# US 2013/0158948 A1 · Techniques for optimized scatterometry

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- Number: US 2013/0158948 A1
- Document ID: US20130158948A1
- Status: Published application
- Kind: A1
- Application number: 13/712,734
- Filed: 2012-12-12
- Published: 2013-06-20
- Earliest claimed: 2011-12-16
- Assignees: KLA-Tencor Corporation
- Inventors: Jonathan Iloreta; Paul Aoyagi; Hanyou Chu; Jeffrey Chard; Peilin Jiang; Mikhail Sushchik; Leonid Poslavsky; Phillip D. Flanner, III
- Web page: https://patentgrep.com/patent/US20130158948A1

## CPC

- G01B 11/00: Measuring arrangements characterised by the use of optical techniques
- G01B 11/24: for measuring contours or curvatures
- G01B 11/30: for measuring roughness or irregularity of surfaces
- G03F 7/70625: {Dimensions, e.g. line width, critical dimension [CD], profile, sidewall angle or edge roughness}
- G06F 7/00: Methods or arrangements for processing data by operating upon the order or content of the data handled (logic circuits H03K19/00)
- G06F 17/16: Matrix or vector computation {, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization (matrix transposition G06F7/78)}
- H10P 74/203: {Structural properties, e.g. testing or measuring thicknesses, line widths, warpage, bond strengths or physical defects}
- G01B 2210/56: Measuring geometric parameters of semiconductor structures, e.g. profile, critical dimensions or trench depth
- G06F 17/40: Data acquisition and logging (for input to computer G06F3/00)
- G16Z 99/00: Subject matter not provided for in other main groups of this subclass
- G01B: MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
- G03F: PHOTOMECHANICAL PRODUCTION OF TEXTURED OR PATTERNED SURFACES, e.g. FOR PRINTING, FOR PROCESSING OF SEMICONDUCTOR DEVICES; MATERIALS THEREFOR; ORIGINALS THEREFOR; APPARATUS SPECIALLY ADAPTED THEREFOR; (phototypographic composing devices B41B; photosensitive materials or processes for photographic purposes G03C; electrophotography, sensitive layers or processes therefor G03G)
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)
- H10P: GENERIC PROCESSES OR APPARATUS FOR THE MANUFACTURE OR TREATMENT OF DEVICES COVERED BY CLASS H10
- G16Z: INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR SPECIFIC APPLICATION FIELDS, NOT OTHERWISE PROVIDED FOR

## Abstract

Provided are optimized scatterometry techniques for evaluating a diffracting structure. In one embodiment, a method includes computing a finite-difference derivative of a field matrix with respect to first parameters (including a geometric parameter of the diffracting structure), computing an analytic derivative of the Jones matrix with respect to the field matrix, computing a derivative of the Jones matrix with respect to the first parameters, and computing a finite-difference derivative of the Jones matrix with respect to second parameters (including a non-geometric parameter). In one embodiment, a method includes generating a transfer matrix having Taylor Series approximations for elements, and decomposing the field matrix into two or more smaller matrices based on symmetry between the incident light and the diffracting structure.

## Claims

### Claim 1

A method of evaluating a diffracting structure, the method comprising:
computing a finite-difference derivative of a field matrix with respect to first parameters, the field matrix comprising material boundary information and the first parameters comprising a parameter describing the material boundaries of the diffracting structure;
computing an analytic derivative of a Jones matrix with respect to the first parameters using the finite-difference derivative of the field matrix with respect to the first parameters;
computing a finite-difference derivative of the Jones matrix with respect to second parameters comprising a non-material boundary parameter; and
determining values of parameters of the diffracting structure based on the analytic derivative of the Jones matrix with respect to the first parameters and the finite-difference derivative of the Jones matrix with respect to the second parameters.

### Claim 2 (depends on 1)

The method of claim 1, further comprising:
concatenating the derivative of the Jones matrix with respect to the first parameters with the derivative of the Jones matrix with respect to the second parameters prior to determining the values of the parameters.

### Claim 3 (depends on 1)

The method of claim 1, further comprising:
computing an analytic derivative of the Jones matrix with respect to third parameters, the third parameters comprising a thickness parameter of the diffracting structure.

### Claim 4 (depends on 3)

The method of claim 3, further comprising:
concatenating the derivative of the Jones matrix with respect to the first parameters with the derivative of the Jones matrix with respect to the second parameters and derivative of the Jones matrix with respect to third parameters prior to determining the values of the parameters.

### Claim 5 (depends on 1)

The method of claim 1, further comprising:
generating a model of the diffracting structure based on the derivative of the Jones matrix with respect to the first parameters and the finite-difference derivative of the Jones matrix with respect to the second parameters, wherein the Jones matrix represents reflectance for the model of the diffracting structure.

### Claim 6 (depends on 1)

The method of claim 1, further comprising:
determining parameters of a second model of the diffracting structure based on the derivative of an error between modeled spectral information and measured spectral information for the diffracting structure, wherein the derivative of the error is based on the derivative of the Jones matrix with respect to the first parameters and the finite-difference derivative of the Jones matrix with respect to the second parameters; and
wherein determining the values of the parameters of the diffracting structure is based on the model if a convergence condition is reached.

### Claim 7 (depends on 1)

The method of claim 1, wherein the first parameters are geometric parameters.

### Claim 8 (depends on 1)

The method of claim 1, wherein the second parameters comprise parameters to be computed with a finite-difference analysis.

### Claim 9 (depends on 1)

The method of claim 1, wherein the first parameters comprise one or more of: a critical dimension of the diffracting structure and dielectric constants of the materials in the diffracting structure.

### Claim 10 (depends on 1)

The method of claim 1, wherein the second parameters comprise the order of Fourier transform used in the electromagnetic solver.

### Claim 11

A machine-readable storage medium having instructions stored thereon which cause a data processing system to perform a method of computation in an optical measurement system to evaluate a diffracting structure, the method comprising:
computing a finite-difference derivative of a field matrix with respect to first parameters, the field matrix comprising material boundary information and the first parameters comprising a parameter describing the material boundaries of the diffracting structure;
computing an analytic derivative of a Jones matrix with respect to the first parameters using the finite-difference derivative of the field matrix with respect to the first parameters;
computing a finite-difference derivative of the Jones matrix with respect to second parameters comprising a non-material boundary parameter; and
determining values of parameters of the diffracting structure based on the analytic derivative of the Jones matrix with respect to the first parameters and the finite-difference derivative of the Jones matrix with respect to the second parameters.

### Claim 12 (depends on 11)

The machine-readable storage medium of claim 11, the method further comprising:
concatenating the derivative of the Jones matrix with respect to the first parameters with the derivative of the Jones matrix with respect to the second parameters prior to determining the values of the parameters.

### Claim 13 (depends on 11)

The machine-readable storage medium of claim 11, the method further comprising:
computing an analytic derivative of the Jones matrix with respect to third parameters, the third parameters comprising a thickness parameter of the diffracting structure.

### Claim 14 (depends on 13)

The machine-readable storage medium of claim 13, the method further comprising:
concatenating the derivative of the Jones matrix with respect to the first parameters with the derivative of the Jones matrix with respect to the second parameters and derivative of the Jones matrix with respect to third parameters prior to determining the values of the parameters.

### Claim 15 (depends on 11)

The machine-readable storage medium of claim 11, wherein the first parameters are geometric parameters.

### Claim 16 (depends on 11)

The machine-readable storage medium of claim 11, wherein the second parameters comprise parameters to be computed with a finite-difference analysis.

### Claim 17 (depends on 11)

The machine-readable storage medium of claim 11, wherein the first parameters comprise one or more of: a critical dimension of the diffracting structure and dielectric constants of the materials in the diffracting structure.

### Claim 18 (depends on 10)

The machine-readable storage medium of claim 10, wherein the second parameters comprise an order of formulation of a Fourier transform.

### Claim 19

An optical measurement system comprising:
first logic for receiving optical reflectance measurements for a diffracting structure;
a processor configured to evaluate the diffracting structure according to a method comprising:
computing a finite-difference derivative of a field matrix with respect to first parameters, the field matrix comprising material boundary information and the first parameters comprising a parameter describing the material boundaries of the diffracting structure;
computing an analytic derivative of the Jones matrix with respect to the first parameters using the finite-difference derivative of the field matrix with respect to the first parameters;
computing a finite-difference derivative of the Jones matrix with respect to second parameters comprising a non-material boundary parameter; and
determining values of parameters of the diffracting structure based on the analytic derivative of the Jones matrix with respect to the first parameters and the finite-difference derivative of the Jones matrix with respect to the second parameters.

### Claim 20 (depends on 17)

The optical measurement system of claim 17, wherein the processor is configured to evaluate the structure according to the method further comprising:
computing an analytic derivative of the Jones matrix with respect to third parameters, the third parameters comprising a thickness parameter of the diffracting structure.

### Claim 21

A method of evaluating a diffracting structure associated with a sample illuminated by incident light, the method comprising:
generating a transfer matrix having Taylor Series approximations for elements, the transfer matrix relating electric fields at a first layer of the sample with electric fields at a second layer of the sample, at least one of the Taylor Series approximations comprising a field matrix representing boundary conditions;
decomposing the field matrix into two or more smaller matrices based on a symmetry between the incident light and the diffracting structure;
computing spectral information for the first layer based on the Taylor Series approximation elements of the transfer matrix with the decomposed field matrix; and
providing a model of the diffracting structure, the model associated with the spectral information.

### Claim 22 (depends on 21)

The method of claim 21, wherein the Taylor Series approximations for elements of the transfer matrix comprise second order terms for diagonal elements and first order terms for non-diagonal elements.

### Claim 23 (depends on 21)

The method of claim 21, wherein the diffracting structure comprises a grating with symmetry in one dimension.

### Claim 24 (depends on 21)

The method of claim 21, wherein the diffracting structure comprises a grating with symmetry in two dimensions.

### Claim 25 (depends on 21)

The method of claim 21, wherein decomposing the field matrix based on the symmetry comprises:
determining a blocked diagonal matrix of the field matrix based a grating plane of the diffracting structure being symmetric about an incident plane of the incident light.

### Claim 26 (depends on 21)

The method of claim 21, wherein:
the field matrix comprises dielectric permittivity, the method further comprising computing Fourier coefficients of distribution of the dielectric permittivity of the first layer.

### Claim 27 (depends on 21)

The method of claim 21, wherein computing the spectral information for the first layer comprises:
substituting the Taylor Series approximation elements of the transfer matrix into an expression for impedance at the first layer; and
substituting the expression for impedance into an expression for reflectance at the first layer.

### Claim 28 (depends on 27)

The method of claim 27, further comprising:
rearranging an inverse term of the expression for impedance to eliminate an asymmetric matrix term in the inverse term.

### Claim 29 (depends on 27)

The method of claim 27, further comprising computing impedance at the first layer based on the Taylor Series approximation elements of the transfer matrix with the decomposed field matrix.

### Claim 30 (depends on 21)

The method of claim 21, further comprising:
replacing a coefficient of at least one of the Taylor series approximation elements of the transfer matrix with a variable to be determined empirically.

### Claim 31 (depends on 21)

The method of claim 21, further comprising:
determining an error between the spectral information associated with the model and measured spectral information for the diffracting structure;
determining parameters of a second model of the diffracting structure based on the error; and
determining values of parameters of the diffracting structure based on the second model if a second error between second spectral information associated with the second model and the measured spectral information is less than a threshold value.

### Claim 32

A machine-readable storage medium having instructions stored thereon which cause a data processing system to perform a method of computation in an optical measurement system to evaluate a diffracting structure, the method comprising:
generating a transfer matrix having Taylor Series approximations for elements, the transfer matrix relating electric fields at a first layer of the sample with electric fields at a second layer of the sample, at least one of the Taylor Series approximations comprising a field matrix representing boundary conditions;
decomposing the field matrix into two or more smaller matrices based on a symmetry between the incident light and the diffracting structure;
computing spectral information for the first layer based on the Taylor Series approximation elements of the transfer matrix with the decomposed field matrix; and
providing a model of the diffracting structure, the model associated with the spectral information.

### Claim 33 (depends on 32)

The machine-readable storage medium of claim 32, wherein the Taylor Series approximations for elements of the transfer matrix comprise second order terms for diagonal elements and first order terms for non-diagonal elements.

### Claim 34 (depends on 32)

The machine-readable storage medium of claim 32, wherein decomposing the field matrix based on the symmetry comprises:
determining a blocked diagonal matrix of the field matrix based a grating plane of the diffracting structure being symmetric about an incident plane of the incident light.

### Claim 35 (depends on 32)

The machine-readable storage medium of claim 32, wherein:
the field matrix comprises dielectric permittivity, the method further comprising computing Fourier coefficients of distribution of the dielectric permittivity of the first layer.

### Claim 36 (depends on 32)

The machine-readable storage medium of claim 32, wherein computing the spectral information for the first layer comprises:
substituting the Taylor Series approximation elements of the transfer matrix into an expression for impedance at the first layer; and
substituting the expression for impedance into an expression for reflectance at the first layer.

### Claim 37 (depends on 36)

The machine-readable storage medium of claim 36, further comprising:
rearranging an inverse term of the expression for impedance to eliminate an asymmetric matrix term in the inverse term.

### Claim 38 (depends on 36)

The machine-readable storage medium of claim 36, further comprising computing impedance at the first layer based on the Taylor Series approximation elements of the transfer matrix with the decomposed field matrix.

### Claim 39 (depends on 32)

The machine-readable storage medium of claim 32, further comprising:
replacing a coefficient of at least one of the Taylor series approximation elements of the transfer matrix with a variable to be determined empirically.

### Claim 40

An optical measurement system comprising:
first logic for receiving optical reflectance measurements for the diffracting structure;
a processor configured to evaluate the diffracting structure according to a method comprising:
generating a transfer matrix having Taylor Series approximations for elements, the transfer matrix relating electric fields at a first layer of the sample with electric fields at a second layer of the sample, at least one of the Taylor Series approximations comprising a field matrix representing boundary conditions;
decomposing the field matrix into two or more smaller matrices based on a symmetry between the incident light and the diffracting structure;
computing spectral information for the first layer based on the Taylor Series approximation elements of the transfer matrix with the decomposed field matrix; and
providing a model of the diffracting structure, the model associated with the spectral information.