First-principles study of transport properties in Os and OsSi

Combining density functional calculations and solutions to the steady-state Boltzmann transport equation, the transport properties of osmium and osmium silicide are studied. The calculations of electrical and thermal resistivity take full account of the ab initio electron-phonon coupling, while the Seebeck coefficient is calculated in the relaxation-time approximation. Our predicted resistivities of Os are in good agreement with experiment at both low and high temperatures, which implies electron-electron scattering is less important than electron-phonon scattering, even for temperatures below 20 K. Intrinsic OsSi is a narrow-gap semiconductor, and our prediction of the band gap confirms the indirect estimation from the measured resistivity. This agreement results from a common cancellation of error in density functional theory with spin-orbit coupling. For the degenerate semiconductor OsSi, we determine the range of carrier concentrations present in experiments, through comparison with transport data. We further predict the power factor and figure of merit for both $n$ and $p$ types, with the dependence on doping concentration and temperature.

Figure 1

Calculated electron band structure of hcp Os along high-symmetry directions. Red solid curves denote results without SOC, and black dashed curves denote results with SOC. The Fermi energy is set to zero, illustrated by a horizontal line.

Figure 2

Phonon dispersion of Os. Red solid curves denote results without SOC, and black dashed curves denote results with SOC.

Figure 3

Left panel, band structure of OsSi. Red solid curves denote results without SOC, black dashed curves denote results with SOC, and dashed-dotted green curves denote results with both SOC and semicore electrons. The energies are aligned at the top of the valence band, which is set to zero and illustrated by a horizontal line. Right panel, density of states with SOC and without semicore.

Figure 4

Phonon dispersion of OsSi. Red solid curves denote results without SOC, and black dashed curves denote results with SOC.

Figure 5

Phonon linewidths of Os, contributed by electron-phonon interactions. (Top) with SOC; (bottom) no SOC. The magnitude of the linewidths is scaled by a factor of 5 for visibility.

Figure 6

Fermi surfaces colored by the magnitude of the Fermi velocities (color legend in atomic units), for hexagonal Os. The panels on the left are side views, and the panels on the right are top views. Two top panels contain all bands crossing the Fermi level. Two middle panels show the second band. Two bottom panels show the first band, which is enclosed by the larger surface of the second band.

Figure 7

Calculated and experimental electrical resistivity of Os. The inset shows the zoom-in figure below 25 K [Expt. 1 (Ref. 10), Expt. 2 (Ref. 30), Expt. 3 (Ref. 9)]. The diagonal components of the resistivity tensor are shown separately.

Figure 8

Calculated and experimental thermal resistivity of Os. The inset shows the zoom-in figure below 25 K [Expt. 1 (Ref. 10), Expt. 2 (Ref. 9)]. The diagonal components of the resistivity tensor are shown separately.

Figure 9

EPC phonon linewidth of OsSi with hole doping of $1\times {10}^{22}$ cm${}^{-3}$, including SOC. The magnitude of the linewidth is scaled by a factor of 2 for visibility.

Figure 10

Fermi surfaces of cubic OsSi with $1\times {10}^{22}$ cm${}^{-3}$ hole doping. The top panel contains all bands intersecting with the Fermi level. Panels below show individual sheets.

Figure 11

Calculated and experimental electrical resistivity of OsSi. The theoretical resistivities with varied levels of hole doping are shown.

Figure 12

Calculated and measured Seebeck coefficient of OsSi. The theoretical Seebeck with varied levels of hole doping are shown.

Figure 13

Total (experimental, dots) and electronic [experimental (squares) and theoretical (solid lines)] thermal conductivity of OsSi. The theoretical thermal conductivity (including the residual experimental resistivity at 0 K) with varied levels of hole doping are shown.

Figure 14

Calculated (a) electrical conductivity, (b) Seebeck coefficient, (c) power factor, and (d) figure of merit $ZT$ of $p$-type (solid lines) and $n$-type (dashed lines) doped OsSi at 50 K (black), 100 K (red), 300 K (green), 500 K (blue), 900 K (violet) as a function of carrier concentration.