Linear and nonlinear optical properties of Li under pressure

The alkali metals, which under ambient conditions are simple in the sense that their electronic structures are free-electron like and the their crystal structures are of high symmetry (body centered cubic), transform to surprisingly complex structures when pressure is applied. The experimentally determined pressure sequence for lithium up to 50 GPa is $bcc\to fcc\to hR1\to cI16$. In the present work the (non)linear optical properties of these Li phases are examined in order to gain better insight into the electronic properties. We propose that the comparison of optical experiments (performed on samples in a diamond anvil cell) with calculations like those presented here can be used to distinguish between various high-pressure phases and thus be a supplement to x-ray diffraction and Raman scattering techniques.

Figure 1

Band structure of the fcc structure of Li at 39.8 GPa. The horizontal line marks ${\mathrm{E}}_F$. The radii of the circles are proportional to the band $s$-character, and inversely to the $p$-character.

Figure 2

Total density of states, and partial $s$ and $p$ DOS of Li-fcc. Zero energy is the Fermi level, ${\mathrm{E}}_F$. Note that the partial DOS functions shown here only include projections onto muffin-tin spheres, i.e., contributions from the interstitial regime are not included. Further, “Total DOS” means here simply the sum of the two partial $s$ and $p$ functions. A true “Total DOS” should be calculated from the energy eigenvalues directly. Therefore, the units are “arbitrary.”

Figure 3

Imaginary part of the frequency dependent dielectric function for Li-bcc and Li-fcc including only the interband contributions.

Figure 4

(Color online) Band structure of Li in the bcc structure at 39.8 GPa. The horizontal line marks ${\mathrm{E}}_F$. The radii of the circles are proportional to the band $s$-character, and inversely to the $p$-character. The figure shows that (at least along the lines considered here) there are no possibilities for interband absorption for low photon energies. The energy scale was chosen so that a comparison to Li-cI16 (Sec. 3C) is easy.

Figure 5

Band structure of Li-hR1 at 39.8 GPa. The horizontal line marks ${\mathrm{E}}_F$. The radii of the circles are proportional to the band $s$-character, and inversely to the $p$-character. The rhombohedral distortion parameter has the value which was found experimentally as well as by theoretical structural relaxation at this pressure, see Refs. 8, 17.

Figure 6

(Color online) Band structure of Li-fcc at 39.8 GPa using the rhombohedral representation, i.e., like Fig. 5, but with the $\mathrm{c}∕\mathrm{a}$ ratio adjusted to match the fcc structure.

Figure 7

Total density of states, and partial $s$- and $p$ DOS of Li-hR1. Zero energy is the Fermi level, ${\mathrm{E}}_F$. Units etc.: See comments in the caption to Fig. 2.

Figure 8

Imaginary part of the frequency dependent dielectric function for Li-hR1 including only the interband contribution. The distinct ${\epsilon }_2^{xx}$ and ${ϵ}_2^{zz}$ principal components are shown. The $z$-axis is the rhombohedral axis, i.e., the direction of the body diagonal in the fcc structure which can be considered to be the strain axis going from fcc to hR1.

Figure 9

Li-hR1 averaged spectrum, $({\epsilon }_2^{zz}+2\times {\epsilon }_2^{xx})∕3$, in comparison with the spectrum for undistorted Li-fcc.

Figure 10

Band structure of the cI16 structure of Li at 39.8 GPa. The horizontal line marks ${\mathrm{E}}_F$. The radii of the circles are proportional to the band $s$-character, and inversely to the $p$-character of the states. The parameter $x$ is 0.05, i.e., the one obtained by total-energy optimization as well as from experiments close to the 39.8 GPa (see Refs. 8, 17).

Figure 11

(Color online) Similar to Fig. 10, but for $x=0.00$, i.e., for the undistoted bcc structure (but in the supercell representation). Compare to Fig. 4, which is the same band structure, but shown in the “standard” representation.

Figure 12

Total and partial $s$- and $p$-DOS of Li-cI16. ${\mathrm{E}}_F$ is at zero energy. As it was mentioned in Ref. 8, the cI16 structure shows a pronounced maximum below ${\mathrm{E}}_F$ and a minimum just above ${\mathrm{E}}_F$. Units etc.: See comments in caption to Fig. 2.

Figure 13

Imaginary part of the frequency dependent dielectric function for Li-cI16 including only the interband contribution for $x=0.01$, 0.03, and 0.05. The value $x=0.05$ is obtained by total-energy minimization at the volume corresponding to $P=39.8\mathrm{GPa}$. The calculated and observed structures agree extremely well, see Refs. 8, 17. For $x=0$ we get the bcc spectrum as shown in Fig. 3.

Figure 14

As Fig. 13 but for a wider range of photon energies.

Figure 15

Li-cI16 $(P=39.8\mathrm{GPa})$. The calculated only nonzero component $\left(xyz\right)$ of the second-order susceptibility. The calculatations are performed for the three different values of $x$, 0.01, 0.03, and 0.05, which were also considered in Figs. 13, 14, $x=0.01$ at the top, $x=0.05$ at the bottom. The $x$-value according to experiment Ref. 8 and calculation Refs. 17, 8 at this pressure is 0.05. The letters and numbers refer to the labels of peaks in the linear spectra in the two previous figures.