Anisotropic optical conductivity and two colors of ${\mathrm{MgB}}_2$

We present the anisotropic optical conductivity of ${\mathrm{MgB}}_2$ between 0.1 and $3.7\mathrm{eV}$ at room temperature obtained on single crystals of different purity by spectroscopic ellipsometry and reflectance measurements. The bare (unscreened) plasma frequency ${\omega }_p$ is almost isotropic and equal to $6.3\mathrm{eV}$, which contrasts some earlier reports of a very small value of ${\omega }_p$. The data suggests that the $\sigma$ bands are characterized by a stronger electron-phonon coupling ${\lambda }_{tr}$ but smaller impurity scattering ${\gamma }_{\mathrm{imp}}$, compared to the $\pi$ bands. The optical response along the boron planes is marked by an intense interband transition at $2.6\mathrm{eV}$, due to which the reflectivity plasma edges along the $a$ and $c$ axes are shifted with respect to each other. As a result, the sample spectacularly changes color from a blueish-silver to the yellow as the polarization is rotated from the in-plane direction toward the $c$ axis. The optical spectra are in good agreement with the published ab initio calculations. The remaining discrepancies can be explained by the relative shift of $\sigma$ bands and $\pi$ bands by about $0.2\mathrm{eV}$ compared to the theoretical band structure, in agreement with the de Haas-van Alphen experiments. The widths of the Drude and the interband peaks are both very sensitive to the sample purity.

Figure 1

(Color online) Magnetization measurements of the superconducting transition of the two single crystals of ${\mathrm{MgB}}_2$ used in this paper.

Figure 2

(Color online) Optical anisotropic spectra of the ${\mathrm{MgB}}_2$ at $300\mathrm{K}$ derived from the ellipsometry and reflectivity measurements on the sample $S1$ as described in the text: the normal-incidence reflectivity $R\left(\omega \right)$(a), optical conductivity ${\sigma }_1\left(\omega \right)$, the dielectric function ${ϵ}_1\left(\omega \right)$ (b), the effective number of carriers $N_{\mathrm{eff}}\left(\omega \right)$, and the effective plasma frequency ${\Omega }_{\mathrm{p},\mathrm{eff}}\left(\omega \right)$ (c). The $a$ axis and the $c$ axis spectra are shown by the solid blue and dashed red lines, respectively.

Figure 3

(Color online) Comparison of the in-plane optical spectra measured on different samples. The same types of spectra are presented as in Fig. 2. The spectra of samples $S1$ and $S2$ are shown by the solid and dashed lines, respectively. Sample $S2$ has presumably a smaller concentration of impurities than sample $S1$.

Figure 4

(Color online) Extended Drude analysis of the optical conductivity of ${\mathrm{MgB}}_2$ (sample $S1$) at $300\mathrm{K}$ along the in-plane and the $c$-axis directions. The symbols are the data, the solid curves show the two-component fit as described in the Sec. 4C.

Figure 5

(Color online) Anisotropic optical spectra of ${\mathrm{MgB}}_2$ at $300\mathrm{K}$, calculated using the results of the published first-principle studies (Refs. 3, 33) as described in the text. The same types of spectra are shown as in Fig. 2. Additionally, the interband contribution along the $a$ axis is presented in (b) as a dotted blue line. The interband conductivity along the $c$ axis is negligible in the shown energy range.

Figure 6

(Color online) Experimental data (symbols) and the multiband fit (solid lines) of the reflectivity (a) and the dielectric function (b) along the $a$ axis (blue) and the $c$ axis (red).

Figure 7

(Color online) Temperature-dependent resistivity curves along the $a$ axis (solid blue) and $c$ axis (dashed red), calculated using the results of the fit of the optical data for ${\gamma }_{\pi \mathrm{imp}}=85.6\mathrm{meV}$ and different values of the scattering rate in the $\sigma$ band: ${\gamma }_{\sigma \mathrm{imp}}=12.4\mathrm{meV}$ $⪡{\gamma }_{\pi \mathrm{imp}}$(left) and ${\gamma }_{\sigma \mathrm{imp}}={\gamma }_{\pi \mathrm{imp}}$ (right).

Figure 8

(Color online) Two images of the same sample ($ac$ plane) made in a polarized light: $E\Vert ab$ (left) and $E\Vert c$ (right). The sample looks golden for $E\Vert c$, due to a sharp reflectivity plasma edge at around $2.5\mathrm{eV}$, while the plasma edge is smeared for $E\Vert ab$ by a strong $\sigma \to \pi$ interband transition at $2.6\mathrm{eV}$.

Figure 9

(Color online) Reflectivity spectra of polycrystalline ${\mathrm{MgB}}_2$ at $300\mathrm{K}$. The spectra, calculated using the effective-medium approximation (spherical or platelike crystallites) and the direct reflectivity averaging, are compared with experimental spectrum from Ref. 20.

Figure 10

(Color online) (a) The general configuration of ellipsometric experiment; (b) three specific sample orientations used in this paper.

Figure 11

(Color online) Ellipsometric spectra $\psi$ and $\Delta$ for three angles of incidence (${60}^{\circ }$, ${70}^{\circ }$, and ${80}^{\circ }$) in the orientations $\left(aca\right)$(a) $\left(caa\right)$(b) and [see Fig. 10b], measured on sample S1. Symbols are the measurement results, the solid lines correspond to the fit as described in the text.

Figure 12

(Color online) Ellipsometric spectra $\psi$ and $\Delta$ for three angles of incidence (${60}^{\circ }$, ${70}^{\circ }$, and ${80}^{\circ }$) in the orientation $\left(aac\right)$ [see Fig. 10b], measured on sample $S2$. Symbols are the measurement results, the solid lines correspond to the fit as described in the text.

Figure 13

The effect of the exposure to the air on the optical properties of ${\mathrm{MgB}}_2$. The time dependence of ellipsometric parameter $\Delta$ at $1.5\mathrm{eV}$ is shown [sample $S1$, geometry (caa), angle of incidence ${70}^{\circ }$]. Initially the sample was kept in a flow of dry nitrogen, which was switched off at the moment designated by the arrow.