AN EXPLANATION OF "UNCERTAINTY" IN ELLIPSOMETRIC MEASUREMENT AND FILM THICKNESS CALCULATION

この文書は、いくつかの基本的 な計測の概念について述べ、標準的 なユーザが、膜厚標準の較正に対し て偏光解析法を計測に用いる際に内 在する不確実性のソース、コンポー ネント、計測をよく理解できるよう にすることを目的としています。

It is based on concepts contained within the "Guide to the Expression of Uncertainty in Measurement", as developed by the International Organizations for Standardization (ISO) in collaboration with other International authorities. (ISO/TAG 4/WG3) and ANSI/NCSL Z540-2-1997.

In most general terms, uncertainty means "doubt" about the exactness or correctness of a measurement and its results.

However, in order to better understand the concept of "uncertainty", it is necessary that other terms, related to a measurement be defined, albeit briefly.

In the most general form, the definition of a standard applicable to every context could be formulated as "A MODEL, TO WHICH OBJECTS OR ACTIONS MAY BE COMPARED", providing a criterion for judgment, and its form depends on how and what is to be judged or evaluated.

A "physical standard" is thus defined as a physical quantity associated with a set of procedures, a recipe if you will, for measuring that quantity and assigning a unit to it. Though the standard as well as the procedures may be arbitrary, the important aspect is that they be practical and achieve universal and international acceptance.

Often, basic primary physical standards are established, such as length, from which alternate ones are created or derived. The selection of those is usually achieved by international agreement.

Once the BASIC standard is established, a set of procedures is also created, allowing the measurement of that quantity for which the standard was originally created.

Accuracy is defined as the correctness of a measurement or series of measurements that consists in most cases of comparisons of a measured (i.e. the value of a specific quantity), to a universally accepted primary standard, thus rendering the measurand traceable to such standard.

Hence, "Uncertainty" is a quantifiable attribute related to the accuracy, (i.e., correctness) of the measurement results stated, as well as a certain level of confidence, at which the values attributed to the quantity measured, are expected to be.

In the context of metrology, traceability means that the results of a measurement or a series of measurements are in a known valid relationship to internationally or nationally recognized standards, and that a thoroughly documented, unbroken chain of reference is established to a specific Measurement Authority.

As with every metrology tool and technique, the "correctness of measurement" depends on a number of parameters, such as:

A.          The precision or power of resolution of the measuring instrument

B.          Environmental conditions.

C.         The correct application of measurement and Calculation techniques.

Like in any form of metrology, a range of values is obtained with repeated measurements. Typically this will result in a normal Gaussian distribution of measurement data about the mean value. This is caused, in the most general terms, by random sources of error such as environmental conditions instrument noise and human error. The effects of this random portion of measurement error can be reduced by averaging the results of many repeated measurements.

Unfortunately, there are some sources of non-random systematic errors that can affect even the mean of repeated measurements. These systematic errors will remain constant from one measurement to another and cannot be reduced by averaging the results of multiple measurements. These are the errors that most often can lead to significant incorrectness, regardless of the precision and repeatability of an instrument. To minimize this portion of measurement error, a system or instrument is "calibrated".

Calibration means that the measurement results are compared with significant precision to an established standard, thus defining the measurement accuracy.

Uncertainty, closely tied to accuracy, is therefore defined as a parameter associated with the results of a measurement that characterizes the dispersion of the value that could be reasonably attributed to the quantity measured (measuand). It is expressed based on the sum of random and systematic errors associated with the calibration of a system using an accepted standard and standardized procedures. It also needs to be expressed at a certain level of confidence, at which the values attributed to the quantity measured, are expected to be.

However, in this "chain" of comparison, a calibration process can never be more accurate than the standard itself and a measurement system cannot be calibrated to an uncertainty less then the uncertainty of the standard to which it is compared.

Film thickness standards are usually artifacts of extremely small physical dimensions - from a few Angstroms, to 1 micron - which prohibits measurement in a classical sense, i.e. direct comparison with a meter. They are measured and calibrated by means of analyzing various changes in the properties of light, as it interacts or propagates through them.

From the behavior of light, such as reflectance or polarization, sensitive to the "optical thickness" or optical path, the pertinent physical quantities have to be extracted using some mathematical model.

This raises a two-fold issue: light propagation and in general interaction of light with matter are strongly dependent on the physical and chemical properties of a specific material and secondly, the measurand (I.e. length) is the result of a mental (conceptual) and mathematical model.

Whereas the mathematical model is in some cases straightforward, there are certain limitations such as extremely thin films, films near or at the "cycle thickness" (ellipsometry) or films which absorb light energy, complex structures.

The conceptual aspect however is more difficult to universally adhere to. Those are surface characteristics such as:

• Specularity
• Smoothness / roughness
• Cleanliness
• Homogeneity of the material
• Interface characteristics and structures, and substrate properties.

All those are assumptions, if not unknown factors, in the model, hence underscoring the difficulty in obtaining the absolute values of the quantities in the question - namely thickness.

A standard for film thickness then, has to provide not only the derived thickness and information of the optical properties of the material used, but also a consensual model which can replicated by the end user, based on the specific measurement method and an accepted mathematical model.

The calibration of the Film Thickness Standards is done ellipsometrically, and like any other metrological process, it is subject to various errors, which can distort the results in a more or less critical way. The sources of random errors include human error in sample alignment, reading the correct values as well as imperfect null searching. In null ellipsometry noise of the photoelectric detection system can add to the random error effect. Systematic errors include errors not known and which cannot be reduced by statistical methods, such as equipment imperfection and alignment, optical components, mechanical tolerances, setup and alignment, stray light, parasitic beams, noise and residual polarization of the light source, polarization dependent detector sensitivity and alignment procedures.

In a conventional research null ellipsometer, errors in measuring delta and psi are considerably reduced because of the capability of a "four-zone" averaging. Since an ellipsometer measures only changes in the state of polarization of light, expressed in DELTA (D) and PSI (y) values, the useful parameters, film thickness and refractive index are mathematically derived. This constitutes an additional source of error.

In the mathematical model for an ideal ellipsometer, DELTA and PSI are functions of at least eight independent parameters: the refractive index of the ambient medium, the complex refractive index of the substrate, the thickness of the films, the complex index of the film, angle of incidence of the light and wavelength as expressed in terms of RHO (p):

Where:
Where:		= wavelength
		= angle of incidence
		= index of immersion medium
		= thickness of the film(s)
		= optical properties of the film(s) - index and extinction coefficient
		= optical properties of the substrate index and extinction coefficient
and
each parameter=

This then becomes a complex system where the various error components cannot be added in a linear fashion.

Therefore the stated "total uncertainty" is modeled using the Monte-Carlo error simulation. (Ref: Karl V. Bury, Statistical Models in Applied Science, cp15).

The result of such an error simulation is a distribution of all possible results within the boundaries of an "error ellipse".

Thus the measurement uncertainty reflects the maximum systematic errors whose multiple possible combinations are treated as random events. The maxima within which all random events occur have been chosen to include the total error limits due to the systematic equipment errors in addition to the uncertainties stated by NIST.

The following tables list those limits and the result of the Monte-Carlo error simulation for each nominal thickness value, based on 5000 iterations.

Note: the errors in delta and psi represent the random error portion, based on a spread of 40 consecutive measurements, combined with the errors stated by NIST.

The DELTA and PSI measurements used for the error simulation were obtained from the NIST PRIMARY STANDARDS, used to check and monitor the RR research ellipsometer or from freshly cleaned monitor samples where the nominal values were not available.

The standard uncertainty for each nominal value is obtained from 5000 iterations, and with simultaneous calculation of thickness and index of refraction for the nominal values of 1000A. The other nominal values use a fixed index of 1.462 and the error due to index calculations is significantly lessened since this (the index of refraction) parameter is eliminated as one of the variables (floating parameters):

The COMBINED STANDARD UNCERTAINTY - u(c), stated on the Certificate of Calibration is calculated using the formula , where u(m) is the uncertainty due to measurement location, number of measurements (n) and film uniformity, while u(s) represents the uncertainty due to modeling (Standard uncertainty ­ type B).

From the above uncertainty components and the degrees of freedom, the effective degree of freedom is calculated, applying the formula below:

Based on this, a multiplier is selected (Student “t” coverage factor) that allows the uncertainty of the measurand to be expressed at a 95% confidence interval. As mentioned earlier in this document, there is a controversy within the scientific community about the structure of the film - substrate system, regarding the existence of a very thin, silicon-rich interlayer.

Therefore, the choice of a single or double layer model will add to the total uncertainty in characterizing a film. The choice of a fixed index of 1.462 adds to the total uncertainty as well, but it is convenient for instruments or ellipsometer setup (such as angle of incidence, wavelength ) with no capability of calculating simultaneously thickness and index of refraction.

At PSI we offer the thickness value using a single layer model, the uncertainties relate only to the thickness obtained from this model, and are individually calculated for each physical standard.

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© 2005 Process Specialties, Inc.   Explanation of Uncertainty  REV01 09.02.05