Site selective 5f electronic correlations in β-uranium

Ruizhi Qiu(邱睿智), Liuhua Xie(谢刘桦), and Li Huang(黄理)

1Institute of Materials,China Academy of Engineering Physics,Mianyang 621907,China

2School of Physical Science and Technology,Southwest University,Chongqing 400715,China

3Science and Technology on Surface Physics and Chemistry Laboratory,Mianyang 621908,China

Keywords: uranium, low-symmetry crystal structure, 5f electronic correlation, site-selectivity, density

1. Introduction

Actinide metals exhibit a wide variety of exotic behaviors due to the complex nature of the 5f states.[1]At ambient pressure, some allotropes of light actinides exhibit quite complex crystal structures,in which the crystal symmetry is low and the number of atoms per unit cell is large.Three extreme cases are given by theαandβphases of plutonium (Pu) andβphase of uranium(U).Bothα-Pu[2]andβ-Pu[3]are monoclinic and there are 16 and 34 Pu atoms in the unit cell,respectively. Forβ-U,[4]the number of U atoms within the unit cell is as large as 30. These complex and low-symmetry crystal structures origin from the extremely narrow 5f band, which favors the Peierls distortion of the high symmetry lattice.[1,5]

The numbers of nonequivalent atoms in the unit cell ofα-Pu,β-Pu, andγ-U are 8, 7, and 5, respectively. This implies that different actinide atoms in these phases probably have different amounts of electronic correlation. Forα-Pu, the 5f electronic structures, including site-resolved density of states and hybridization functions, were investigated by means of a combination of density functional theory (DFT) and dynamical mean-field theory (DMFT).[6]It was argued thatα-Pu’s 5f electrons show apparent site dependence, which is interesting for a pure single-element material. Very recently, the same method (DFT+DMFT)[7]was employed to study the electronic structures ofβ-Pu.[8]The site-selective electronic correlations inβ-Pu were confirmed,though its site selectivity is less prominent than that inα-Pu. The two pioneering works are quite important,but their conclusions are not solid.In their calculations,the same interaction parameters(Coulomb repulsion interactionUand Hund’s exchange interactionJH) are adopted to describe the correlation effects on nonequivalent Pu atoms,which will introduce some sorts of non-consistency.In addition,the value of Coulomb interactionUused in these DFT+DMFT calculations is much larger than those from the first-principle evaluation of effective interaction.[9,10]

Here we concentrate onβ-U,which is also an ideal playground for exploring how the electronic correlation varies with different sites. The determination ofβ-U structure is beset with difficulty because of lack of the single crystals of pureβ-U.Thus,there is a controversy over the crystal structure ofβ-U during 1948–1974,although it was well-established thatβ-U is tetragonal with 30 atoms per unit cell.[11]Three suggested space groups are noncentrosymmetricP42nm(No.102),P¯4n2(No.118)and centrosymmetricP42/mnm(No.136). The debate stems from the alloying elements and intrinsically complicate high-temperature x-ray powder diffraction. By using the advanced methods including neutrons and Rietveld profile,the controversy is resolved by Lawson and Olsen in 1988.[4]It was found thatβ-U has the centrosymmetric space groupP42/mnm,as shown in Fig.1. The 30 U atoms in the unit cell ofβ-U can be grouped into five nonequivalent types(UI,UII,UIII,UIV,and UV). Their Wyckoff positions and atomic coordinates are collected in Table 1. Note that the space group ofα-U structure (Cmcm) is also centrosymmetric and it is the subgroup ofP42/mnmrather thanP42nmandP¯4n2. This crystal structure is also supported by the recent theoretical calculation.[12]Besides the structural properties, the elastic properties, electronic structures, and high-pressure behavior are investigated in the previous theoretical calculations.[12–14]To the best of our knowledge, the 5f electronic correlations inβ-U has not been well studied, the site-resolved electronic structure is unexplored,and the degree of freedom of 5f localization in this phase remains unknown.

In this work, the linear response approach[15]is used to determine the effective Coulomb interactionUforβ-U and it is found that the valuesULRvary with different sites and are moderate. Our calculated results suggest that similar toα-Pu andβ-Pu,β-U is another example that exhibits site selective 5f electronic correlations. Furthermore,the experimental lattice parameters of complexβ-U could be well reproduced from the structural optimization within DFT+ULR.

Fig.1. (a)Schematic crystal structure of β-U.The uranium atoms at non-equivalent positions(UI,UII,UIII,UIV,and UV)are represented by blue,red,cyan,purple,and green balls,respectively. The U–U bonds with bond length dU-U ＜3.0 ˚A are visualized. (b)Schematic view of the z=0 layer. (c)Schematic view of the z=1/2 layer.

Table 1. The Wyckoff positions and atomic coordinates of β-U.[4]

2. Methods

Note that the DFT+DMFT method may be the most powerful approach ever established to study the electronic structures and physical properties of actinides.[7]It has been widely employed to study Pu and its compounds.[6,16,17]However,it requires huge computer resources and is extremely timeconsuming. If there are multiple non-equivalent atoms in the unit cell, the situation becomes even worse. The reason is that each non-equivalent atom is described by a multi-orbital quantum impurity model, which should be solved by quantum impurity solvers repeatedly in the DMFT iterations. In the previous works aboutβ-Pu,a simplified quantum impurity solver based on the one-crossing approximation was used,[8]with which the computational accuracy is restricted. In the present work, we adopt a traditional but more economic approach, namely DFT+U, to study the electronic structure ofβ-U.[18,19]Within this formalism, the structure optimization of complexβ-U could be performed. This approach has been successfully applied to a large number of actinide metals and compounds.[10,20–24]

All the electronic structure calculations are performed using the Viennaab initiosimulation package (VASP).[25]The relativistic effect, i.e., spin–orbit coupling, is included in all the calculations. For the exchange-correlation term in the DFT Hamiltonian, the local-density approximation(LDA) of the Ceperley–Alder functional[26]and the generalized gradient approximation (GGA) of the Perdew–Burke–Ernzerhof (PBE) functional[27]are used. Although the interatomic distance is slightly larger, the fluctuation of the charge distribution in this bulk metallic system is not prominent enough to induce the dynamical correlation and thus the van der Waals interaction is not considered here. The Kohn–Sham equation is solved within the projector augmented wave formalism.[28,29]The valence electronic configuration for the U atom is 6s26p66d15f37s2. From the convergence tests, the optimal cutoff energy for plane wave basis is 450 eV and the division of Monkhorst–Packk-mesh is 3×3×7.

The 5f electronic correlations in U atoms are very important. We use a static mean-field scheme (i.e. the Hubbard-Ucorrection method) to capture their contributions. According to the formalism of DFT+U, an on-site interaction term is added to the Hamiltonian,i.e.,[19]

HereJis the index of correlated atom,UJis the effective Coulomb interaction,andnJis the occupation matrix;nJcan be evaluated by the projection of Kohn–Sham orbitals (ψk,ν)into the states of 5f localized orbitals():

wherek,ν,andmare Bloch wavevector,band index,and index of localized orbitals, respectively, andfk,νis the Fermi-Dirac distribution of the Kohn–Sham states.

Forβ-U,the values of effective interactionUJfor the five non-equivalent atoms is determined using the linear response approach (i.e.,UJLR).[10,15]The Hubbard parameterUJcould be evaluated as the second derivative of the ground state total energy with respect to the occupation numbernJ=Tr[nJ].The perturbed potential only acts on the localized orbitals of a Hubbard atomJ,αJ|〉〈| in whichαJis the amplitude.All occupation number{nJ}vary in response to the change ofαJand one can use this response function and the Legendre-Fenchel transformation to evaluate the HubbardUJ. In numeric,an 1×1×2 supercell is used and the response function is derived by settingαto±0.1,±0.2,±0.3,±0.4,±0.5 eV.

The occupation matrixnJis Hermite and the eigenvalues of this matrix could be denoted as{}, which could be used to determine the degree of 5f electronic localization or delocalization. In terms of{},the Hubbard-Ufunctional in equation (1) could be transformed as ∑J,mUJ(1-)/2.Clearly, the fully localized electrons (～1) and empty occupation (～0) are energetically favorable. Thus, the Hubbard-Ucorrection scheme favors Mott localization (i.e.,insulating feature), instead of fractional occupation of localized orbitals(i.e.,the metallic-like hybridization). We can use the following quantity to measure the degree of 5f delocalization for each correlated atom:

If the 5f electrons tend to be fully localized,～1 or 0 andDJ～0. On the contrary, if the 5f electrons favor itinerant states,is far away from 1.0 andDJis much greater than 0.

In the practical DFT+Ucalculations, electronic metastable states often emerge.[23]For the system with various interaction parameters,the problem is much worse.To circumvent this problem, we employ theU-ramping scheme[30]with increment in steps of 0.05 eV. Some quantities such asDJcould be shown as a function of the effective interactionU.For other physical quantities,we consider five different cases:U=ULR,0.0 eV,1.0 eV,2.0 eV,2.5 eV.

3. Results and discussion

3.1. Perspective from atomic structure

Let us first pay close attention to the atomic structure ofβ-U.As is shown in Fig.1(a),the lattice ofβ-U is tetragonal and centrosymmetric. There are two planar atomic configurations in thez=0[Fig.1(b)]andz=1/2[Fig.1(c)]layers of the unit cell. According to the space group ofβ-U(No.136),these two layers are equivalent and could be transformed to each other by using the rotation of 90°alongcaxis and translation of (1/2,1/2,0). In the atomic layer, the configuration consists of UI, UII, UIV, and UVatoms and they form two connected hexagons. Note that the hexagons are not regular.The connected points of the two hexagons are occupied by the UIatom. The four neighbor positions of connected points are occupied by UIVatoms, and the UVatoms occupy the four extended positions. The two terminal vertices of the two connected hexagons are occupied by UIIatoms. Besides the planar configurations, the residual UIIIatoms constitute the bars which intersect the hexagons.

The calculated distribution of neighbors for each nonequivalent atoms are illustrated in Fig.2.Here only the U–U interatomic distances up to 3.6 ˚A were taken into account.Note that there are no bonds with bond length ranging from 3.15 ˚A to 3.22 ˚A, as indicated by the dashed lines in Fig. 2.Thus we could classify the bonds in a similar way toβ-Pu.[3,8]One group of short bonds with length ranging from 2.6 ˚A to 3.15 ˚A and the other group of long bonds with length ranging from 3.22 ˚A to 3.6 ˚A are defined.Within this classification,the number of bonds and the corresponding average bond lengths for each U site are computed,as listed in Table 2.

Fig.2. Distribution of neighbors for each U site in β-U. The height of the bar means the number of neighbors and the color represents the bonding nonequivalent atom. The nonequivalent atoms UI, UII, UIII,UIV,and UV are represented by blue,red,cyan,purple,and green bars,respectively. Note that the UV–UV bond around 3.01 ˚A was absent in Ref.[4]. Two groups of bonds,i.e.,short and long bonds,are separated by the dashed line.

Table 2. The number of short and long bond lengths in β-U.Here,ULR denotes the Coulomb interaction parameter obtained from the linear response approach. See the main text for more details. Note that this experimental structure is different from the structure used in our previous calculation,[10,31] which is obtained from Pearson’s Handbook.[32]

Note that the number of short bonds for UI,III,IVis large,which implies that these sites may be relatively clustered. In Fig.2,one can find that the first and second nearest-neighbors of UIare UIVand UIII,respectively. In addition,UIIIand UIVare bonded to each other around 3.13 ˚A. In contrast, there is only three short bonds for UIIsite. This means that the site UIIis relatively isolated. The intermediate case is UV.

3.2. Site-resolved interaction parameters

Now we employ the linear response approach to determine the interaction parametersUJfor the five non-equivalent U atoms.[10,15]The calculated results using the PBE exchangecorrelation functional and 1×1×2 supercell are also collected in Table 2. The other exchange–correlation functionals such as LDA[26]and the solid variation of the PBE functional(PBEsol) are also examined. The calculations are also performed using the conventional unit cell. All calculations yield the same results within the accuracy of 0.1 eV. These benchmark calculations suggest that our calculations are reliable and the dominant effect is indeed the local atomic environment.

From Table 2, one can find that the UIIatoms have the largestULR(=2.7 eV), which means that their 5f electrons are more localized than the others. The UI,III,IVatoms have the smallestULR(=2.5 eV), which means that their 5f electrons are more itinerant and more easily hybridized with conduction electrons than the others. The intermediate case of UVhas the intermediate valueULR=2.6 eV. These results also suggest that the interaction parameter has the similar sequence with the number of short bonds in Table 2. The categories as proposed above are confirmed.

Note that the range of calculated interaction parametersUJ, i.e., [2.5 eV,2.7 eV],is small. As can be seen in the following, it is also appropriate to use a same value of HubbardU, such as 2.5 eV, for the DFT+Ucalculations. However, it is physically consistent to use different interaction parameters for different sites since it results from different local atomic environments. In addition, the difference ofUamong different nonequivalent sites is not small for the other low-symmetry structure,such asα-Pu andβ-Pu.[10]In those cases,it is necessary to use different values ofUfor different nonequivalent sites.

3.3. Degree of delocalization

From the self-consistent DFT and DFT+U(U=ULR,1.0 eV,2.0 eV,and 2.5 eV)calculation,the occupation matricesnJand the corresponding eigenvaluesfor each nonequivalent atoms UI-Vare obtained. These eigenvalues are shown in Fig.3. For DFT and DFT+UwithU=1.0 eV,the site dependence of ¯nJmis not obvious. All the values ofare small and approximately uniform, which implies that all the 5f electrons are delocalized. In contrast, the profile of eigenvalues manifests its site dependence for DFT+UwithU=2.0 eV.There is one large eigenvalue in the cases of UIVand UVand two relatively large eigenvalues for UII. For UIand UIII, the values ofare still close to uniform. This indicates that the Hubbard interaction gradually induces the localization of 5f electrons for UII,IV,V. For larger value ofU,i.e.,U=2.5 eV,the Hubbard interaction driven localization of 5f electrons is more obvious. For UII, all the eigenvalues are approximately equal to 1 or 0. The 5f localization in this site is evident. The site dependence of 5f delocalization could also be found within DFT+UwithU=2.5 eV. As comparison,the two largest eigenvalues for UIare smaller than those for UII. In particular, the site-dependent HubbardUJLRresults in huge differences for each nonequivalent site. From the profile of eigenvalues in the last row of Fig. 3, it is clear that the 5f electrons of UIand UIIare delocalized and localized,respectively. In addition, the two 5f electrons around UIVand UVbecome localized in this case.

Fig.3. The eigenvalues of occupation matrix for the five nonequivalent U atoms(from left to right)of β-U obtained by DFT+U calculations. The nonequivalent atoms UI,UII,UIII,UIV,and UV are represented by blue, red, cyan, purple, and green bars, respectively. Here five different cases are considered. They are U =0.0 eV,1.0 eV,2.0 eV,2.5 eV,and ULR (from top to bottom).

Table 3. The degree of delocalization D (3)for UI-V within DFT+U calculations. Here we considered five cases:U =0.0 eV,1.0 eV,2.0 eV,2.5 eV,ULR. Note that the order of D is same as that of ¯nm in Fig.3.

Using the definition of the degree of delocalizationD(3), one can quantitatively describe the dependence of localization with respect to atomic site and HubbardU. Table 3 presents the values ofDfor the five nonequivalent atoms and DFT+Ucalculation withU=0.0 eV,1.0 eV,2.0 eV,2.5 eV,andULR. The value ofDdecreases with the increasingU,which confirms the Hubbard interaction driven localization of 5f electrons. In addition, the differences ofDamong the five nonequivalent sites are very slightly different for DFT and DFT+U=1.0 eV,but prominent for DFT+Uwith largerU.For DFT+UwithU=ULR,the value ofDfor UIIis the smallest and that for UIis the largest. This means that the 5f electrons of UIIare the most localized and those of UIare the most delocalized. For all DFT+Ucalculations,the degree of delocalization has the same sequence:

3.4. The 5f partial density of states

In Fig.4,the calculated 5f partial density of states(pDOS)for the five non-equivalent U atoms are shown. Both DFT and DFT+U(UJ=ULR, 1.0 eV, 2.0 eV, and 2.5 eV) are considered. For DFT or DFT+U=1.0 eV,the differences of spectra among UI–V sites are inapparent.The main peaks of pDOS situate above the Fermi level. There are more small peaks with lower intensities below the Fermi level. The appearance of the multi-peak feature indicates that there is a strong hybridization between 5f and other conduction electrons. In addition,the increase of pDOS towards the Fermi level is approximately monotonous.

Fig.4. The 5f partial electronic density of states for the five nonequivalent U atoms (from left to right) of β-U obtained by DFT+U calculations. The vertical dashed line denotes the Fermi level. The nonequivalent atoms UI, UII, UIII, UIV, and UV are represented with blue,red,cyan,purple,and green bars,respectively. Here,five different cases are considered. They are U =0.0 eV,1.0 eV,2.0 eV,2.5 eV,and ULR (from top to bottom).

In contrast,a large peak appears below the Fermi level for DFT+Uwith largerU. This is the consequence of Hubbard interaction, and the peak corresponds to the so-called lower Hubbard bands. WhenUis increased,the Hubbard bands are pushed away from the Fermi level, and the spectral weights near the Fermi level are reduced. The site dependence of pDOS also manifests itself here.

From the profile of pDOS in the last row of Fig. 4 from DFT+ULR,while the Hubbard peak is far from the Fermi level for UII, it is still close to the Fermi level for UI. This is a remarkable feature of the 5f electron localization for UII.For UI,the Hubbard bands are shifted below the Fermi level slightly.The distance of Hubbard peak from the Fermi level is closely related to the 5f electronic localization. Clearly, the 5f electrons of UIIis much localized than that of UI. From the pDOS profile,the degree of localization for UIV,Vis close to that for UIIwhile the degree of localization for UIIIis close to that for UI. This is consistent with the sequence ofD(4).

3.5. Structural property

Forβ-U, the structural properties were already well described by GGA calculations.[12–14]This is confirmed by our PBE calculation of lattice parameters,as shown in Table 4.Table 4 also shows the calculated lattice parameters from LDA,LDA+UwithU=2.5 eV andULRcalculations. One can find that the lattice constants are underestimated by LDA due to the over binding.This discrepancy could be remedied by the Hubbard correction. The lattice constants and the internal lattice parameters could be well reproduced by LDA+U=2.5 eV and LDA+ULRcalculations. In particular,for the internal parameterszIIIandxIV,the PBE results deviate from the experimental data while LDA+Uperforms well. This improvement of DFT+Uover PBE already found in the literature[10,24]is confirmed here,particularly for a complicate structure.

Table 4. Lattice parameters from experiment,[4] PBE, LDA, and LDA+U with U =2.5 eV, and ULR calculations. The percent in the bracket represent the error of calculated results from experimental data.

4. Conclusion

In summary,we have investigated the 5f electronic structures ofβ-uranium (β-U) using the DFT and DFT+Uapproaches. We determine the site-resolved interaction parameters,and confirm the existence of site-selective electronic correlation inβ-U.For the atoms in the connect point of the two connected hexagons(i.e.,UI),the 5f electrons are highly delocalized. For the atoms in the terminal vertices of the hexagons(i.e.,UII),the 5f electrons are strongly correlated. The degree of localization on each position is closely related to the local atomic environment.This conclusion is supported by the analyses of the neighbor distance,the occupation matrix,and the 5f partial density of states ofβ-U. Obviously, this site-selective electronic correlation may be a generic feature for those actinide metals which have multiple non-equivalent sites in the unit cell. Thus, we believe that the high-pressure phases of Am,[33]Cm,[34,35]and Cf[16,36]probably exhibit some sorts of site selectivity. Further theoretical works are highly desired.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 22176181, 11874329,11934020, and U1930121), the Foundation of the President of China Academy of Engineering Physics (Grant No. YZJJZQ2022011), and the Foundation of Science and Technology on Surface Physics and Chemistry Laboratory(Grant No.WDZC202101).