anisotropic B–T Phase Diagram of Non-Kramers System PrRh2Zn20

We have investigated the low temperature quadrupolar phenomena of the non-Kramers system PrRh2Zn20 under magnetic fields in the [100] and [110] directions. Our experiments reveal the B-T phase diagram of PrRh2Zn20 involving four electronic states regardless of the field direction, namely, a non-Fermi liquid (NFL) state, an antiferro-quadrupolar (AFQ) ordered state, a novel heavy-fermion (HF) state, and a field-induced singlet (FIS) state. In the wide range of the NFL state, the resistivity can be well scaled by a characteristic temperature, suggesting the realization of the quadrupole Kondo effect. The HF state exhibits a Fermi liquid behavior with a large A coefficient of the T^2 term in the resistivity, suggesting the formation of nontrivial heavy quasi-particles. The FIS state results from the considerable splitting of a non-Kramers doublet by a magnetic field. The phase diagram shows a large anisotropy with respect to the field direction. It is found that the anisotropy of the phase diagram can be explained in terms of that of the energy splitting of the non-Kramers doublet by a magnetic field. This indicates that the low temperature properties of PrRh2Zn20 are governed by the non-Kramers doublet, namely, quadrupole degrees of freedom. Since a similar phase diagram has been obtained for the related compound PrIr2Zn20, it is expected that the B-T phase diagram constructed in this work is universal throughout non-Kramers systems.

Magnetic resonance anisotropy in CeB6: an entangled state of the art.

Electron spin resonance (ESR) in strongly correlated metals is an exciting phenomenon, as strong spin fluctuations in this class of materials broaden extremely the absorption line below the detection limit. In this respect, ESR observation in CeB6 provides a unique chance to inspect Ce3+ magnetic state in the antiferroquadrupole (AFQ) phase. We apply the original high frequency (60 GHz) experimental technique to extract the temperature and angular dependences of g-factor, line width and oscillating magnetization. Experimental data show unambiguously that the modern ESR theory in the AFQ phase considering the Γ8 ground state of Ce3+ ion completely fails to predict both the g-factor magnitude and its angular dependence. Alignment of the external magnetic field along [100] axis induces a strong (more than twofold) broadening of ESR line width with respect to the other crystallographic directions and results also in the anomalous temperature dependences of the g-factor and oscillating magnetization. In this experimental geometry the latter parameter surprisingly exceeds total static magnetization by 20% at T* ~ 2.5 K. We argue that the unusual physical picture of ESR in CeB6 may be strongly affected by spin fluctuations and dynamic collective effects predominantly pronounced in [100] direction.

Classical Monte Carlo Study for Antiferro Quadrupole Orders in a Diamond Lattice

We investigate antiferro quadrupole orders in a diamond lattice under magnetic fields by Monte Carlo simulations for two types of classical effective models. One is an XY model with Z_3 anisotropy, and the other is a two-component phi^4 model with a third-order anisotropy. We confirm that the universality class of the zero-field transition is that for the three-dimensional XY model. Magnetic field corresponds to a Z_3 field in the effective model, and under this field, we find that collinear and canted antiferro-quadrupole orders compete. Each phase is characterized by symmetry breaking in the sector of (sublattice Z_2)x(reflection Z_2 for the order parameter). When Z_3 anisotropy and magnetic field vary, it turns out that this system is a good playground for various multicritical points; bicritical and tetracritical points emerge in a finite field. Another important finding is about the scaling of parasitic ferro quadrupole order at the zero-field critical point. This is the secondary order parameter induced by the primary antiferro order, and its critical exponent beta'=0.815 clearly differs from the expected value that is twice the value for the primary order parameter. The corresponding correlation length exponent is also different, nu'=0.597(12). We also discuss relation of the present effective quadrupole models with the 3-state Potts model as well as implication to undertanding of orbital orders in Pr-based 1-2-20 compounds.

Non-Fermi Liquid Properties in a Cubic Pr-Based Compound PrPb3 under Magnetic Fields

We have studied the electrical resistivity ρ of the Pr-based cubic compound PrPb3 with the Γ3 doublet ground state. The temperature dependence of the resistivity is found to exhibit a non-Fermi liquid behavior with convex curve at high temperatures, as suggestive of the putative realization of the quadrupole Kondo effect. At low temperatures under magnetic fields, we observe anomalies associated with antiferroquadrupole (AFQ) ordering, incommensurate/commensurate transition inside the AFQ phase, and field-induced phase transition. The constructed phase diagram based on these observations well reproduces the one determined by the specific heat and magnetization measurements. Deep inside each phase, the resistivity turns to show the Fermi liquid behavior with the quadratic temperature dependence. The estimated slope of the T2 term is almost independent of field in contrast to our previous work of PrIr2Zn20 and PrRh2Zn20 in which the slope is largely enhanced at the edge of the AFQ phase.

Perturbation on hyperfine-enhancedPr141nuclear spin dynamics associated with antiferroquadrupolar order inPrV2Al20

The nature of multipolar order and hyperfine-enhanced (HE) $^{141}$Pr nuclear spin dynamics in PrV$_2$Al$_{20}$ was investigated using the muon spin relaxation technique. No explicit sign of time-reversal symmetry breaking was found below the multipolar order temperature $T_Q\sim 0.6$ K in a zero applied field as anticipated on the basis of the antiferroquadrupolar (AFQ) order picture proposed by Sakai and Nakatsuji [J. Phys. Soc. Jpn. 80, 063701 (2011)]. Further evidence of the nonmagnetic ground state was obtained from the observation of HE $^{141}$Pr nuclear spin fluctuations in the MHz scale. A marked increase in the muon spin-lattice relaxation rate (1/$T_{\rm 1,\mu}$) was observed below 1 K with decreasing temperature, which was attributed to the perturbation on the HE $^{141}$Pr nuclear spin dynamics associated with the development of AFQ correlations. The longitudinal field dependence of 1/$T_{\rm 1,\mu}$ revealed that the enhanced $^{141}$Pr nuclear spin accidentally has an effective gyromagnetic ratio close to that of the muon.