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.
Actinide dioxides form a group of important materials from technological viewpoints of a nuclear reactor fuel and a heterogeneous catalyst. On the other hand, this material group has also been actively investigated from a viewpoint of basic science because of its high symmetry of the fluorite structure of the space group Fm3m [1-3]. In the circumstance of such high symmetry of crystal structure, it is possible to observe peculiar ordering of multipole higher than dipole, when we change the kind of actinide ions. Among several magnetic properties of actinide dioxides, a mysterious low-temperature ordered phase of NpO2 has attracted continuous attention in the research field of condensed matter physics.
The phase transition in NpO2 has been confirmed in 1953 from the observation of a peak in the specific heat around 25 K [
Since the multipole moments originate from 5f electrons, it seems to be natural to consider the Hubbard-like model of 5f electrons. However, from the crystal structure of actinide dioxides, it is also important to include explicitly 2p electrons, since actinide ion is surrounded by eight oxygens and the main hopping process between nearest neighbor sites should occur from the f-p hybridization. In this sense, f-p model is more realistic Hamiltonian for actinide dioxides. In fact, the f-p model for actinide dioxides has been analyzed in the fourth-order perturbation theory in terms of f-p hybridization [
In order to clarify the role of f-p hybridization for the appearance of octupole ordering, Maehira and Hotta have performed the band-structure calculations for actinide dioxides by a relativistic linear augmented-plane-wave method with the exchange-correlation potential in a local density approximation [
In this paper, in order to clarify the roles of hybridization between actinide 5f and oxygen 2p electrons for the electronic structure of actinide dioxides, we analyze the tight-binding f-p model in detail. Except for the Slater-Koster integrals of (fpπ) and (fpσ), we determine the parameters in the model from the comparison with experimental results and band-structure calculations. In order to reproduce the result of the relativistic bandstructure calculations and obtain the electronic structure consistent with the local CEF state, we find that the Slater-Koster parameters for f-p hybridization should be limited in a certain range. A typical result is found for (fpπ) 0 and (fpσ) 1 eV, which is consistent with the condition for the appearance of the octupole ordering.
The organization of this paper is as follows. In Section 2, in order to make this paper self-contained, we briefly review the relativistic band-structure calculations for actinide dioxides. It is meaningful to define the problems included in the band-structure calculations. In Section 3, we explain a way to construct the f-p model in the tightbinding approximation. Then, we determine the parameters of the model, except for (fpσ) and (fpπ), from the comparison with the experimental and band-structure calculation results. In Section 4, we depict the energy band structure of the f-p model by changing the values of f-p hybridization. We deduce the reasonable regions for (fpσ) and (fpπ). In Section 5, we discuss some future problems concerning the electronic structure of actinide dioxides. Finally, we summarize this paper. Throughout this paper, we use such units as = kB = 1.
Let us briefly review the band-structure calculation results in order to clarify the problem in the understanding of electronic structure of actinide dioxides. As for details, readers should consult Ref. [
We have performed the calculations by using the relativistic linear augmented-plane-wave (RLAPW) method. We assume that all 5f electrons are itinerant and perform the calculations in the paramagnetic phase. Note that we should take into account relativity even in the calculations for solid state physics because of large atomic numbers of the constituent atoms. The spatial shape of the one-electron potential is determined in the muffin-tin approximation. We use the exchange and correlation potential in a local density approximation (LDA). The self-consistent calculation is carried out for the experimental lattice constant for actinide dioxides.
In
eV. Narrow bands lying in the region 4.5 - 7.5 eV are the 5f bands which are split into two subbands by the spinorbit interaction. The spin-orbit splitting in the 5f states is estimated as 0.95 eV, which is consistent with that for isolated neutral Np atom. Note that in the LDA calculation, we find the metallic state for NpO2, not the insulating state. This point will be discussed later.
Here we remark that Г7 doublet and Г8 quartet levels appear around EF at the Г point. It should be noted that the Г7 level is lower than the Г8 in our band-structure calculations. However, from the CEF analysis on the basis of the j-j coupling scheme, Г8 becomes lower than Г7 in actinide dioxides. When we accommodate 5f electrons in Г8 orbitals, we obtain Г5 triplet for n = 2, Г8 quartet for n = 3, and Г1 singlet for n = 4, as experimentally found in the CEF ground states of UO2 [
In order to resolve the problems, it is necessary to improve the method to include the effect of CEF potentials beyond the simple estimation of the Madelung potential energy. However, it is a difficult task to perform such improvement concerning the formulation of the bandstructure calculation. Thus, in this paper, we choose an alternative method to exploit the tight-binding f-p model for the purpose to understand the role of f-p hybridization for the change of CEF states in the tight-binding model. By changing the parameters in the f-p model, we attempt to clarify the key quantities which characterize the electronic structure of actinide dioxides.
Before proceeding to the construction of a tight-binding model for actinide dioxides with the fluorite structure, first let us define the unit cell including one actinide ion and two oxygen ions, as shown in the
The positions of eight nearest-neighbor oxygen ions are given by b1 = (a/4, a/4, a/4), b2 = (−a/4, a/4, a/4), b3 = (a/4, −a/4, a/4), b4 = (−a/4, −a/4, a/4), b5 = (−a/4, −a/4, −a/4), b6 = (−a/4, a/4, −a/4), b7 = (a/4, −a/4, −a/4), and b8 = (a/4, a/4, −a/4). Note that the two oxygens, O1 and O2, in the same unit cell are specified by b1 for O1 and b5 for O2, respectively.
Now we define the basis of f electrons when we consider the electronic model for actinide dioxides with the fluo-
rite structure. For the purpose, we solve the problem of one f electron in the CEF potential. The CEF Hamiltonian is written as
where fimσ is the annihilation operator of f electron at site i with spin σ in the orbital specified by m. Note that m is the z-component of angular momentum l = 3. Note also that the spin-orbit coupling is not included at this stage.
Since the fluorite structure belongs to Oh point group, Bm,m’ is given by using a couple of CEF parameters and for angular momentum l = 3 as [25,26]
Note the relation of Bm,m' = Bm',m.
After performing the diagonalization of HCEF, we obtain three kinds of CEF states: Г2 singlet [xyz], Г4 triplet [x(5x2−3r2), y(5y2−3r2), z(5z2−3r2)], and Г5 triplet [x(y2−z2), y(z2−x2), z(x2−y2)]. The corresponding CEF energies are given by,
, and
. Note that these seven states are elements of cubic harmonics for l = 3. In the traditional notation, we express CEF parameters and as and with F(4) = 15 and F(6) = 180 for angular momentum l = 3 [
The values of W and x will be discussed later.
Since we will construct the model in the cubic system, it seems to be natural to use these cubic harmonics as f-electron basis function. Thus, in the following, we define μ as the index to distinguish the orbitals of cubic harmonics. Note that μ takes 1 - 7 and the definitions are as follows: 1: xyz, 2: x(5x2 − 3r2), 3: y(5y2 − 3r2), 4: z(5z2 − 3r2), 5: x(y2 − z2), 6: y(z2 − x2), and 7: z(x2 − y2). The corresponding energy Eμ is given by the above equations.
The Hamiltonian is given by
where Hf and Hp denote fand p-electron part, respectively, while Hfp indicates f-p hybridization term. In the following, we explain the construction of each term in detail.
The f-electron part is given by
where fkμσ is the annihilation operator of f electron with spin σ in the orbital μ, is the f-electron dispersion due to the hopping between nearest neighbor actinide ions, Ef is the f-electron level, Eμ denotes the CEF potential energy of μ orbital, λ is the spin-orbit interaction, and ζ is the spin-orbit matrix element.
Concerning the expression of the spin-orbit coupling, it is necessary to step back to the basis of the spherical harmonics. On the basis labelled by m, the spin-orbit interaction ζm,σ,m',σ' is expressed as
and zero for the other cases. By transforming the basis from m to μ, we obtain ζμ,σ,μ',σ' in Equation (5).
The f-electron dispersion in Equation (5) is expressed as
where a denotes the vectors connecting twelve nearest neighbor sites of the fcc lattice and indicates the f-electron hopping amplitude between μ and μ’ orbitals along the direction of a. Here we note that a runs among, , , , , and The hopping integral is expressed by using the Slater-Koster table [27,28]. Here we consider only the f-electron hopping through σ bond (ffσ).
The f-p hybridization term is written as
where pjkνσ is the annihilation operator of p electron with spin σ in the orbital ν of j-th oxygen and j denotes the label of oxygen ions in the unit cell, as shown in
and
where denotes the hopping amplitude between f and p orbitals along b direction. Here we note that b runs among b1 - b8. The hopping integral is represented in terms of (fpσ) and (fpπ) by using the Slater-Koster table [27,28].
The p-electron part is expressed as
where is the p-electron dispersion, i and j denote the label of oxygen ions in the unit cell, as shown in
The diagonal part is given by
where the hopping amplitudes are given by
Here, where (ppσ)' and
(ppπ)' denote the Slater-Koster integrals of p electron between next-nearest neighbor oxygen sites.
As for the off-diagonal parts, we obtain
Other off-diagonal components are all zeros.
The tight-binding Hamiltonian includes many parameters. Here we try to fix some of them from the experimental and band-structure calculations results.
CEF parameters. It should be noted that it is possible to reproduce the CEF states of actinide dioxides, when we accommodate plural numbers of f electrons in the level scheme in which Г8 is lower than Г7. As already mentioned in Section 2, we obtain Г5 triplet for n = 2, Г8 quartet for n = 3, and Г1 singlet for n = 4, as experimentally found in the CEF ground states of UO2 [
Spin-orbit coupling. From the relativistic band-structure calculation for actinide atom, the splitting energy between j = 5/2 and j = 7/2 states has been found to be about 1 eV. Since the splitting energy is given as 7λ/2 with the use of spin-orbit coupling λ, we fix it as λ = 0.3 eV.
fand p-electron levels. In this paper, the f-electron level Ef is set as the origin of energy, leading to Ef = 0. On the other hand, the p-electron level Ep is considered to be Ep = −4 eV from the comparison of the relativistic band-structure calculation results [
Slater-Koster integrals. In the model, we use seven Slater-Koster integrals as (ffσ), (fpσ), (fpπ), (ppσ), (ppπ), (ppσ)', and (ppπ)'. Among these values, concerning the p-electron hoppings, we introduce the ratio between nearest and next nearest neighbor hopping amplitudesgiven by. From the ratio of the distances of nearest and next nearest neighbor sites, we set [
(ppσ) and (ppπ), we determine them as (ppσ) = 0.4 eV and (ppπ) = −0.4 eV, after several trials to reproduce the structure of the wide p bands in the relativistic band structure calculations.
Concerning (ffσ), we note that it is related with the bandwidth Wf of f electrons in the j = 5/2 states on the fcc lattice. In the limit of infinite λ, we have obtained Wf as [
In the following calculations, due to the diagonallization of the Hamiltonian, we depict the tight-binding bands by changing (fpσ) and (fpπ), which are believed to be key parameters to understand the electronic structure of actinide dioxides.
Now we show our results of the diagonalization of the tight-binding model. Note that in the following figures of the band structure, “0” in the vertical axis indicates the origin of the energy, not the Fermi level EF. If it is necessary to draw the line of EF, we set it from the condition of for tetravalent Np ion in NpO2, where denotes the average number of f electrons per actinide ion. In the present paper, we do not take care of the difference in actinide ions.
First we consider the case in which the f-p hybridization is simply suppressed. In
In our first impression, in spite of the simple suppression of the f-p hybridization, the overall structure of f and p bands seems to be similar to that of the relativistic band-structure calculations in
For instance, we find the level crossing in the p-band structure of
Next we include the f-p hybridization as (fpσ) = 1 eV and (fpπ) = 0.1 eV in
Let us now consider the cases of negative (fpπ) by keeping the value of (fpσ) = 1 eV. In Figures 4(a) and (b), we show the results for (fpπ) = −0.1 eV and −0.6 eV, respectively. For (fpπ) = −0.1 eV, we do not find significant difference in the band structure from the case of (fpπ) = 0.1 eV. However, for (fpπ) = −0.6 eV, we find that Г7
is lower than Г8 at the Г point. Regarding the CEF states at the Г point, the f-p model with (fpπ) = 1 eV and (fpπ) = −0.6 eV seems to reproduce the relativistic band-structure calculation results. Note that in the p-band structure, we find the level crossing of two low-energy bands along the line between W and L points, which has not been observed in the band-structure calculation. However, as mentioned above, we do not further pursue the difference in the p-band structure.
Here we turn our attention to the f-electron states at the Г point. In the relativistic band-structure calculations for NpO2 [
In the fluorite crystal structure of AnO2, actinide ion is surrounded by eight oxygen ions in the [
From the viewpoint of the overlap integral between actinide 5f and oxygen 2p electrons, we expect that the hybridization of Г7 orbital is larger than that of Г8. Thus, due to the effect of f-p hybridization, the Г7 level becomes lower than Г8, even if the local CEF ground state is Г8. When the effect of f-p hybridization is relatively larger than that of the CEF potential, it is possible to observe that Г7 is lower than Г8, as actually found in the relativistic band-structure calculation results. We emphasize that it is one of the key points concerning the f-p hybridization to understand the electronic structure of actinide dioxides.
In
positive in the region of |(fpπ)|~1.2 eV. When we change the value of (fpσ), it is found that the f-p hybridization between actinide Г7 and oxygen 2p orbitals vanishes for the case of (fpπ) = −(fpσ). When (fpπ) is larger or smaller than −(fpσ), the energy of the Г7 state is decreased and ∆ is decreased in any case. Thus, ∆ becomes maximum at (fpπ) = −(fpσ), as shown in
Readers may consider that the absolute value of (fpπ) should not be so small only for the purpose to keep the order of the local CEF states. However, if we increase the absolute value of (fpπ) for (fpσ) = 1 eV, we should remark that the fand p-electron bands are significantly changed from those in the relativistic band-structure calculation results. Thus, from the viewpoints of the local CEF states and the comparison with the band-structure calculations, the reasonable parameters are found in the case of small |(fpπ)| in comparison with (fpσ) = 1 eV.
In this paper, we have analyzed the tight-binding model for AnO2 in comparison with the local CEF states and the result of the relativistic band-structure calculations. We have concluded that |(fpπ)| should be small for the case of (fpσ) = 1 eV in our tight-binding model in order to keep the CEF levels at the Г point. We have also emphasized that such a condition coincides with that for the octupole ordering on the basis of the f-p model [
Here we provide a comment on the local CEF state in the band-structure calculations. As long as we perform the band-structure calculations with in the LDA, it is found that the Г7 state is lower than the Г8 at the Г point, in contrast to the local CEF state expected from the experiment. In this paper, we have proposed the scenario to control the effect of f-p hybridization on the CEF state, but it should be remarked that in the LDA calculation, we could not obtain insulating state corresponding to the multipole ordering for NpO2 [
Although we have not discussed the difference in electronic structure due to the change of actinide ions in this paper, it is naively expected that the difference between Ef and Ep becomes small in the order of Th, U, Np, Pu, Am, and Cm from the chemical trends in actinide ions and the previous band-structure calculations. On the other hand, the change of f-p hybridization among actinide dioxides may play more important roles to explain the effect of the difference in actinide ions. It is an interesting future problem to clarify the key issue to understand the difference in electronic structure of actinide dioxides.
In summary, we have constructed the f-p model in the tight-binding approximation. We have determined the parameters by the experimental results and the relativistic band-structure calculations. It has been concluded that the absolute value of (fpπ) should be small for (fpσ) = 1 eV in order to reproduce simultaneously the local CEF states and the band-structure calculation results. The small value of |(fpπ)| is consistent with the condition to obtain the octupole ordering in the previous analysis of the f-p model. We believe that the present tight-binding model will be useful to extract the essential point of the electronic structure of actinide dioxides from the complicated band-structure calculation results.
The authors thank S. Kambe, K. Kubo, and Y. Tokunaga for fruitful discussions on actinide dioxides. This work has been supported by a Grant-in-Aid for Scientific Research on Innovative Areas “Heavy Electrons” (No. 20102008) of The Ministry of Education, Culture, Sports, Science, and Technology, Japan and a Grant-in-Aid for Scientific Research (C) (No. 24540379) of Japan Society for the Promotion of Science. The computation in this work has been partly done using the facilities of the Supercomputer Center of Institute for Solid State Physics, University of Tokyo.