Crystalline Electric Field States Research Articles

Dependence of transport coefficients of Yb(Rh1−xCox)2Si2 intermetallics on temperature and cobalt concentration

Dependence of transport coefficients of the Yb(Rh$_{1-x}$Co$_x$)$_2$Si$_2$ series of alloys on temperature and cobalt concentration is explained by an asymmetric Anderson model which takes into account the exchange scattering of conduction electrons on ytterbium ions and the splitting of 4$f$-states by the crystalline electric field (CEF). The substitution of rhodium by cobalt is described as an increase of chemical pressure which reduces the exchange coupling and the CEF splitting. The scaling analysis and numerical NCA solution of the model show that the effective degeneracy of the 4$f$-state at a given temperature depends on the relative magnitude of the Kondo scale and the CEF splitting. Thus, we find that dependence of the thermopower, $S(T)$, on temperature and cobalt concentration can be understood as an interplay of quantum fluctuations, driven by the Kondo effect, and thermal fluctuations, which favor a uniform occupation of the CEF states. The theoretical model captures all the qualitative features of the experimental data and it explains the evolution of the shape of $S(T)$ with the increase of cobalt concentration.

Superconductivity in a multiorbital model for the Γ3 crystalline electric field state

Superconductivity in a three-orbital model for the Γ3 crystalline electric field (CEF) ground state is investigated by applying the random phase approximation. In this model, we assume an antiferromagnetic interaction between the Γ7 and Γ8 orbitals to stabilize the Γ3 CEF state among f2 states. This interaction may induce spin-singlet pairing between electrons in these orbitals. If we consider the onsite pairing state composed of the Γ7 and Γ8 orbitals, it has the Eg symmetry. Indeed, we find that the Eg spin-singlet superconducting state is realized in a wide parameter region. Possible coexistence of the superconductivity and quadrupole order is also discussed.

Anisotropic Superconductivity Emerging from the Orbital Degrees of Freedom in a Γ3 Non-Kramers Doublet System

We study superconductivity in a three-orbital model for $f^2$ ions with the $\Gamma_3$ crystalline electric field (CEF) ground state. An antiferromagnetic interaction between the $\Gamma_7$ and $\Gamma_8$ orbitals is introduced to stabilize the $\Gamma_3$ CEF state. This interaction also works as an on-site attractive interaction for spin-singlet pairing between electrons in these orbitals. The interorbital pairing state composed of the $\Gamma_7$ and $\Gamma_8$ orbitals on the same site has the $E_g$ symmetry. Indeed, by applying the random phase approximation, we find that the $E_g$ spin-singlet superconducting state is realized over a wide parameter range.

Slave Boson Analysis of f2-Configuration Systems with Γ1 Singlet Crystalline-Electric-Field Ground State under the Cubic Symmetry

We investigated electron phases in the $f^2$-configuration system with the $\Gamma_1$ crystalline-electric-field (CEF) ground state in cubic symmetry. An extended three-orbital periodic Anderson model, which consists of $j=5/2$ states, is evaluated by using the rotationally invariant slave boson formalism. As a result, we found three phases at zero temperature: the $\Gamma_1$ CEF singlet phase, and two kinds of Fermi liquid (FL). One of the FL arises owing to the presence of the $\Gamma_5$ CEF state, while the other forms a quasi-degenerated CEF state. All these phases are separated by first-order transitions: the charge-transfer transition and the CEF energy-level crossing. We also found heavy quasiparticle behaviors in the intermediate valence region between the $f^2$ and $f^3$ configurations. In this region, the mass enhancement factor exhibits a peak structure by increasing the hybridization. Physical properties of these phases and transitions are discussed.

Multipole interactions of Γ3 non-Kramers doublet systems on cubic lattices

We investigate the multipole interactions between adjacent f2 ions with the Γ3 crystalline electric field (CEF) ground state. In this study, we consider only the Γ8 states as the one-electron states. We construct the Γ3 doublet state of the electrons in the Γ8 states, and indeed, it can become the CEF ground state of a simple model Hamiltonian. To derive the multipole interactions, we apply the second-order perturbation theory with respect to the intersite hopping. We obtain no multipole interaction for a simple cubic lattice and octupole interactions for bcc and fcc lattices. We discuss the similarities and differences in the multipole interactions among the models with the local Γ3 CEF states.

Key Role of Hybridization between Actinide 5<i>f</i> and Oxygen 2<i>p</i> Orbitals for Electronic Structure of Actinide Dioxides

In order to promote our understanding on electronic structure of actinide dioxides, we construct a tight-binding model composed of actinide 5f and oxygen 2p electrons, which is called f-p model. After the diagonalization of the f-p model, we compare the eigen-energies in the first Brillouin zone with the results of relativistic band-structure calculations. Here we emphasize a key role of f-p hybridization in order to understand the electronic structure of actinide dioxides. In particular, it is found that the position of energy levels of Г7 and Г8 states determined from crystalline electric field (CEF) potentials depends on the f-p hybridization. We investiagte the values of the Slater-Koster integrals for f-p hybridization, (fpσ) and (fpπ), which reproduce simultaneously the local CEF states and the band-structure calculation results. Then, we find that the absolute value of (fpπ) should be small in comparison with (fpσ) = 1 eV. The small value of |(fpπ)| is consistent with the condition to obtain the octupole ordering in the previous analysis of the f-p model.

Transport, thermal, and magnetic properties in heavy-fermion superconductor Ce 2CoIn 8

We have grown high purity single crystals of Ce 2CoIn 8 using the self-flux technique, and have investigated its transport, thermal, and magnetic properties, including the anisotropic features. Single crystals of Ce 2CoIn 8 were grown in the lower temperature region to avoid the formation of un-wanted phases such as CeCoIn 5 and CeIn 3 impurities. The results of the structural and physical measurements imply that the present single crystals have high purity. The electrical resistivity and specific heat measurements demonstrate that Ce 2CoIn 8 has a superconducting ground state with a distinct non-Fermi liquid character. This indicates that the superconductivity in Ce 2CoIn 8 arises out of the verge of the underlying quantum critical instability mediated by the antiferromagnetic correlations. Additionally, we investigate the crystalline electric field (CEF) energy scheme based on the temperature dependence of the specific heat and the anisotropic features in the susceptibility. We propose one of the CEF level schemes calculated on the basis of the CEF model that the first and second CEF states are located at Δ 1 = 82 K and Δ 2 = 102 K above the ground state doublet, respectively.

Field-dependent mass enhancement inPr1−xLaxOs4Sb12from aspherical Coulomb scattering

The scattering of conduction electrons by crystalline electric field (CEF) excitations may enhance their effective quasiparticle mass similar to scattering from phonons. A well-known example is Pr metal where the isotropic exchange scattering from inelastic singlet-singlet excitations causes the mass enhancement. An analogous mechanism may be at work in the skutterudite compounds ${\text{Pr}}_{1\ensuremath{-}x}{\text{La}}_{x}{\text{Os}}_{4}{\text{Sb}}_{12}$ where close to $x=1$ the compound develops heavy quasiparticles with a large specific heat $\ensuremath{\gamma}$. There the low-lying CEF states are a singlet ground state and a triplet at $\ensuremath{\Delta}=8\text{ }\text{K}$. Due to the tetrahedral CEF the main scattering mechanism must be the aspherical Coulomb scattering. We derive the expression for mass enhancement in this model including also the case of dispersive excitations. We show that for small to moderate dispersion there is a strongly field-dependent mass enhancement due to the field-induced triplet splitting. It is suggested that this effect may be seen in ${\text{Pr}}_{1\ensuremath{-}x}{\text{La}}_{x}{\text{Os}}_{4}{\text{Sb}}_{12}$ with suitably large $x$ when the dispersion is small.

Theory of induced quadrupolar order in tetragonalYbRu2Ge2

The tetragonal compound $\mathrm{Yb}{\mathrm{Ru}}_{2}{\mathrm{Ge}}_{2}$ exhibits a nonmagnetic transition at ${T}_{0}=10.2\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and a magnetic transition at ${T}_{1}=6.5\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ in zero magnetic field. We present a model for this material based on a quasiquartet of ${\mathrm{Yb}}^{3+}$ crystalline electric field (CEF) states and discuss its mean-field solution. Taking into account the broadening of the specific heat jump at ${T}_{0}$ for magnetic field perpendicular to [001] and the decrease of ${T}_{0}$ with magnetic field parallel to [001], it is shown that ferroquadrupole order of either ${\mathrm{O}}_{2}^{2}$- or ${\mathrm{O}}_{xy}$-type are prime candidates for the nonmagnetic transition. Considering the matrix element of these quadrupole moments, we show that the lower CEF states of the level scheme consist of a ${\ensuremath{\Gamma}}_{6}$ and a ${\ensuremath{\Gamma}}_{7}$ doublet. This leads to induced type of ${\mathrm{O}}_{2}^{2}$ and ${\mathrm{O}}_{xy}$ quadrupolar order parameters. The quadrupolar order introduces exchange anisotropy for planar magnetic moments. This causes a spin-flop transition at low fields perpendicular [001] which explains the observed metamagnetism. We also obtain a good explanation for the temperature dependence of magnetic susceptibility and specific heat for fields both parallel and perpendicular to the [001] direction.

Electronic Higher Multipoles in Solids

This tutorial review discusses basic features of electronic multipoles and their ordering phenomena. Starting with classification of electric and magnetic higher multipoles under spherical symmetry, we proceed to two representative cases with strong spin-orbit interaction. First, staggered octupole orders in Ce1−xLaxB6 and SmRu4P12 are discussed using the crystalline electric field states. Secondly, a scalar component of even higher multipoles in Pr skutterudites is studied where ordering of fourth-rank tensor (hexadecapole) and sixth-rank one (hexacontatetrapole) has been observed. Possibility of a reentrant transition from the multipole ordered state is discussed. Briefly mentioned are multipole aspects of Pomeranchuk instability of itinerant electrons and collective motion of nucleus. It is argued that the multipoles provide a new framework to study the metal-insulator transition.

Relativistic band-structure calculations for electronic properties of actinide dioxides

Energy band structures of actinide dioxides AnO 2 ( An = Th , U, Np, and Pu) are investigated by a relativistic linear augmented-plane-wave method with the exchange–correlation potential in a local density approximation (LDA). It is found in common that the energy bands in the vicinity of the Fermi level are mainly due to the hybridization between actinide 5f and oxygen 2p electrons. By focusing on the crystalline electric field states, we point out the problem in the application of the LDA to AnO 2 .

Effective Crystalline Electric Field Potential in a j– j Coupling Scheme

We propose an effective model on the basis of a $j$-$j$ coupling scheme to describe local $f$-electron states for realistic values of Coulomb interaction $U$ and spin-orbit coupling $\lambda$, for future development of microscopic theory of magnetism and superconductivity in $f^n$-electron systems, where $n$ is the number of local $f$ electrons. The effective model is systematically constructed by including the effect of a crystalline electric field (CEF) potential in the perturbation expansion in terms of $1/\lambda$. In this paper, we collect all the terms up to the first order of $1/\lambda$. Solving the effective model, we show the results of the CEF states for each case of $n$=2$\sim$5 with $O_{\rm h}$ symmetry in comparison with those of the Stevens Hamiltonian for the weak CEF. In particular, we carefully discuss the CEF energy levels in an intermediate coupling region with $\lambda/U$ in the order of 0.1 corresponding to actual $f$-electron materials between the $LS$ and $j$-$j$ coupling schemes. Note that the relevant energy scale of $U$ is the Hund's rule interaction. It is found that the CEF energy levels in the intermediate coupling region can be quantitatively reproduced by our modified $j$-$j$ coupling scheme, when we correctly take into account the corrections in the order of $1/\lambda$ in addition to the CEF terms and Coulomb interactions which remain in the limit of $\lambda$=$\infty$. As an application of the modified $j$-$j$ coupling scheme, we discuss the CEF energy levels of filled skutterudites with $T_{\rm h}$ symmetry.

Orbital-Controlled Superconductivity in f-Electron Systems

We propose a concept of superconductivity controlled by orbital degree of freedom taking CeMIn5 (M= Co, Rh, and Ir) as typical examples. A microscopic multiorbital model for CeMIn5 is analyzed by fluctuation exchange approximation. Even though the Fermi-surface structure is unchanged, the ground state is found to change significantly among paramagnetic, antiferromagnetic, and d-wave superconducting phases, depending on the dominant orbital component in the band near the Fermi energy. We show that our picture naturally explains the different low-temperature properties of CeMIn5 by carefully analyzing the crystalline electric field states.

Specific Heat Study on Sm-based Filled Skutterudite Phosphides SmT4P12 (T=Fe, Ru and Os)

Sm-based filled skutterudite phosphides SmFe 4 P 12 , SmRu 4 P 12 and SmOs 4 P 12 synthesized at high temperatures and high pressures have been studied by specific heat measurements in magnetic fields. SmFe 4 P 12 shows an anomaly at 1.6 K attributed to a ferromagnetic transition. SmRu 4 P 12 shows a metal–insulator transition at 16 K. This transition consists of two successive transitions, which suggests orbital and antiferromagnetic orderings. SmOs 4 P 12 shows a sharp anomaly at 4.5 K due to an antiferromagnetic ordering. We discuss the characteristic behavior of those phase transitions and the crystalline electric field state in Sm T 4 P 12 ( T =Fe, Ru and Os). Interestingly, the magnetic entropy of SmFe 4 P 12 have only 0.16 R ln 2 below the ferromagnetic ordering temperature. This suggests a strong spin fluctuation effect. On the other hand, the variation of magnetic entropy in SmRu 4 P 12 and SmOs 4 P 12 reaches to nearly R ln 4 below the ordering temperature.

Elastic constants of PrNi 5

The elastic constants of intermetallic compound PrNi 5 with a hexagonal crystal structure have been investigated by the ultrasonic measurement. For the present single crystal we have observed the elastic softening of C 44 below 40 K. It is ascribed to the quadrupolar response for the crystalline electric field state of PrNi 5.

Dimer splitting of Er 3+ in Cs 3Er 2X 9 (X=Cl,Br): mystery solved

Inelastic neutron scattering (INS) measurements were performed on the rare earth dimer compounds Cs 3Er 2X 9 (X=Cl,Br). The crystalline electric field (CEF) parameters were determined. The lack of any exchange splitting associated with the CEF transitions can be explained by the large single-ion anisotropy for Er. Specific heat and susceptibility data support the picture of an almost pure Ising system, which splits the lowest CEF state in the presence of the dimer interaction into a non-magnetic ground-state doublet and a magnetic excited doublet without any allowed transition between these two dimer states.

Intermultiplet Crystal-Field Transitions in SmBa2Cu3O7.

Inelastic neutron scattering has been employed to study the crystalline-electric-field (CEF) excitations of ${\mathrm{Sm}}^{3+}$ ions in SmB${\mathrm{a}}_{2}$C${\mathrm{u}}_{3}$${\mathrm{O}}_{7}$. We have been able to assign the energetic ordering of all the three and four CEF states in the ground-state and first-excited $J$ multiplets $^{6}H_{5/2}$ and $^{6}H_{7/2}$, respectively. A set of CEF parameters was derived which excellently describes the observed energy spectra.

Resistivity anomalies in heavy-fermion CeCu 2Sb 2 and CeNi 2Sb 2

Temperature dependences of the electrical resistivity, magnetic susceptibility and specific heat of the tetragonal compounds CeCu 2Sb 2 and CeNi 2Sb 2 have been studied. The resistivity of both compounds displays a sharp maximum followed by a shallow minimum at higher temperature, indicative of heavy-fermion behavior. The possibility of anomalous low-energy crystalline electric field states is discussed by comparison with other cerium-based 1-2-2 compounds.

Crystal field, phase separation, and percolative superconductivity in ErBa2Cu3Ox (6<x<7) (abstract)

Inelastic neutron scattering has been employed to study the crystalline-electric-field (CEF) level scheme of Er3+ in ErBa2Cu3Ox (6<x<7). We have been able to resolve all the seven ground-state CEF transitions for a series of samples covering the superconducting as well as the semiconducting phases. All the CEF transitions are characterized by a monotonic behavior of their energies and intensities versus the oxygen content x, with the exception of the lowest-lying CEF state which exhibits a most unusual behavior: when going from x=7 to x=6, we observe a major shift to lower energies and a significant decrease of the intensity: moreover, there is evidence for real substructures associated with this CEF transition as shown in Fig. 1. More specifically, we find the observed energy spectra to be the result of a superposition of three stable states, whose spectral weights distinctly depend on the oxygen content x. We identify the three stable states A1, A2, and A3 by two local regions of metallic (Tc≊90 K, Tc≊60 K) and a local region of semiconducting character, respectively, i.e., there is clear experimental evidence for phase separation. The superconducting phases are shown to result from the formation of a two-dimensional percolative network. A bond percolation model correctly predicts the critical oxygen concentrations associated with the two-plateau structure of Tc. A preliminary account of the present work has recently been published.1

Crystal field and magnetic properties of the tetragonal TbNi2Si2 compound

The magnetic properties of a TbNi2Si2 single crystal have been investigated experimentally by specific heat, anisotropy of the susceptibility and magnetization, and inelastic neutron scattering, leading to the determination of its fundamental coupling parameters. The lowest crystalline electric field states are two singlets close to each other, that explains the different magnetic structures evidenced in TbNi2Si2: an amplitude modulated structure, close to T N , a simple antiferromagnetic one at low temperature and an antiphase one when an external magnetic field is applied along the c direction.