Aspherical and covalent bonding character of d electrons of molybdenum from synchrotron x-ray diffraction

Electrons in the d-orbitals of transition metals and their complexes govern their properties and functions. The magnetism of a simple transition metal is caused by the interaction between its d-electrons. Exotic properties such as superconductivity, multiferroicity, and colossal magnetoresistance were found in transition metal oxides. The properties are closely related to their electronic structure of the d-electron. The d-electrons have both an itinerant and localized character in the system. Characterization of the d-electron in the system is one of the main topics for condensed matter physics and considerable amounts of studies have been carried out to investigate the d-electron during the past one hundred years [1]. In particular, considerable research has been carried out for 3d-transition metal oxides during the last three decades after the discovery of the high-Tc superconductivity of copper oxide [2]. The heavier 4d- and 5d-elements and their complexes had been ignored until the discovery of the exotic superconductivity of Sr₂RuO₄ [3].

The spatial and energetic structures of d-electrons have been largely investigated both experimentally and theoretically. The distribution of d-electrons in 3d-transition metals [4–9] and their complexes [10, 11] have been observed by experimental charge density studies. Spectroscopic studies of 3d-transition metals [12–14] and their complexes [1, 15, 16] have also been carried out using optical [12, 14, 15, 17], photoemission, [1, 13, 16, 18] and x-ray absorption spectroscopies [19], among others. The spatial and energetic structures of the 3d-electrons have been revealed by the measurements. The energetic structure of the 4d- and 5d-system has also been investigated by the spectroscopies [20]. However, the spatial structure of the 4d- and 5d-system has never been revealed experimentally except for one example [8], as the contribution of the 4d- and 5d-electrons to x-ray diffraction is much lower than that of the 3d-system.

We have conducted accurate structure factor measurements for the charge density study from high energy x-ray diffraction (HXRD) of one of the largest third generation synchrotron radiation (SR) facility SPring-8. The highest precision of structure factor using the technique exceeds 0.1%, which is comparable to the extremely accurate Pendellosung fringe method [21] and quantitative convergent beam electron diffraction [22]. The spatial distribution of small amounts of electrons such as the interlayer bonding electron of TiS₂ [23] and the conductive π-like electron of LaB₆ [24] have been revealed experimentally by SR-HXRD. It is essential to verify a performance of SR-HXRD for the visualization of 4d- and 5d-electrons. Typical materials with 4d- and/or 5d-electrons are required for this purpose.

Molybdenum is one of the simplest 4d-system. The electron configuration of molybdenum is 4d⁵5s¹. The electronic structure of molybdenum was investigated by both theoretical and experimental studies [25–28]. The Fermi surface was investigated using de Haas-van Alphen measurements by several research groups [27, 28]. The band structure was determined by theoretical calculations [25, 26]. The experimental Fermi surface was consistent with that calculated from theory. Zunger et al [25] demonstrated that the d-electrons in the molybdenum comprise bonding orbital d_xy+yz+xz and antibonding orbitals d_z2 and d_x2y2. The electronic structure of molybdenum was investigated by the liner combination of Gaussian orbitals method (LCGO) [26]. The density of states, Fermi surface, charge form factors, Compton profiles, and optical conductivity were theoretically estimated by this method. The electron density distribution in real space from the experimental results will provide a further understanding of molybdenum. In this study, we completed a charge density study of molybdenum using the SR-HXRD technique [29].

Molybdenum powder with 99.9% purity and 3–5 μm average particle size was used as a sample. The powder was sealed in a 0.2 mm Lindemann glass capillary with argon gas. Synchrotron powder x-ray diffraction data were measured at SPring-8 BL02B2. Imaging Plate (IP) was used as a detector. The wavelength of the incident x-ray was 37.7 keV calibrated by the lattice constant of the National Institute of Standards and Technology (NIST) CeO₂ standard sample. The temperature of the sample was controlled at 30 K using a He gas flow low-temperature device. Two two-dimensional powder images were measured. One of which was measured by moving detector position to a high scattering angle region in 2θ to improve the counting statistics and to extend the reciprocal resolution.

The size of the perfect crystal region for molybdenum is estimated less than 1 μm from peak width of powder profiles. In the case of 1 μm, the largest extinction factor is 0.2% at hkl = 110, where the extinction factor is approximated by y ≈ exp [−(l/2l_L)], l is the size of the perfect crystal region, and l_L is the extinction length [30]. It is estimated by l_L =(πv_ccosθ_B)/(2P r_eλF), where v_c is the volume of unit cell, θ_B is Bragg angle, P is polarization factor, r_e is classical electron radius, and F is absolute value of structure factor. In synchrotron x-ray source, P can be approximated by 1.

Molybdenum emits huge amounts of fluorescence and characteristic x-rays when it receives high energy beam. The x-rays increase the background scattering in the powder diffraction data as shown in figure 1(A). Figure 1(A) shows the powder profile of the 620 Bragg reflection. The ratio of the standard uncertainty to the Bragg intensity exceeds 1.6%. In this study, the combination of copper and nickel foils attached to the front of the IP was used to reduce the x-ray fluorescence from the molybdenum. Figure 1(B) shows the powder profile of the 620 Bragg reflection using metal foils. The ratio of the standard uncertainty to the Bragg intensity improved to 0.92%. The multiple overlaid measurements with the metal foils was effective for improving the precision of the measured structure factors. The ratios of the uncertainties and structure factors of the lowest 16 reflections were better than 0.004.

Zoom In Zoom Out Reset image size

The Rietveld refinements using multiple datasets were carried out using the program Synchrotron Powder (SP) [29]. The reciprocal resolution in the analysis corresponds to $\sin \,\theta /\lambda$ = 2.32 Å⁻¹. The observed structure factors were initially extracted from the results of the Rietveld refinements based on the independent atom model (IAM). The reliability factors based on the weighted profile R_wp and the Bragg intensity R_I of the final pattern fitting were 0.0253 and 0.0133, respectively. The determined lattice constants, a, and the isotropic atomic displacement parameter, u_iso, were 3.142 600(1) Å and 0.000 837(3) Ų, respectively. The estimated isotropic atomic displacement parameter using ${u}_{{\rm{iso}}}^{D}=\left(3{h}^{2}T\right)/\left(4{\pi }^{2}m{k}_{B}{\theta }_{D}\right)$ [31] is 0.001 04, where h is the Planck constant, T is temperature, m is atomic mass, ${k}_{B}$ is the Boltzmann constant, and ${\theta }_{D}$ is the Debye temperature ${\theta }_{{\rm{D}}}$ = 380 K. ${u}_{{\rm{iso}}}/{u}_{{\rm{iso}}}^{{\rm{D}}}$ = 0.805 is consistent with the case of aluminum which value is 0.804 using 0.002 893(8) and 0.003 597 of ${u}_{{\rm{iso}}}$ and ${u}_{{\rm{iso}}}^{{\rm{Theo}}},$ respectively.

The intensity ratio of completely overlapped Bragg reflections was determined by the multipole refinement. Table 1 shows the reliability factor and multipole parameters by XD2016 [32] for the experimental structure factors. The electron configuration of molybdenum was 1s² 2s² 2p⁶ 3s² 3p⁶ 3d¹⁰ 4s² 4p⁶ 4d⁵ 5s¹. We set 4d⁵ valence electron shell. The local axes for the molybdenum atom were parallel to the [100], [010], and [001] directions. The scale factor s, isotropic thermal displacement, u_iso, radial expansion/contraction parameters for the spherical valence, κ, aspherical valence, κ', and the hexadecapole parameter, H0, were refined in the analysis. There is a relationship between H0 and H4+, where H4+ = 0.74048H0.

We also prepared theoretical structure factors with the same reciprocal resolution of the observed data using the WIEN2k program [33]. The first principle calculation based on the density functional theory was performed using the full potential-linearized augmented plane wave (FP-LAPW) with the generalized gradient approximation (GGA) in the package. Experimental lattice constants were used for the calculations. We used 1000 k points with a plane-wave cutoff parameter of R^MTK_max = 7.0. The theoretical structure factors were calculated by the lapw3 program. The charge density from the theoretical structure factors was also determined by a multipole modelling. The reliability factor and multipole parameters are also listed in table 1.

3.1. Structure factors

The present experimental and theoretical structure factors are listed in table 2. The structure factors of the IAM, f_IAM and LCGO, f_LGCO, by Jani et al [26] are also listed in the table. The values are listed as form factors divided by the phase factor. The sixteen lower resolution values are also shown in the table. We call the present observed structure factors f_OBS, and the theoretical structure factors by WIEN2k f_WIEN. The first two f_OBS, f_WIEN, and f_LCGO were smaller than or equal to the corresponding f_IAM.

Table 2.  The lowest 16 structure factors of the present study and LCGO. f_OBS and f_WIEN denote the present experimental and theoretical structure factors. f_IAM was calculated by XD2016. f_LCGO is the theoretical results [26].

hkl	fOBS	fWIEN	fIAM	fLCGO
110	31.31(4)	31.62	31.84	31.59
200	27.11(5)	27.56	27.56	27.49
211	24.71(4)	25.10	24.91	24.98
220	22.73(5)	23.23	23.07	23.11
310	21.19(4)	21.72	21.68	21.62
222	20.24(6)	20.69	20.58	20.58
321	19.20(4)	19.70	19.66	19.60
400	18.35(8)	18.80	18.86	18.70
330	17.55(6)	18.16	18.15	18.05
411	17.56(5)	18.12	18.15	18.01
420	16.92(5)	17.49	17.50	17.37
332	16.35(5)	16.92	16.90	16.81
422	15.84(5)	16.35	16.35	16.23
431	15.24(4)	15.82	15.83	15.70
510	15.24(5)	15.79	15.83	15.67
521	14.12(4)	14.85	14.87	14.73

Figure 2 shows plots of the relative ratio of the structure factors to f_IAM for f_OBS, f_WIEN, and f_LCGO. The deviations from f_IAM in the lowest two f_OBS, f_WIEN and f_LCGO are also well recognized in the figure. The structure factors with resolutions better than 0.4 Å⁻¹ were almost the same as those of f_IAM within experimental uncertainties. The key features that deviated from the IAM were mainly included in the first two reflections. The maximum deviation of the structure factors from the f_IAM was less than 2% in the f_OBS, f_WIEN and f_LCGO. The deviations include information on the aspherical distribution of the d-electrons.

Zoom In Zoom Out Reset image size

3.2. Valence charge density map

Figure 3 shows valence charge density maps for 110 plane from the multipole refinements of the (A) present observed and (B) theoretical (WIEN2k) structure factors. Contour lines were drawn from 0.0 to 2.0 with a step width of 0.1 e Å⁻³. The centers and corners of the figures present the atomic sites. The map of the same section was reported by [27]. There are four peaks around the atomic sites in figure 3(A) and (B). These peaks were also found in the previous study [27]. The distances between the peaks and the atomic site for observation and WIEN2k were 0.574 and 0.557 Å, respectively. The charge densities at the maxima for observation and WIEN2k were 1.1 and 1.3 e Å⁻³, respectively. The features of the present observation are well-consistent with the theory. The numerical differences were 0.017 Å in distance and 0.2 e Å⁻³ in charge density.

Zoom In Zoom Out Reset image size

3.3. Static deformation density map and d-orbital population

Figure 4 shows static deformation density maps for 110 plane of (A) observation and (B) WIEN2k. Contour lines were drawn from −0.3 to 0.3 with a step width of 0.05 e Å⁻³. The static deformation density is the difference between the multipole model density and the IAM without effects of thermal smearing. The d_3z2-r2 shaped negative regions along the up-down direction were found in both figures. In addition, an excess of the charge density was found in the diagonal directions. We have numerically estimated the electron occupancies of the 4d-orbitals of molybdenum. The quantization axes were parallel to the crystal axes as shown in figure 4.

Zoom In Zoom Out Reset image size

Table 3 lists the d-orbital occupancies of molybdenum of observation and WIEN2k. The d-electrons of molybdenum can occupy two types of orbitals. One is triply generate Γ^'₂₅, dγ and the other is doubly generate Γ₁₂, dε. Γ^'₂₅ is d_xy, d_yz, and d_zx and Γ₁₂ is d_x2y2 and d_3z2r2. Occupancies of the two orbitals are also listed in the table. It was found that almost 0.5 electron decreased from the Γ₁₂ orbital in the result of the observation. The numbers of deficient and excess electrons of observation was approximately 0.2 electrons different from those of WIEN2k indicating the more aspherical feature of the valence electron of observation.

Table 3.  The d-orbital populations for observation and WIEN2k.

Orbital	OBSdpop	OBSdocc	WIENdpop	WIENdocc
z2	0.767 19	15.3%	0.861 08	17.2%
xz	1.158 41	23.1%	1.095 82	21.9%
yz	1.158 41	23.1%	1.095 82	21.9%
x2-y2	0.766 34	15.3%	0.857 55	17.1%
xy	1.159 09	23.1%	1.099 25	21.9%

The topological properties of the charge density for observation and WIEN2k were calculated. The charge densities and Laplacians at the bond critical point (BCP) are listed in table 4. The charge density and Laplacian of observation were 0.04 e Å⁻³ higher and 0.06 e Å⁻⁵ lower than those of WIEN2k, respectively. These facts suggest the more covalent bonding character of observation than that of WIEN2k.

Table 4.  Charge density ρ and Laplacian ${{\rm{\nabla }}}^{2}\rho$ at BCP for observation and WIEN2k.

ρOBS	∇2ρOBS	ρWIEN	∇2ρWIEN
0.373	3.123	0.326	3.174

We completed an experimental charge density study of a 4d-transition metal, molybdenum, using state of the art SR-HXRD at SPring-8. Sufficient deviations from the IAM in the structure factors were observed in the first two reflections and the origin of the deviations was revealed by the charge density study by multipole modelling. Solid crystalline molybdenum was formed by the covalent bonding of the Γ'₂₅ d-orbitals. The bonding contributes to the hardness of the molybdenum solid. The present charge density study supports this picture of solid molybdenum as a hard material. The present study also reveals that molybdenum has more covalent bonding character than the theoretical calculation by WIEN2k with the GGA basis set. We have recently observed a small amount of tight-binding like electron in pure aluminum by SR-HXRD [34]. The chemical bonding was similar to the presently observed covalent bonding character. These studies imply that valence electrons in a pure metal system have a more atomic orbital like character than that expected by the DFT theory.

The less than 0.5 electron deficiency of the orbitals was clearly recognized by the d-orbital population analysis and the spatial distribution of the 4d-electrons was well recognized in the valence and static deformation density maps in the present study. These facts suggest that the spatial structure of a 4d- system can be experimentally revealed by the present SR-HXRD. Novel physical properties are found in 4d- and 5d-system such as the superconductivity of Sr₂RuO₄ [35] and the metal-insulator transition in Cd₂Os₂O₇ [36]. The present experimental and analytical techniques easily apply to these systems by changing the sample and temperature.

The quality of high-energy quantum beam x-ray and electron beam has been drastically improved throughout the past decade such as with x-ray laser, etc. A state of the art high-energy quantum beam enables us to open a new door in subatomic scale studies. The 4d- and 5d-system with novel physical properties will be a promising target of high-energy quantum beam science.

This work was supported by JSPS KAKENHI Grant Numbers JP17J06100(T. S.), JP17H05328 (E. N.), 18H04499 (E. N.) and JP18K14136 (H. K.). This work was also partly supported by CASIO SCIENCE PROMOTION FOUNDATION. The synchrotron experiments were performed at SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2016B1754, 2017A0074, 2017B0074, 2018A0074, 2018B0074 and 2019A0068).