Theoretical Modeling of Optical Properties of Materials

An example of our current capabilities is shown in Figure 1, where the real and imaginary parts of the dielectric function are shown for magnesium oxide (MgO). The bottom curve shows experimental results, as obtained by Roessler and Walker [D.M. Roessler and D.R. Huffman, "Magnesium Oxide MgO," in Handbook of Optical Constants of Solids II, edited by E.D. Palik (Academic Press, New York, 1998), pp. 919ff.]. The bottom curve shows theoretical results (offset vertically for sake of presentation) that neglect the electron-hole interaction, and the middle curve shows analogous results that include the interaction. We wish to emphasize the correspondence of measured and calculated spectral features.

Figure 1.

As a different example, Figure 2 shows the electron probability distribution as a scatter plot for a Frenkel (Figure 2a) and charge-transfer (Figure 2b) exciton in solid C₆₀. This probability distribution is shown for when the hole is located on the molecule at the center of each picture. (A portion of the face-centered-cubic lattice is shown.) In a Frenkel exciton, an excited electron remains primarily on the molecule from which it originated. In a charge-transfer exciton, the electron is located primarily on the first coordination shell of neighboring molecules. One theory suggests that the different colors of C₆₀ in the solid versus in benzene solutions originates from the absence of charge-transfer-exciton absorption features in the liquid, where the C₆₀ molecules are isolated from each other.

Modeling Excitations of a Molecular Solid:

Figure 2a. Lowest Frenkel ecitons absorbing below 677 nm (red) present in benzene solution.

Figure 2b. Lowest charge-transfer eciton absorbing below 470 nm (blue-green) present only in solid state.

CORE EXCITATION SPECTRA

The same theory that allows calculation of optical constants because of excitation of valence electrons can also be applied to core excitation spectroscopy, an important analytical tool for characterizing materials. In Figure 3, an example of our present capabilities is shown for six lithium halides. (For LiAt, the short half-life of astatine allows only the theoretical result to obtained.) For each halide, the low-lying unoccupied band structure is shown on the same energy scale as an absorption spectrum that neglects the electron-hole interaction (dashed line, labeled "n.i.") and a spectrum that includes it (solid line, labeled "inter."). Where available, the corresponding measured absorption spectrum is shown. Everything is presented on a common energy scale. (The Li 1s edge is actually around 60 eV). It may be noted that the different features in the various absorption spectra, a consequence of the halogen atomic shell structure, exhibit a close correspondence between experiment and theory.

Figure 3.

OPTICAL CONSTANTS AND BIREFRINGENCE IN CUBIC, UNIAXIAL, AND BIAXIAL MATERIALS

The capability to model optical constants allows us to examine more complicated optical properties as well. This includes ordinary birefringence in uniaxial and biaxial crystals. However, a particularly important, recent example of complicated optical property has been the birefringence in cubic materials that are candidates for use in lenses in the deep ultraviolet region that is relevant to photolithography. As a function of wave vector q of light, the dielectric tensor governing optical properties of a cubic-symmetry solid has an expansion of the form,

The first term would suggest isotropic optical properties, while the second term indicates that anisotropies occur at second order in the components of q. This can give rise to a small but significant birefringence that depends on the direction of propagation. In Figure 4, the maximum birefringence (for light propagating along a <110> direction) is shown for several semiconductors and insulators. The curves indicate our theory, while the points are experiment results measured by Burnett et al. or by others and cited by them [J.H. Burnett, Z.H. Levine, and E.L. Shirley, Phys. Rev. B 64, R241102 (2001)]. The discovery by Burnett et al. of such a large birefringence in calcium fluoride and barium fluoride has had a large impact on the semiconductor manufacturing industry.

Figure 4

References

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