First-principles investigations of the electronic and magnetic structures and the bonding properties of uranium nitride fluoride (UNF)

1 Introduction

From the iso-electronic relationship for valence shell states: 2O (2s², 2p⁴)≡N (2s², 2p³)+F (2s², 2p⁵), nitride fluorides of formulation M^IVNF type can be considered as pseudo-oxides and isoelectronic with M^IVO₂ (M^IV stands for a generic tetravalent metal). Compared to homologous oxides, some relevant physical properties can be expected due to differentiated bonding of M with nitrogen and fluorine qualified as less and more ionic, respectively. A small number of tetravalent metal nitride fluorides exist such as transition metal-based TiNF [1] and ZrNF [2] on one hand and heavier actinide equiatomic ternaries such as ThNF [3] and UNX (X=halogen) [4], [5] on the other hand. Besides the actinide-based compounds, only the rare-earth ternary CeNCl was evidenced by Ehrlich et al. [6]. Recently, CeNF [7] was proposed with potential experimental synthesis routes besides a full account of physical properties based on extended theoretical works within well-established density functional theory (DFT) [8], [9]. In fact, it has been shown in recent decades that this theory with DFT-based methods allowed us not only to explain and interpret experimental results resolved at the atomic chemical constituent scale but also to operate it as a predictive tool to propose new compositions with targeted specific properties. As an example of largely investigated compounds within the BCN pseudo phase diagram, binary carbon nitrides and ternary boron carbon nitrides were identified theoretically with high hardness close to well-known ultra-hard diamond and potentially enabled to replace diamond in tooling machinery, industry and forage applications [10], [11], [12].

In continuation of our investigations of nitride fluorides (cf. [7] and references therein) we focus herein on the electronic and magnetic properties of uranium nitride fluoride (UNF) isoelectronic with UO₂ [13]. It needs to be mentioned here that UO_2+x is known, and a complete study on the average structure and local configuration of oxygen excess with pertaining physical effects has recently been reported [14]. However, such an investigation is out of the scope of present work.

UNF is tetragonal and crystallizes in the space group P4/n [4]. The structure is shown in Fig. 1. In spite of its overall three-dimensional character, it can be considered along the c axis as successions of [U₂N₂]²⁺-like motifs separated by [F₂]²⁻-like layers; this is supported by the shorter U–N versus U–F distances: 2.29 Å versus 2.61 Å. It will be shown that this structural setup has an important influence on the electronic distribution (cf. Fig. 3, see below).

Fig. 1:

Tetragonal structure of UNF showing [F2]²⁻ planes at 0, 0, 1/2 interlayering [U₂N₂]²⁺ blocks (also cf. Fig. 3).

2 Computational methodology

Within DFT we first used the Vienna ab initio simulation package (VASP) code [15], [16] for geometry optimization, total energy calculations as well as establishing the energy-volume equations of state. The projector-augmented wave (PAW) method [16], [17] is used with atomic potentials built within the generalized gradient approximation (GGA) scheme following Perdew et al. [18]. This exchange-correlation scheme was preferred to the local density approximation [19], which is known to be underestimating interatomic distances and energy band gaps. The conjugate-gradient algorithm [20] is used in this computational scheme to relax the atoms of the different crystal setups. The tetrahedron method with Blöchl corrections [21] as well as a Methfessel-Paxton [22] scheme was applied for both geometry relaxation and total energy calculations. Brillouin zone (BZ) integrals were approximated using a special k-point sampling of Monkhorst and Pack [23]. The optimization of the structural parameters was performed until the forces on the atoms were <0.02 eV Å⁻¹ and all stress components <0.003 eV Å⁻³. The calculations were converged at an energy cut-off of 400 eV for the plane-wave basis set with respect to the k-point integration in the BZ with a starting mesh of 6×6×6 up to 12×12×12 for best convergence and relaxation to zero strains.

The charge density issued from the self-consistent calculations can be analyzed using the atoms in molecules theory (AIM) approach developed by Bader [24]. Such an analysis can be useful when trends between similar compounds are examined; it does not constitute a tool for evaluating absolute ionizations. Bader’s analysis is performed using a fast algorithm operating on a charge density grid arising from high-precision VASP calculations and generates the total charge associated with each atom.

From the calculations, we also extract information on the electron localization (EL) at atomic sites thanks to the EL function (ELF) [25], [26]. Normalizing the ELF between 0 (zero localization) and 1 (strong localization) – with the value of 1/2 corresponding to a free electron gas behavior – enables analyzing the contour plots following a color code: blue zones for zero localization, red zones for full localization and green zone for ELF=1/2, corresponding to a free electron gas.

Then for a full account of the electronic structure, the site-projected density of states (PDOS) and the properties of chemical bonding based on the overlap matrix (S_ij) with the COOP criterion [27] within DFT, we used the scalar relativistic full potential augmented spherical wave (ASW) method [28], [29] with the GGA scheme [18]. The basis set, limited in the ASW method, was chosen to account for the outermost shells to represent the valence states for the band calculations. The matrix elements were constructed using partial waves up to l_max+1=4 for U and l_max+1=2 for N, O and F. F-2s states at low energy (much lower than corresponding O and N 2s states) were considered as core states, i.e. not included in the valence basis set; in the limited ASW basis set they were replaced by 3s states. Self-consistency was achieved when charge transfers and energy changes between two successive cycles were such that ΔQ<10⁻⁸ and ΔE<10⁻⁶ eV, respectively. The BZ integrations were performed using the linear tetrahedron method within the irreducible tetragonal wedge following Blöchl [21].

3 Geometry optimization and energy-dependent results

Table 1 shows the starting experimental and calculated lattice parameters and z_U coordinate for both spin degenerate (NSP: non-spin-polarized) as well as spin-resolved (SP: spin-polarized) configurations. Better agreement with experiment is observed with SP calculations. These calculations lead – expectedly – to a magnetization of 4 μ_B (Bohr magnetons) per unit cell or 2 μ_B per formula unit (FU) which arises from the presence of two unpaired electrons in the U 5f states of tetravalent uranium. Note that these calculations merely indicate the trend of developing magnetization from the present PAW-GGA calculations, i.e. they do not point to the long-range magnetic order or the ground state which is searched for and precised in the next section.

In fact, UO₂, in which uranium is also tetravalent, is known to be an insulating antiferromagnet in the ground state [30] provided that the Hubbard U [31] method is used in further calculations, as is shown for UNF here below.

The trend to magnetic polarization can be checked as a function of volume by establishing the energy-volume equation of state (EOS) in both NSP and SP configurations. We also verify this for UO₂. The NSP and SP E(V) curves are shown in Fig. 2. They all exhibit quadratic behavior with systematically lower SP energy minima. The SP solution is favored for larger volumes, but both NSP and SP curves merge together at small volumes or high pressure. The fit of the curves with a Birch EOS [32] up to the third order:

Fig. 2:

Energy volume curves and fit values from Birch EOS in non-spin-polarized (NSP) and spin-polarized (SP) configurations.

E(V)=E0(V0)+98V0B0[(V0/V)2/3−1]2+916B0(B′−4)V0[​(V0/V)2/3−1]3

provides equilibrium parameters E₀, V₀, B₀ and B′, respectively, for the energy, the volume, the bulk modulus and its pressure derivative. The obtained values with goodness of fit χ² magnitudes are displayed in the insets of Fig. 3. The equilibrium volumes for both compounds UNF (Table 1) and UO₂ come close to experiment: V(UO₂)=163.73 ų cell⁻¹ or 40.93 ų per FU [33].

Fig. 3:

UNF electron localization function slice along the (101) plane with a projection over four adjacent cells showing the succession of [U₂N₂]²⁺-like blocks and [F₂]²⁻-like planes. Blue, green and red spheres represent U, N and F atoms, respectively (see the text).

The SP volumes are larger than the NSP ones and the corresponding energies are lower. Also ΔE_UO2(SP–NSP)=−0.23 eV FU⁻¹ whereas ΔE_UNF(SP–NSP)=−0.29 eV FU⁻¹, meaning that somehow the U–N bond less ionic than U–O prevails. This can be verified from the trend of charge transfer between the two compounds which can be rationalized from the analysis of charge density resulting from the calculations within AIM theory based on Bader’s work [24]. Such an analysis is particularly relevant when it comes to comparing two electronically close compounds such as UNF and UO₂ here. The results of computed charge changes Q between neutral and ionized elements and resulting overall ΔQ in the structure are as follows:

UNF: Q(U)=+2.29; Q(N)=−1.47; Q(F)=−0.82; ΔQ=±2.29UO2: Q(U)=+2.48; Q(O)=−1.24; ΔQ=±2.48

Charge transfer is as expected from U to N, O, F with different magnitudes: N^−1.47, O^−1.24, F^−0.82 not translating their formal ionizations but proportional with the electronegativities χ increasing along N, O and F, i.e. χ(N)=3.04 <χ(N)=3.44<χ(F)=3.98. The overall ΔQ translating the total±transfer is smaller for UNF which stresses further the covalent role brought by N through the formation of [U₂N₂]²⁺ layers as illustrated below.

Also it is interesting to note the large difference of magnitude of the bulk moduli B₀ pointing to more compressible UNF than UO₂. This is partly due to the larger volume of UNF (43.5 ų FU⁻¹) versus UO₂ (39.95 ų FU⁻¹); i.e. the larger the volume the more compressible the compound; but this could also be due to the rather layered nature of the UNF structure versus tri-dimensional fluorite-type UO₂.

At this point, it is interesting to show the 2D-like structure from the point of view of EL which is expected to illustrate further the different chemical behaviors of N and F versus U with smaller U–N versus U–F distances observed experimentally and by calculations. The EL with the ELF based on the kinetic energy [25], [26] is used here. In the projections, blue, green and red contours represent zero, free electron like and strong localizations, respectively. Figure 3 shows ELF slices along the (101) plane with a projection over four adjacent cells. The succession of [U₂N₂]²⁺-like and [F₂]²⁻-like planes along the tetragonal c axis is clearly observed. The isolated fluorine is displayed by the blue zones of no localization around it.

4 Electronic structure and bonding

All-electron full potential scalar relativistic ASW calculations were then undertaken for assessing the electronic band structure and qualitative analysis chemical bonding. A comparison between UNF and UO₂ was done with spin-degenerate (NSP) calculations in order to examine the role of each chemical constituent in the site PDOS as well as in the chemical bonding. For UNF and UO₂, the top panels in Fig. 4 show the NSP site PDOS. The zero energy along the x axis is with respect to the Fermi level E_F which crosses the lower energy part of the U(5f) states within the valence band (VB). The main part of U(5f) is centered in the empty conduction band (CB) above E_F due to the low filling of 5f states. Nevertheless, the crossing occurs at a relatively high PDOS. This is connected with an instability of the electronic system in such a spin degenerate configuration of both compounds and with the expected onset of magnetic polarization as shown in the next paragraph. Large differences characterize the VB where N(2s) is at −15 eV versus O(2s) at −20 eV; these states show little mixing with uranium itinerant states. Oppositely, the hybridization (mixing) between uranium itinerant states and those of N, O p states is identified, respectively, in the energy windows {−5.5; −3 eV} and {8; −5.4 eV}; this shift of energies is due to the larger electronegativity of O versus N. In agreement with the observation above on the [U₂N₂]²⁺-like layers separated by [F₂]²⁻-like layers characterizing the structure of UNF (cf. text and Fig. 3), there is little mixing to be noted between U and F states at −7.5 eV. This aspect should be confirmed from the qualitative analysis of the chemical bonding based on the overlap integral S_ij (i and j designate two chemical species) as implemented in the ASW method with the COOP criterion. The comparative bonding strengths (U–N versus U–F as well as U–O) is qualitatively estimated with the integrated COOP, iCOOP shown in Fig. 4 (lower panels). In both panels, little bonding can be identified in the VB lower part where where s-like states are dominant; and significant bonding is found above −10 eV with p states. Comparing the areas below the iCOOP shows larger U–N iCOOP versus U–F iCOOP leading to prevailing U–N bonding. The U–O bond in UO₂ shows closely similar behavior to U–N albeit with slightly larger iCOOP (note that there are 2 FU in UNF and 4 FU in fluorite UO₂ with only 1 FU accounted for in the calculations due to the F centering). Nevertheless, U–N iCOOP keeps positive bonding behavior above E_F, whereas U–O iCOOP drops rapidly to negative magnitude within the CB. Again this is due to the covalent U–N bond versus rather ionic U–O.

Fig. 4:

Non-spin-polarized (NSP) calculations for UNF (left) and UO₂ (right) displaying site-projected DOS (up) and chemical bonding from unit-less integrated COOP iCOOP (bottom).

Subsequent SP calculations lead to an onset of magnetization in both UNF and UO₂ with M=2 μ_B FU⁻¹. From Fig. 5, showing the corresponding PDOS along the two spin channels (↑;↓), the integer value is due to the full polarization of electrons in ↑ spin PDOS with a gap appearing in ↓ PDOS and an energy shift between ↑ and ↓ U PDOS signaling the onset of magnetic polarization. The non-metal s, p states do not show energy shifts. However, the calculations were conducted with plain GGA calculations and it is known that for uranium-based compounds, such as UO₂, a Hubbard U repulsive parameter is required to be added [33]. With U=4.1 eV, the magnetization M=2 μ_B FU⁻¹ is reproduced with a small gap opening in ↑ spin PDOS, as shown in Fig. 6 (top panel). The compound exhibits a magnetic semi-conductor-like behavior. Yet in view of the antiferromagnetic isoelectronic UO₂, further calculations assuming two magnetic sub-cells, one considered as UP SPINS and the second as DOWN SPINS, lead to ±2 μ_B per magnetic sub-cell and to larger opening of the band gap (~2 eV) with an insulator behavior as shown in the bottom panel of Fig. 6. The energy is further lowered by −3.36 eV with respect to the plain SP calculations discussed above. Then the ground state of UNF is predicted to be an insulating antiferromagnet.

Fig. 5:

Spin-polarized calculations for UNF and UO₂ displaying site-projected DOS (up) and chemical bonding (bottom).

Fig. 6:

Spin-polarized calculations for UNF with GGA+U scalar relativistic calculations in implicitly ferromagnetic (top) and antiferromagnetic (bottom) configurations. Notice the gap opening.

5 Conclusion

In this paper, we have focused on the original properties brought by the changing of the chemistry between two isoelectronic compounds: UO₂ and UNF. Particularly the U–O bonding in the three-dimensional fluorite UO₂ is differentiated into covalent [U₂N₂]²⁺-like layers separated by ionic [F₂]²⁻-like in the two-dimensionally assimilated nitride fluoride. Using complementary programs within DFT, this has been quantitatively identified through electronic structure calculations in both non-magnetic spin-degenerate and SP configurations and illustrated by electron localization mapping, charge changes, site- and spin-projected density of states and chemical bonding based on overlap integrals describing the magnitudes of U–O, U–N and U–F bonding. UNF is found relative to be an insulating antiferromagnet, likewise UO₂ in the ground state.