Optical properties of bcc transition metals in the range $0–40\mathrm{eV}$

We present a systematic analysis of the optical properties of bcc transition metals in the groups VB: V, Nb, and Ta, and VIB: paramagnetic Cr, Mo, and W. For this we use our formulation of time-dependent current-density-functional theory for the linear response of metals. The calculated dielectric and electron energy-loss functions are compared with our ellipsometry measurements and with data reported in literature, showing an overall good agreement. The experimental data of the dielectric functions presented by Nestell and Christy and by Weaver et al. differ mostly in the low-frequency region. However, we found that their reflectivity data are in very good agreement up to about $3\mathrm{eV}$. We attribute this apparent discrepancy to the Drude-like extrapolation model used by Weaver et al. in the Kramers-Kronig procedure to extract the optical constants from their reflectivity data. Our experiments are in good agreement with Nestell and Christy’s data. The calculated absorption spectra show some deviations from the experiments, in particular in the $3d$ metals. We assign the spectra in terms of transitions between pairs of bands and we analyze which parts of the Brillouin zone are mainly involved in the absorption. Our results suggest that the blueshift of some spectral features in our calculations can be attributed mainly to the incorrect description of the virtual $d$ bands by the approximations used for the ground state exchange-correlation functional. These virtual bands are too weakly bound by the local density and generalized gradient approximations, in particular in the $3d$ metals. We calculate separately the inter- and intraband contributions to the absorption and we show using a $\mathbf{k}\bullet \mathbf{p}$ analysis that, within the scalar-relativistic approximation, interband transitions contribute to the absorption already at frequencies well below $0.5\mathrm{eV}$. This finding makes questionable the Drude-like behavior normally assumed in the experimental analysis of the linear response. We find that the combination of the Drude model in which we use the calculated plasma frequency and an optimized relaxation time, and the calculated interband response can well describe the experimental spectra. The electron energy-loss spectra are very well reproduced by our calculations showing in each metal a dominant plasmon peak at about $22–24\mathrm{eV}$, well above the corresponding Drude-like free-electron plasma frequency, and additional features in the range $10–15\mathrm{eV}$. We show that the renormalization of the plasma frequency is due to the interplay between inter- and intraband processes, and that the additional features arise from the rich structure in the dielectric function caused by interband transitions.

Figure 1

Theoretical LDA ground-state band structures. The left panel shows vanadium (solid bold line) and niobium (dashed thin line), whereas the right panel shows chromium (solid bold line) and molybdenum (dashed thin line). The band structures reported for Nb and Mo refer to scalar-relativistic calculations.

Figure 2

Theoretical scalar-relativistic LDA ground-state band structures. The left panel shows niobium (solid bold line) and tantalum (dashed thin line), whereas the right panel shows molybdenum (solid bold line) and tungsten (dashed thin line).

Figure 3

Theoretical Fermi surface cross sections for V, Nb, Ta (left panel), and Cr, Mo, and W (right panel). The Brillouin zones have been plotted at the correct relative scale. The solid, the long dashed and the short dashed lines represent the contributions to the Fermi surface due to the bands 2, 3, and 4, respectively. The shaded regions indicate the regions in the Brillouin zone that contribute most to the absorption due to the transitions between the respective bands.

Figure 4

The calculated (solid bold line) and measured (dashed thin line) dielectric functions for vanadium, niobium, tantalum, chromium, molybdenum, and tungsten. Our measurements have been performed as described in Sec. 1, the other experimental results are taken from Refs. 1, 2, 3, 10, 11. The theoretical curves reported for Nb, Ta, Mo, and W are results of scalar-relativistic calculations.

Figure 5

Comparison between nonrelativistic and scalar-relativistic calculations of the dielectric functions of niobium, tantalum, molybdenum, and tungsten.

Figure 6

Comparison between the measured and calculated interband contribution to the absorption spectra of vanadium, niobium, tantalum, chromium, molybdenum, and tungsten. The Drude absorption (intraband contribution to the experimental absorption, see text) is also reported. The experimental spectra used are taken from Ref. 10.

Figure 7

Assignment of the absorption spectra. The figure shows the total intensity (bold line) and the decomposition in terms of the different contributions due to direct transitions between the indicated pairs of bands.

Figure 8

Electron energy-loss spectra for vanadium, niobium, tantalum, chromium, molybdenum, and tungsten. The experimental results are taken from Refs. 1, 2, 3, 46. The calculated real and imaginary parts of the dielectric function are also reported in the range $6–40\mathrm{eV}$. The calculated results reported for Nb, Ta, Mo, and W refer to scalar-relativistic calculations.