# Scatterometry

**DOI:**https://doi.org/10.1007/978-3-642-35950-7_16855-1

## Synonyms

## Definitions

Scatterometry is a technique for measuring periodic structures on a surface with dimensions from a few nanometers to tens of micrometers. Scatterometry is a fast, nondestructive technique and is, therefore, suitable for in-line metrology.

## Theory and Applications

The basic working principle of scatterometry is to use the information in the interference of light interacting with periodic structures, for example, a diffraction grating, to characterize a surface. The intensities of the resulting diffraction orders are used as a unique fingerprint for a given surface. Scatterometry is often applied where imaging techniques cannot be used due to a lack of resolution and can be considered a super-resolution technique.

### The Principal Workflow

The experimental quantities to be measured in scatterometry are the intensities of the diffracted light as a function of the angle of incidence, the reflection angles of the diffracted orders, and/or the wavelength of the diffracted light. Quantities of interest are represented by the diffraction efficiencies calculated as the ratio of the incident intensity and scattered intensity for a given order.

Simulation of the diffraction efficiencies is typically performed using software for solving Maxwell’s equations, such as rigorous coupled-wave analysis (RCWA) (Moharam and Gaylord 1981) or finite element methods (FEMs) (Humphries 1997). A degree of a priori information about the sample being measured is needed in order to simulate the diffraction efficiencies. This includes the material properties of the periodic structures and substrate and an estimate of the dimensions of the structure. Such a priori information is almost always available, for example, when scatterometry is used for quality control of fabricated structures.

The last step is to compare the measured diffraction efficiencies to the simulated ones. Performing many simulations of structures with small dimensional variations can produce a database of diffraction efficiencies. The database approach greatly reduces the time for the comparison. Instead of performing the lengthy calculations described in step (2), a simple database lookup process is used. On a normal desktop computer, this can be accomplished in a few milliseconds.

### Applications

Continuous advancement in the semiconductor industry and development of large volume production technologies (i.e., injection molding, microinjection molding, roll-to-roll manufacturing) have introduced new fabrication methods to produce micro- and nanoscale periodic structures enabling different functionalities. To characterize nanostructured surfaces, scanning electron microscopy (SEM) or atomic force microscopy (AFM) is generally used to ensure the required lateral and vertical resolutions. However, AFM and SEM are far too slow to use for in-line characterization, unless highly costly parallel realizations are employed. As such, due to its potential for high-speed measurement, scatterometry is a competitive technique for in-line applications (Calaon et al. 2015). With scatterometry, it is possible to characterize relatively large areas (up to square centimeters) with nanometer precision for the geometry of periodic structures.

The ability to only measure periodic structures may seem limiting. However, many structures, especially in the semiconductor industry, are periodic by design. Furthermore, periodic structures for the purpose of characterization can often be placed near the real structures of interest, for example, in the dice lines on a wafer. Instead of measuring on the real structure, measurements can be performed on the test structure and correlated to the real design. This method is also known from the printing industry, for example, test fields at the edge of a newspaper.

### Semiconductor Industry

Scatterometry is used for in-line dimensional measurements of critical geometries during the fabrication of silicon components, for example, measuring the etch depth before continuing to the next process step. The increasing use of multiple patterning steps extended the initial adoption of scatterometry to monitor etch processes on integrated circuit masks to complex features, where as many as ten parameters are included in the model (e.g., line-width, height, sidewall angles, roughness, and pitch measurements). In some applications, the use of multi-patterned features (e.g., antireflective coatings) can prevent ultraviolet light from penetrating into deeper layers. Recent developments have focused on applying scatterometry to overlay metrology, but challenges in multilayer measurements need to be resolved (Hsu et al. 2015).

The semiconductor industry is currently adopting scatterometry for line shape metrology, providing statistically valid average values for large numbers of increasingly smaller features. Recent technological advances have demonstrated scatterometry to be a viable solution for fast nondestructive in-line process control and monitoring for extreme ultraviolet lithography in electronic circuit fabrication (Li et al. 2013).

Instruments are commercially available for the semiconductor industry, but they are highly expensive and only target large production facilities.

### Polymer Industry

With the development in nano-texturing of surfaces with materials other than semiconductors, scatterometry has found applications in fields from microfluidic channels (Calaon et al. 2015) to roll-to-roll applications (Petrik et al. 2014). Scatterometry has recently been used for in-line optical critical dimension quantification of injection molded nanostructures on transparent polymer surfaces (Madsen et al. 2017). Quantification of replication fidelity between master geometries and polymer structures can be used as direct feedback for continuous quality control of injection molded production of nanostructured products. Scatterometers can characterize the critical geometries of different nanostructures with an accuracy of a few nanometers in less than a second for single acquisitions.

### Scatterometers

Scatterometers can be divided into two main categories: (1) those that scan the incoming and/or outgoing angles, referred to as “angular scatterometers,” and (2) those that scan a range of wavelengths, referred to as “spectroscopic scatterometers.”

The simplest scatterometer comprises a static setup with the incoming beam and detector kept at fixed angles, limiting the intensity measurement acquisition to a single diffraction order (Kleinknecht and Meier 1978). Due to its simplicity, this setup is suitable for in-process characterization.

Spectroscopic scatterometry scans the wavelength of the incoming light, filtered at the detector side with, for example, a spectrometer. Specular reflection is studied, instead of the first (and higher) orders of the diffracted light. In the simple configuration (Fig. 2 right), both the light source and the detector are positioned normal to the sample. The setup of the normal incidence configuration, without any moving parts, provides simple alignment and good vibration isolation properties. The possibility of imaging specific areas of interest has been demonstrated by building a scatterometer into a conventional optical microscope (Madsen et al. 2015).

In imaging scatterometry, a set of multispectral images acquired at different wavelengths is used to determine the diffraction efficiencies of different structures at single pixels in the images. Reference images (e.g., a dark image and an image of a blank sample) are acquired for each wavelength and used to characterize the diffraction efficiency of a single pixel. The diffraction efficiency from a single pixel is then used for reconstruction of the grating shape parameters in that local area.

### Modeling

For modeling of grating structures, RCWA is the most widespread technique to calculate diffraction efficiencies. The introduction of RCWA enables scatterometers to determine the dimensions of gratings. RCWA decomposes periodic structures into smaller boxes characterized by representative parameters that describe the grating, such as pitch, height, width, sidewall angles, and, for more advanced models, the roundness of the top and bottom corners of the grating. The computer model solves the electromagnetic field of the different boxes composing each single object of the periodic structure. Consequently, the boundaries of the different rectangular boxes are matched to find the electromagnetic field outside the grating structure.

FEM can also be applied for simulation of diffraction efficiencies. With FEM, more complex structures can be handled with the support of a CAD model in the design phase and a meshing procedure for the determination of Maxwell’s equations (Jin 2002). However, the drawback of this method is the need for much longer computational times, which can only be reduced by the use of supercomputers.

### Comparison of Experimental and Numerical Data

Diffraction efficiencies are compared with the simulated data and the best match yields the dimensions of the structures on the sample (Fig. 1). Direct optimization or library search approaches are the main approaches for the comparison.

Direct optimization is generally used for simple geometries, and all the free parameters (e.g., height and width) are optimized using, for example, a differential evolution algorithm (Storn and Price 1997). The optimization identifies the best fit using either scalar diffraction theory, RCWA, or FEM between selected free parameters of the structure under analysis and reference measurements.

For the library search approach, all values are computed for different elements in a pre-generated database. The size of the generated database containing scattering intensity profiles for different structures is limited by the time available to generate it. The subsequent comparison is based on a very fast algorithm, such as chi-squared minimization, where *χ*^{2} can be defined as

where *η*_{i} is the *i*th measured diffraction efficiency, *N* is the number of measured diffraction efficiencies, *δη* are the experimental uncertainties, and *f*(Ω_{i}, *α*) are the modeled diffraction intensities for a given set of parameters.

Typically, the minimization only takes a few milliseconds, and direct feedback using the library search approach is a valuable tool when used on a production line. The approach will always provide a best fit within the produced library database. The validity of the generated “true value” can be subject to errors coming from values outside the pre-generated database and faulty data calculated using incorrect optical material properties.

### Uncertainty Analysis

Type A: Detector noise, light fluctuations

Type B: Long-term stability of the light source, polarization of the light, incident angle variations, wavelength variations of the detected light, orientation of the sample, light losses in the system, incomplete collection of the scattered light

Sources of random error can be reduced by increasing the integration time or averaging the dataset, and their quantification and computation are well addressed by least-squares approaches. Type B influence factors require instrument calibration to identify and quantify their magnitude.

The uncertainties resulting from the model are more challenging to handle, and a full uncertainty analysis for the combined uncertainties has yet to be presented in the literature. For the advanced scatterometry technique known as “Mueller polarimetry,” where changes in light polarization are measured, traceable measurements have recently been demonstrated (Hansen et al. 2017). Here, a general least-squares analysis approach was used to estimate the combined instrument and model uncertainties.

## Cross-References

## References

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