Spatial dispersion in silicon

Spatial dispersion (SD) is a nonlocal effect that can introduce optical anisotropy in an otherwise isotropic material, causing the electromagnetic response at a given point to depend not only on the local field, but also on the field in the vicinity of that point. In this study, we investigate the impact of SD on a cubic semiconductorlike silicon, which is typically considered a negligible effect due to the small size of the lattice parameters with respect to the wavelength of light. However, our findings demonstrate that SD can be significant above the band gap, where transmission measurements are not feasible and reflection measurements are required for characterization. We utilize Mueller matrix ellipsometry spectroscopy to quantify the anisotropy caused by SD in (110) and (100) silicon wafers, and determine the complete permittivity tensor of silicon when spatial dispersion is included. In the most general case, this tensor is found to depend on two complex parameters.

Figure 1

Calculated $D$ values for different simulation conditions: (a) (110) plane at $0{}^{\circ }$ azimuth angle over AOIs from $0{}^{\circ }$ to $89{}^{\circ }$; (b) (110) wafer at $70{}^{\circ }$ AOI over azimuths from $0{}^{\circ }$ to $180{}^{\circ }$; (c) (100) plane at $0{}^{\circ }$ azimuth angle over AOIs from $0{}^{\circ }$ to $89{}^{\circ }$; (d) (100) plane at $70{}^{\circ }$ AOI over azimuths from $0{}^{\circ }$ to $180{}^{\circ }$.

Figure 2

MM of (110) silicon measured at $70{}^{\circ }$ AOI, at the azimuth angles over wavelengths from 200 to 800 nm. The off-diagonal elements are magnified by 100.

Figure 3

Spectroscopic MM of (100) silicon ($\varphi =0$), over AOIs from $60{}^{\circ }$ to $85{}^{\circ }$. The off-block-diagonal elements are magnified by 100 to improve the visualization.

Figure 4

Jones matrix elements corresponding to the MM given in Fig. 2. Experimental data (dashed line) and best fits (solid line) of (110) plane at $70{}^{\circ }$ AOI over the azimuth angle from $0{}^{\circ }$ to $180{}^{\circ }$. The scale of ${\rho }_{ps}$ and ${\rho }_{sp}$ is magnified by 200.

Figure 5

Fit results of silicon (110). (a) Real and imaginary parts of the permittivity $\varepsilon$ values, i.e., one when considering spatial dispersion (dots) and the other when disregarding spatial dispersion (line) ${\varepsilon }_{NoSD}$. (b) Difference between $\varepsilon$ and ${\varepsilon }_{NoSD}$. (c) Real part and imaginary parts of $p_1$. (d) ${\chi }^2$ of the fits with or without considering SD.

Figure 6

Standard deviations of $p_1$ extracted from (110) and (100) wafers (red and blue lines, respectively). The error in the determination from (100) is much larger than that from (110).

Figure 7

Relative reflectance difference spectra for normal incidence light polarized along $\left[1\overline{1}0\right]$ (reflectance $R_{1\overline{1}0}$) and [001] (reflectance $R_{001}$) in a naturally oxidized (110) Si surface. $R$ is the average of these two reflectances. Black circles are experimental values from Ref. [24] and the red line is calculated from our measured values of $\varepsilon$ and $p_1$.