Numerical calibration of the angle of incidence in ellipsometers

Abstract

The experimental error in the angle of incidence strongly affects the film optical constants and thickness sought from the ellipsometric data, particularly when the film is thinner than 200 Å. This angle should therefore be accurately measured and this has been a frequently faced difficulty, especially in fixed-angle-of-incidence and in situ ellipsometers. We describe in this paper two numerical calibration procedures evaluating the angle of incidence from the ellipsometric data and measured on well-identified film-substrate reflecting structures either with a known or with an unknown film refractive index. The procedures rapidly determine the angle of incidence independently of the film thickness. An error analysis shows that the angle of incidence can thus be determined with an uncertainty of a few hundredths of a degree from ellipsometers inducing the same order of angular errors. The thickness ranges of films yielding such lower uncertainties are deduced from the error analysis. The effect of substrate real index error on this uncertainty is studied in particular. The supplementary results produced therein may be used in data interpretation while checking the angle of incidence if it is accurately measured by some other method. The procedures may also be easily adopted to variably damped least-squares (VDLS) algorithms employed in real-time monitoring of films during growth. The calibration procedures are illustrated for all these aspects with experimental data from real reflecting systems.

Error propagation in fixed-point ellipsometric inversions

“…The thicknesses of thin films such as those of native oxides (≈20 Å) can thus be accurately determined. Moreover, the angle of incidence of the polarized light on samples in some delicate situations can be advantageously precalibrated [12,13].…”

“…We now take up a second fixed-point inversion, called inversion ' d', which can be used to calibrate the angle of incidence [12,13] using a sample whose film index is known. This inversion allows the determination of along with the film thickness d 1 from a data set of and .…”

“…2 ] and C Z is the 2 × 2 covariance matrix of 1 and 2 which are considered independently measured with random errors given in Z T = [ n,k,d ) are literally the same as those of J nkd of equation (13). With independent 1 and 2 measured, the 4 × 2 Jacobian J is now: The first and last two rows of J nkd M AI and J are evaluated respectively at…”

“…The logical assumption made therein is that parameter correlation favours the propagation of the errors in , and to these parameters. Surprisingly though, very thin film thicknesses can be found with greater accuracy along with the angle of incidence from a data set of and when the film index value is assumed [12,13]. Another factor is the ability to retrieve with precision the optical parameters of thinner films from the MSA data [10,11].…”

Nulling ellipsometry in the study of chemically treated Si surfaces

“…The random errors could build up in either sense (±) in each of the four quantities , , n s and k s intervening and consequently there will be 16 (= 2 4 ) equally probable solutions for each and d, because these four variables could combine in 16 Assuming now that the Si substrate optical constants n s and k s are known exactly ( n s = 0 and k s = 0), the experimental errors of the order of 0.01 • and 0.02 • respectively in and bring out ±0.006 • in and ±0.1 Å (2%) in the surface layer thickness [16]. Adding an error of ±0.002 to n s with k s = 0 shifts to ±0.011 • from ±0.006 • while d remains almost unchanged.…”