CN111122459B - Method and device for correcting depolarization effect caused by uneven thickness in spectroscopic ellipsometry - Google Patents

Method and device for correcting depolarization effect caused by uneven thickness in spectroscopic ellipsometry Download PDF

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CN111122459B
CN111122459B CN201911369701.1A CN201911369701A CN111122459B CN 111122459 B CN111122459 B CN 111122459B CN 201911369701 A CN201911369701 A CN 201911369701A CN 111122459 B CN111122459 B CN 111122459B
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CN111122459A (en
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张传维
刘贤熠
郭春付
李伟奇
刘世元
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Wuhan Eoptics Technology Co ltd
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N21/00—Investigating or analysing materials by the use of optical means, i.e. using sub-millimetre waves, infrared, visible or ultraviolet light
    • G01N21/17—Systems in which incident light is modified in accordance with the properties of the material investigated
    • G01N21/25—Colour; Spectral properties, i.e. comparison of effect of material on the light at two or more different wavelengths or wavelength bands
    • G01N21/27—Colour; Spectral properties, i.e. comparison of effect of material on the light at two or more different wavelengths or wavelength bands using photo-electric detection ; circuits for computing concentration
    • G01N21/274—Calibration, base line adjustment, drift correction
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
    • G01B11/00—Measuring arrangements characterised by the use of optical techniques
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
    • G01B21/00—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant
    • G01B21/02—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant for measuring length, width, or thickness
    • G01B21/04—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant for measuring length, width, or thickness by measuring coordinates of points
    • G01B21/045—Correction of measurements
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    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N21/00—Investigating or analysing materials by the use of optical means, i.e. using sub-millimetre waves, infrared, visible or ultraviolet light
    • G01N21/17—Systems in which incident light is modified in accordance with the properties of the material investigated
    • G01N21/21—Polarisation-affecting properties
    • G01N21/211—Ellipsometry
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N21/00—Investigating or analysing materials by the use of optical means, i.e. using sub-millimetre waves, infrared, visible or ultraviolet light
    • G01N21/17—Systems in which incident light is modified in accordance with the properties of the material investigated
    • G01N21/21—Polarisation-affecting properties
    • G01N21/211—Ellipsometry
    • G01N2021/213—Spectrometric ellipsometry
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Abstract

The invention provides a source depolarization effect correction modeling method and device for spectrum ellipsometry measurement of uneven thickness. And then, by reasonably assuming a distribution function of uneven thickness, fully considering the unevenness of the thickness of each layer, and selecting reasonable nodes and weight accumulation to obtain the sample forward modeling average Mueller matrix. When the thickness distribution of each layer of the sample is random, nodes and weights in Gaussian distribution are adopted, multidimensional calculation is simplified into one-dimensional calculation, and the calculation efficiency is greatly improved. And finally, fitting and extracting information such as optical constants, thickness values and the like of the parameters of the sample to be detected through a nonlinear regression algorithm. The invention can realize the correction of the depolarization effect of the spectrum ellipsometry thickness unevenness, can perform rapid data analysis on various optical thin film devices and on-line measurement in various nanometer manufacturing processes, and can extract the optical and collective parameters of the nanometer material.

Description

Method and device for correcting depolarization effect caused by uneven thickness in spectroscopic ellipsometry
Technical Field
The embodiment of the invention relates to the technical field of ellipsometry, in particular to a method and a device for correcting and modeling an depolarization effect of a source with uneven thickness in spectroscopic ellipsometry.
Background
The ellipsometry method is a method for measuring nanometer parameters by using the polarization characteristics of light, has simple and rapid measuring process, can effectively represent the geometric and optical parameters of thin film materials, has the advantages of rapidness, accuracy, no damage and the like, and is an indispensable technical means in industries such as flexible electronics, flat panel display, solar cells and the like. The basic principle is that a beam of light with known polarization state is incident on the surface of a sample to be measured, and the optical constant and the thickness of the film are obtained through the change (amplitude ratio and phase difference) of the polarization state before and after the reflection or transmission of the sample.
Non-uniform areas along the thin film substrate are defects that are often encountered in practice. For example, various plasma chemistry techniques produce films that exhibit such defects. In addition to possible variations in process parameters such as input power, gas flow, operating pressure, substrate temperature stability, and other fundamental effects, as well as electric field dispersion near substrate edges and corners, the film thickness can be non-uniform during growth. For transparent and weakly absorbing films, this non-uniform area is an important defect from an optical point of view. The main effect associated with the non-uniform areas of the film is the coherence of the light, i.e. the difference between the maximum and minimum of the measured optical quantity. If the size of the irradiation spot on the substrate is small, the influence of the uneven area on the measured optical quantity is reduced. However, in commercial optical devices employed in practice, the illumination spot is relatively large. In addition, reducing the spot size may require increasing the detector acquisition light intensity integration time in order to achieve an acceptable signal-to-noise ratio, which is generally undesirable.
Existing methods generally analyze wedge-shaped thickness non-uniformities for special cases, such as the wedge-shaped thickness distribution model analysis of grown zinc oxide (ZnO) on silicon substrates by Shakil Pittal et al, usa. Generally, the thickness of the film with larger thickness is not uniform in the growth process. It is therefore not reasonable to use the above special assumptions in the case of complex thickness inhomogeneities and rapid measurements of large industrial areas.
Disclosure of Invention
The embodiment of the invention provides a source depolarization effect correction modeling method and device for spectrum ellipsometry thickness unevenness, which are used for solving the problems that the existing ellipsometry analysis method analyzes wedge-shaped thickness unevenness under a special condition and uses the special assumption unreasonable under the conditions of complicated thickness unevenness and industrial large-area rapid measurement.
In a first aspect, an embodiment of the present invention provides a method for correcting and modeling a depolarization effect of a source with nonuniform thickness in spectroscopic ellipsometry, including:
and Step1, measuring the sample to be measured by using a Mueller spectrum ellipsometer, and obtaining a reflection Mueller matrix of the sample to be measured.
Step2, deriving the average Mueller matrix reflected by the samples to be detected with different thicknesses according to the reflected Mueller matrix of the samples to be detected;
step3, establishing a multilayer film stack thickness uneven optical model based on the average Mueller matrixes reflected by samples to be measured with different thicknesses, and obtaining a Mueller matrix corresponding to the thickness uneven optical model;
and Step4, performing Gaussian numerical integration on the Mueller matrix corresponding to the optical model with uneven thickness to obtain a Mueller matrix numerical solution of the optical model with uneven thickness.
Step5, based on the Mueller matrix numerical solution of the optical model with uneven thickness, utilizing a 4 x 4 transmission matrix method to carry out forward modeling, calculating a Jones matrix of each node of the optical model with uneven thickness, and converting the Jones matrix into a sample forward modeling average Mueller matrix;
and Step6, matching the measured spectrum after the scattering depolarization effect is separated with the modeling spectrum by using a nonlinear regression algorithm, and extracting the optical parameters and the geometric parameters of the sample to be detected.
Further, in Step1, measuring the sample to be measured by using a muller spectrum ellipsometer, and obtaining a reflection muller matrix of the sample to be measured, specifically including:
and measuring the sample to be measured by the dual-rotation Mueller matrix ellipsometer to obtain a reflection Mueller matrix of the sample to be measured, thereby obtaining depolarization information of the sample to be measured.
Further, in Step2, deriving an average mueller matrix reflected by samples to be measured with different thicknesses according to the reflected mueller matrix of the sample to be measured specifically includes:
assuming that the incident light of the muller spectrum ellipsometer is completely polarized light, the longitudinal height fluctuation characteristic dimension of the film under the elliptical light spot is obviously different, light beams in different polarization states are reflected by the sample, and finally the average muller matrix of the sample is collected by the polarization analysis end of the muller spectrum ellipsometer
Figure BDA0002339337320000031
Comprises the following steps:
Figure BDA0002339337320000032
wherein < R > is the average light intensity reflectance; < psi > is the average polarization ratio angle; < Δ > is the average phase difference angle.
Further, in Step3, establishing an optical model of multilayer film stack thickness unevenness based on the average mueller matrix reflected by samples to be measured with different thicknesses, specifically including:
assuming that the sample to be tested is a stack structure of m layers of thin films, each layer has an average thickness
Figure BDA0002339337320000033
And non-uniform standard deviation σ i (i-1, 2 … m); assuming that the distribution density function is w (t), the Mueller matrix corresponding to the optical model of uneven thickness
Figure BDA0002339337320000034
Comprises the following steps:
Figure BDA0002339337320000035
Figure BDA0002339337320000036
wherein S is an elliptical spot area, M (t) is a Mueller matrix of an ideal thickness stack t；
Figure BDA0002339337320000037
Is the thickness t of the ith layer i Film average thickness of i-th layer
Figure BDA0002339337320000038
Distribution standard deviation of sigma i A distribution density function of (a); sigma i％ The percentage of the thickness distribution standard deviation of the ith layer of film to the average thickness of the ith layer of film is shown;
further, in Step4, the gaussian integral formula obtained by performing gaussian numerical integration on the mueller matrix corresponding to the optical model with uneven thickness is as follows:
Figure BDA0002339337320000041
Figure BDA0002339337320000042
Figure BDA0002339337320000043
in the formula, S is an elliptical light spot area; n is a radical of 1 The number of integral nodes of the thickness of the first layer of film; n is a radical of 2 The number of integral nodes of the thickness of the second layer of film; n is a radical of m The number of integral nodes of the thickness of the mth layer of film; when the thickness of the m layers selects the same node number and distribution form, N is the simplified integral node number; m (T) i ) Corresponding thickness vector T for ith node i The lower ideal sample mueller matrix; w is a i Is the weight of the ith node;
Figure BDA0002339337320000044
is the average thickness vector of m layers of films; sigma ％ As a vector of the percentage of standard deviation of the thickness distribution of the m layers of the film.
Further, in Step5, forward modeling is performed by using a 4 × 4 transmission matrix method, a jones matrix of each node of the optical model with uneven thickness is calculated, and then the jones matrix is converted into a sample forward modeling average mueller matrix, which specifically comprises the following steps:
Figure BDA0002339337320000045
Figure BDA0002339337320000046
wherein T is a thin film transmission matrix; m is the number of layers of the film stack; jones is the Jones matrix; p is the vibration direction of the polarized light parallel to the incident plane, and s is the vibration direction of the polarized light vertical to the incident plane; r is pp The Fresnel reflection coefficient in the pp direction; r is ps Fresnel reflection coefficient in ps direction; r is a radical of hydrogen sp The fresnel reflection coefficient in the sp direction; r is ss The Fresnel reflection coefficient in the ss direction;
Figure BDA0002339337320000047
an incident matrix of an environmental layer; t is jp (-d j ) Transmitting a matrix for the jth film portion; l is t Setting a matrix for the substrate;
Figure BDA0002339337320000048
is kronecker product; j is the complex conjugate matrix of Jones matrix J, A -1 Is the inverse of matrix A; m is a sample forward modeling average Mueller matrix;
the matrix A is:
Figure BDA0002339337320000051
in a second aspect, an embodiment of the present invention provides a modeling apparatus for correcting a depolarization effect of a source of uneven thickness in spectroscopic ellipsometry, including:
and the measuring module is used for measuring the sample to be measured by using the muller spectrum ellipsometer to obtain the reflection muller matrix of the sample to be measured.
The average Mueller matrix derivation module is used for deriving the average Mueller matrices reflected by the samples to be detected with different thicknesses according to the reflected Mueller matrices of the samples to be detected;
the model establishing module is used for establishing a multilayer thin film stack thickness-nonuniform optical model based on the average Mueller matrixes reflected by samples to be measured with different thicknesses, and obtaining a Mueller matrix corresponding to the thickness-nonuniform optical model;
and the Gaussian numerical integration module is used for performing Gaussian numerical integration on the Mueller matrix corresponding to the uneven thickness optical model to obtain a Mueller matrix numerical solution of the uneven thickness optical model.
The ellipsometry parameter calculation module is used for forward modeling by utilizing a 4 x 4 transmission matrix method based on the Mueller matrix numerical solution of the uneven thickness optical model, calculating each node Jones matrix of the uneven thickness optical model, and converting the Jones matrix into a sample forward modeling average Mueller matrix;
and the parameter extraction module is used for matching the measurement spectrum after the separation of the scattering depolarization effect with the modeling spectrum by using a nonlinear regression algorithm, and extracting the optical parameters and the geometric parameters of the sample to be detected.
In a third aspect, an embodiment of the present invention provides an electronic device, which includes a memory, a processor, and a computer program stored in the memory and executable on the processor, where the processor executes the computer program to implement the steps of the multi-source data transmission method according to the second aspect of the present invention.
In a fourth aspect, an embodiment of the present invention provides a non-transitory computer-readable storage medium, on which a computer program is stored, where the computer program, when executed by a processor, implements the steps of the multi-source data transmission method according to the embodiment of the first aspect of the present invention.
The embodiment of the invention provides a source depolarization effect correction modeling method and device for spectrum ellipsometry thickness unevenness, and the traditional ellipsometry analysis method is a simple wedge-shaped thickness distribution hypothesis, or utilizes micro light spots to reduce the depolarization influence of the thickness unevenness even without considering the depolarization effect, or mechanically polishes the surface of a sample and introduces a new surface layer. In the case of complex thickness inhomogeneities and rapid measurement of large industrial areas, it is not reasonable to use the special assumptions mentioned above. On the basis of the traditional ellipsometry analysis method, the invention provides a modeling method for the thickness unevenness of each layer, and fully considers the depolarization influence of the thickness unevenness of the film, thereby more reasonably extracting the optical and geometric parameters of the film.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, and it is obvious that the drawings in the following description are some embodiments of the present invention, and those skilled in the art can also obtain other drawings according to the drawings without creative efforts.
Fig. 1 is a schematic structural diagram of a source depolarization effect correction modeling method for spectroscopic ellipsometry thickness unevenness according to an embodiment of the present invention;
FIG. 2 is a diagram of an optical model of a multilayer thin film stack having non-uniform thickness according to an embodiment of the present invention;
FIG. 3 is a node weight graph of Gaussian distribution, rectangular distribution, and triangular distribution provided by an embodiment of the present invention;
FIG. 4 is a plot of the depolarization index spectra for an example single layer film;
fig. 5 is a block diagram of a source depolarization effect correction modeling apparatus for spectroscopic ellipsometry thickness unevenness according to an embodiment of the present invention;
fig. 6 is a schematic structural diagram of an electronic device according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Reference herein to "an embodiment" means that a particular feature, structure, or characteristic described in connection with the embodiment can be included in at least one embodiment of the application. The appearances of the phrase in various places in the specification are not necessarily all referring to the same embodiment, nor are separate or alternative embodiments mutually exclusive of other embodiments. It is explicitly and implicitly understood by one skilled in the art that the embodiments described herein can be combined with other embodiments.
The traditional ellipsometry analysis method is a simple wedge-shaped thickness distribution hypothesis, or utilizes micro light spots to reduce or even not consider the depolarization effect of uneven thickness, or mechanically polishes the surface of a sample and introduces a new surface layer. In the case of complex thickness inhomogeneities and rapid measurement of large industrial areas, it is not reasonable to use the special assumptions mentioned above.
Therefore, the embodiment of the invention provides a source depolarization effect correction modeling method for uneven thickness in spectroscopic ellipsometry. The following description and description will proceed with reference being made to various embodiments.
Fig. 1 is a schematic structural diagram of a source depolarization effect correction modeling method for spectroscopic ellipsometry thickness unevenness according to an embodiment of the present invention, as shown in fig. 1, the method includes:
the embodiment of the invention provides a source depolarization effect correction modeling method for uneven thickness in spectroscopic ellipsometry, which comprises the following steps:
and Step1, measuring the sample to be measured by using a Mueller spectrum ellipsometer, and obtaining a reflection Mueller matrix of the sample to be measured.
In particular, spectroscopic ellipsometers can be classified into a rotating device type and a phase modulation type. Wherein, only the dual-rotation mueller matrix ellipsometer can obtain all mueller matrix elements by one-time measurement, thereby obtaining the depolarization information of the sample. Therefore, in the embodiment of the invention, the dual-rotation mueller matrix ellipsometer is selected to measure the sample to be measured, all mueller matrix elements of the sample to be measured are obtained at one time, and then the depolarization information of the sample to be measured is obtained. Wherein the depolarization index DI is:
Figure BDA0002339337320000081
in formula (1), DI represents a depolarization index, DI ═ 0, and represents a completely unpolarized mueller matrix, and DI ═ 1, and represents a completely polarized mueller matrix.
And Step2, deriving the average Mueller matrix reflected by the samples with different thicknesses according to the reflected Mueller matrix of the sample to be detected. In the embodiment of the invention, the sample refers to a sample to be detected.
In this embodiment, the stokes vector S of the outgoing beam out Comprises the following steps:
Figure BDA0002339337320000082
in the formula (2), S in And S out Stokes vectors for the incident and emergent beams, respectively; m sample An average mueller matrix for the sample reflections; i is in And I out Total light intensity of the incident light beam and the emergent light beam respectively; q in And Q out The difference between the incident light intensity and the emergent light intensity of the p light beam and the s light beam; u shape in And U out The light intensity difference of incident and emergent + 45-degree linearly polarized light beams and-45-degree linearly polarized light beams is respectively; v in And V out The light intensity difference of the incident and emergent right-handed circularly polarized light beams and the left-handed circularly polarized light beams is respectively; m 11 、M 21 ……M 44 Each represents 16 elements of the mueller matrix of the sample.
Figure BDA0002339337320000083
In the formula (3), M is an isotropic sample reflection or transmission Mueller matrix; psi is the amplitude ratio angle and delta is the phase difference angle.
Assuming that the incident light of the muller spectrum ellipsometer is completely polarized light, the longitudinal height fluctuation characteristic dimension of the film under the elliptical light spot is obviously different, light beams in different polarization states are reflected by the sample, and finally the average muller matrix of the sample is collected by the polarization analysis end of the muller spectrum ellipsometer
Figure BDA0002339337320000091
Comprises the following steps:
Figure BDA0002339337320000092
wherein < R > is the average light intensity reflectance; < psi > is the average amplitude ratio angle; < delta > is the average phase difference angle.
And Step3, establishing a multilayer film stack thickness uneven optical model based on the average Mueller matrix reflected by the samples to be measured with different thicknesses, and obtaining the Mueller matrix corresponding to the thickness uneven optical model.
In particular, the thickness unevenness is a common defect in the growth process of the nano thin film, and is a common depolarization source in ellipsometry. FIG. 2 is a diagram of an optical model of non-uniform thickness of a multilayer thin film stack according to an embodiment of the present invention, in which a sample to be tested is a multilayer thin film stack structure, and the thickness (d 1-dm) of each layer of thin film has an average thickness
Figure BDA0002339337320000093
And non-uniform standard deviation σ i (ii) a Assuming that the distribution density function is w (t), the Mueller matrix corresponding to the optical model of the uneven thickness is
Figure BDA0002339337320000094
Comprises the following steps:
Figure BDA0002339337320000095
Figure BDA0002339337320000096
Figure BDA0002339337320000097
Figure BDA0002339337320000098
wherein S is an elliptical spot area, and M (t) is a Mueller matrix of an ideal thickness stack t;
Figure BDA0002339337320000099
is the thickness t of the ith layer i Film average thickness of i-th layer
Figure BDA00023393373200000910
Distribution standard deviation of sigma i A distribution density function of (a); sigma i％ The percentage of the thickness distribution standard deviation of the ith layer of film to the average thickness of the ith layer of film is shown;
fig. 3 is a node weight graph of gaussian distribution, rectangular distribution and triangular distribution according to an embodiment of the present invention. Common non-uniform thickness distributions are rectangular, triangular, parabolic and gaussian. In this example, for randomly distributed thicknesses, a gaussian distribution density function is chosen. To facilitate determination of the thickness non-uniformity, the standard deviation of the thickness non-uniformity is normalized to a percentage of the average thickness.
And Step4, performing Gaussian numerical integration on the Mueller matrix corresponding to the optical model with uneven thickness to obtain a Mueller matrix numerical solution of the optical model with uneven thickness.
Obtaining an accurate mueller matrix numerical solution of the optical model with uneven thickness by using high-precision Gaussian numerical integration;
the interpolation type numerical integration adopts n equidistant nodes, so the algebraic precision is n-1. Gaussian integration cancels this constraint, algebraic precision is 2n-1, so Gaussian integration formula is chosen. Assuming that the thicknesses of the thin films all satisfy the same thickness distribution and are Gaussian distribution, and the corresponding orthogonal polynomial is a Hermit polynomial, different Gaussian integral nodes and weights can be obtained.
Figure BDA0002339337320000101
Figure BDA0002339337320000102
Figure BDA0002339337320000103
In the formula, S is an elliptical light spot area; n is a radical of 1 The number of integral nodes of the thickness of the first layer of film; n is a radical of 2 The number of integral nodes of the thickness of the second layer of film; n is a radical of m The number of integral nodes of the thickness of the mth layer of film; when the thickness of the m layers selects the same node number and distribution form, N is the simplified integral node number; m (T) i ) Corresponding thickness vector T for ith node i The lower ideal sample mueller matrix; w is a i Is the weight of the ith node;
Figure BDA0002339337320000104
is the average thickness vector of m layers of films; sigma ％ Is a vector of the percentage of standard deviation of the thickness distribution of the m layers of film.
And Step5, based on the Mueller matrix numerical solution of the optical model with uneven thickness, utilizing a 4 x 4 transmission matrix method to carry out forward modeling, calculating each node Jones matrix of the optical model with uneven thickness, and further converting the Jones matrix into a sample forward modeling average Mueller matrix.
The embodiment of the invention is not limited to isotropic materials, and the anisotropic material modeling method is also applicable, so that the 4 × 4 transmission matrix method is selected in the embodiment to obtain 4 elements of the jones matrix at one time.
Selecting reasonable initial thickness t 0 Distribution markInitial value of tolerance sigma 0 And substituting the complex refractive index N (N + ik) (N is the refractive index, and k is the extinction coefficient) of the material into a 4 x 4 transmission matrix method to perform forward modeling to calculate the Jones matrix of each node of the sample to be detected, and then converting the Jones matrix into a Mueller matrix.
Figure BDA0002339337320000111
Figure BDA0002339337320000112
In formula (12), T is a thin film transmission matrix; m is the number of layers of the film stack; jones is the Jones matrix; p is the vibration direction of the polarized light parallel to the incident plane, and s is the vibration direction of the polarized light vertical to the incident plane; r is pp The Fresnel reflection coefficient in the pp direction; r is a radical of hydrogen ps Fresnel reflection coefficient in ps direction; r is sp The fresnel reflection coefficient in the sp direction; r is a radical of hydrogen ss The Fresnel reflection coefficient in the ss direction; l is i -1 An incident matrix of an environmental layer; t is jp (-d j ) Transmitting a matrix for the jth film portion; l is t Setting a matrix for the substrate; in the formula (13), the reaction mixture is,
Figure BDA0002339337320000113
is kronecker product; j is the complex conjugate matrix of Jones matrix J, A -1 Is the inverse of matrix A; m is an average Mueller matrix of the forward modeling of the sample;
the matrix A is:
Figure BDA0002339337320000114
and Step6, matching the measured spectrum after the scattering depolarization effect is separated with the modeling spectrum by using a nonlinear regression algorithm, and extracting the optical parameters and the geometric parameters of the sample to be detected. The measured spectrum is obtained by measuring a sample to be measured by a Step1 through a dual-rotation Mueller matrix ellipsometer, and the modeled spectrum is a Mueller matrix corresponding to an optical model with uneven thickness in a Step 3-Step 5.
Figure BDA0002339337320000115
Figure BDA0002339337320000116
Figure BDA0002339337320000117
Psi and delta are the amplitude ratio and phase difference of the sample p-light and s-light reflection or transmission fresnel coefficients, respectively. In the formula (17), M is the number of fitting spectra, K is the number of fitting parameters, and y correct (λ i θ, φ) is experimental correction data for fixed angle of incidence and azimuth at a single wavelength, y cal (p,λ i And theta, phi) is a calculated model at a single wavelength. p is a K-dimensional vector formed by parameters to be measured of the nano-structure film, omega is the value range of the parameters to be measured,
Figure BDA0002339337320000121
extracting values for the final parameters to be measured; δ y represents the standard deviation of ellipsometric data.
The invention aims to model and analyze depolarization optics of an optical film with rough and uneven surface and extract optical and geometric parameters of the optical film. And accumulating partial polarized light on the elliptic light spot by utilizing numerical integration through the assumption of a reasonable thickness distribution function to obtain an average sample Mueller matrix, and iterating optical and geometric parameters in a numerical inversion mode to obtain an optimal solution.
The method measures the sample to be measured through the muller matrix ellipsometer, obtains all the muller matrix elements of the sample to be measured at one time, and further obtains the depolarization information of the sample. Thickness non-uniformity is a common defect in the growth process of nano thin films, and is a common depolarization source in ellipsometry. And selecting reasonable nodes and weight accumulation to obtain the sample forward modeling average Mueller matrix by reasonably assuming a distribution function of uneven thickness and fully considering the unevenness of each layer thickness. When the thickness distribution of each layer of the sample is random, nodes and weights in Gaussian distribution are adopted, multidimensional calculation is simplified into one-dimensional calculation, and the calculation efficiency is greatly improved. And finally, fitting and extracting information such as optical constants, thickness values and the like of the parameters of the sample to be detected through a nonlinear regression algorithm. The invention can realize the correction of the depolarization effect of the uneven thickness in spectroscopic ellipsometry measurement, can quickly analyze data of various optical thin film devices and online measurement in various nanometer manufacturing processes, and can extract optical and geometric parameters of nanometer materials.
In one single layer thin film example of the present invention (Si + SiO2), the present invention obtains an depolarization spectrum by mueller matrix ellipsometry. Then, we model and fit the ideal model and the thickness unevenness model by the methods in the above embodiments, respectively, to obtain the depolarization index spectrogram in fig. 4. The nodes and weights of the gaussian distribution in fig. 3 are used in consideration of the randomness of the uneven distribution of the thickness. Regardless of the depolarization effect of the thickness non-uniformity, the image fit is not ideal. Considering the thickness non-uniformity optical model in the above embodiment, the measured spectrum and the modeled spectrum match well. Thus proving the rationality and utility of the present invention.
TABLE 1
Fitting value Ideal Thickness Non-Uniformity
SiO2-thickness/nm 3156.66 2988.13
A 1.45055 1.5012
B 0.00513331 0.00529
C 0.00006432 0.000112913
σ t％ 0 2.5019
χ 2 122.286 29.86
Figure BDA0002339337320000131
Table 1 shows the fitting results of the single-layer film example (Si + SiO2), where Ideal represents an Ideal model and Thickness Non-Uniformity represents a Thickness Non-Uniformity optical model in table 1 and fig. 4; DI denotes a depolarization index. A, B and C in Table 1 are three parameters of the refractive index of Cauchy model, σ t Is the standard deviation of the uneven distribution of thickness. In formula (18), n is a refractive index; λ is the wavelength.
Fig. 5 is a block diagram of a modeling apparatus for correcting a depolarization effect of a source of uneven thickness in spectroscopic ellipsometry according to an embodiment of the present invention, and referring to fig. 5, the modeling apparatus for correcting the depolarization effect of the source of uneven thickness in spectroscopic ellipsometry according to an embodiment of the present invention includes:
the measuring module 501 is configured to measure a sample to be measured by using a muller spectrum ellipsometer, and obtain a reflection muller matrix of the sample to be measured.
An average mueller matrix derivation module 502, configured to derive, according to the reflected mueller matrices of the samples to be measured, average mueller matrices reflected by the samples to be measured with different thicknesses;
the model establishing module 503 is configured to establish an optical model with uneven thickness of the multilayer thin film stack based on the average mueller matrices reflected by the samples to be measured with different thicknesses, and obtain a mueller matrix corresponding to the optical model with uneven thickness;
a gaussian numerical integration module 504, configured to perform gaussian numerical integration on the mueller matrix corresponding to the optical model with uneven thickness, so as to obtain a mueller matrix numerical solution of the optical model with uneven thickness.
An ellipsometry parameter calculation module 505, configured to calculate a jones matrix of each node of the uneven-thickness optical model by using a 4 × 4 transmission matrix method for forward modeling based on the mueller matrix numerical solution of the uneven-thickness optical model, and further convert the jones matrix into a sample forward modeling average mueller matrix;
and the parameter extraction module 506 is configured to match the measurement spectrum after the separation of the scattering depolarization effect with the modeling spectrum by using a nonlinear regression algorithm, and extract optical parameters and geometric parameters of the sample to be detected.
Specifically, the modeling apparatus for correcting depolarization effect of spectrum ellipsometry thickness non-uniform source provided in the embodiment of the present invention is specifically configured to perform the steps of the modeling method for correcting depolarization effect of spectrum ellipsometry thickness non-uniform source in the embodiment of the present invention, and since the modeling method for correcting depolarization effect of spectrum ellipsometry thickness non-uniform source has been described in detail in the embodiment, no description is given to functional modules of the modeling apparatus for correcting depolarization effect of spectrum ellipsometry thickness non-uniform source here.
Fig. 6 illustrates an electronic device structure diagram, and as shown in fig. 6, the server may include: a processor (processor)601, a communication Interface (Communications Interface)602, a memory (memory)603 and a communication bus 604, wherein the processor 601, the communication Interface 602 and the memory 603 complete communication with each other through the communication bus 604. The processor 601 may call logic instructions in the memory 603 to perform the following method: and measuring the sample to be measured by using a Mueller spectrum ellipsometer to obtain a reflection Mueller matrix of the sample to be measured. And deducing the average Mueller matrixes reflected by the samples to be detected with different thicknesses according to the reflection Mueller matrixes of the samples to be detected. And establishing an optical model of the multilayer film stack with uneven thickness based on the average Mueller matrix reflected by the samples to be measured with different thicknesses, and obtaining the Mueller matrix corresponding to the optical model with uneven thickness. And performing Gaussian numerical integration on the Mueller matrix corresponding to the optical model with uneven thickness to obtain a Mueller matrix numerical solution of the optical model with uneven thickness. Based on the numerical solution of the Mueller matrix of the optical model with uneven thickness, forward modeling is carried out by using a 4 x 4 transmission matrix method, and each node Jones matrix of the optical model with uneven thickness is calculated and then converted into a sample forward modeling average Mueller matrix. And matching the measured spectrum after the separation of the scattering depolarization effect with the modeling spectrum by using a nonlinear regression algorithm, and extracting optical parameters and geometric parameters of the sample to be detected.
The present embodiment also provides a non-transitory computer readable storage medium having a computer program stored thereon, wherein the computer program is configured to, when executed by a processor, implement the steps of the method as described in the embodiments above. Examples include: and measuring the sample to be measured by using a Mueller spectrum ellipsometer to obtain a reflection Mueller matrix of the sample to be measured. And deducing the average Mueller matrixes reflected by the samples to be detected with different thicknesses according to the reflection Mueller matrixes of the samples to be detected. And establishing an optical model with uneven thickness of the multilayer film stack based on the average Mueller matrix reflected by the samples to be measured with different thicknesses, and obtaining the Mueller matrix corresponding to the optical model with uneven thickness. And performing Gaussian numerical integration on the Mueller matrix corresponding to the optical model with uneven thickness to obtain a Mueller matrix numerical solution of the optical model with uneven thickness. Based on the numerical solution of the Mueller matrix of the optical model with uneven thickness, forward modeling is carried out by using a 4 x 4 transmission matrix method, and each node Jones matrix of the optical model with uneven thickness is calculated and then converted into a sample forward modeling average Mueller matrix. And matching the measured spectrum after the separation of the scattering depolarization effect with the modeling spectrum by using a nonlinear regression algorithm, and extracting optical parameters and geometric parameters of the sample to be detected.
In summary, the embodiments of the present invention provide a source depolarization effect correction modeling method and apparatus for spectroscopic ellipsometry thickness unevenness, and the conventional ellipsometry analysis method is a simple wedge-shaped thickness distribution assumption, or utilizes micro light spots to reduce the depolarization effect of the thickness unevenness and even does not consider the depolarization effect, or mechanically polishes the surface of a sample to introduce a new surface layer. In the case of complex thickness inhomogeneities and rapid measurement of large industrial areas, it is not reasonable to use the special assumptions mentioned above. On the basis of the traditional ellipsometry analysis method, the invention provides a modeling method for the thickness unevenness of each layer, and fully considers the depolarization influence of the thickness unevenness of the film, thereby more reasonably extracting the optical and geometric parameters of the film.
The above-described method embodiments are merely illustrative, wherein the units described as separate parts may or may not be physically separate, and the parts displayed as units may or may not be physical units, may be located in one place, or may be distributed on a plurality of network units. Some or all of the modules may be selected according to actual needs to achieve the purpose of the solution of the present embodiment. One of ordinary skill in the art can understand and implement it without inventive effort.
Through the above description of the embodiments, those skilled in the art will clearly understand that each embodiment can be implemented by software plus a necessary general hardware platform, and certainly can also be implemented by hardware. With this understanding in mind, the above-described technical solutions may be embodied in the form of a software product, which can be stored in a computer-readable storage medium such as ROM/RAM, magnetic disk, optical disk, etc., and includes instructions for causing a computer device (which may be a personal computer, a server, or a network device, etc.) to execute the methods described in the embodiments or some parts of the embodiments.
Finally, it should be noted that: the above examples are only intended to illustrate the technical solution of the present invention, but not to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and such modifications or substitutions do not depart from the spirit and scope of the corresponding technical solutions of the embodiments of the present invention.

Claims (9)

1. A method for correcting and modeling depolarization effect of a source with uneven thickness in spectroscopic ellipsometry is characterized by comprising the following steps:
step1, measuring the sample to be measured by using a Mueller spectrum ellipsometer, and obtaining a reflection Mueller matrix of the sample to be measured;
step2, deriving the average Mueller matrix reflected by the samples to be detected with different thicknesses according to the reflected Mueller matrix of the samples to be detected;
step3, establishing a multilayer film stack thickness uneven optical model based on the average Mueller matrixes reflected by samples to be measured with different thicknesses, and obtaining a Mueller matrix corresponding to the thickness uneven optical model;
step4, performing Gaussian numerical integration on the Mueller matrix corresponding to the optical model with uneven thickness to obtain a Mueller matrix numerical solution of the optical model with uneven thickness;
step5, based on the Mueller matrix numerical solution of the optical model with uneven thickness, utilizing a 4 x 4 transmission matrix method to carry out forward modeling, calculating a Jones matrix of each node of the optical model with uneven thickness, and further converting the Jones matrix into a sample forward modeling average Mueller matrix;
and Step6, matching the measured spectrum after the scattering depolarization effect is separated with the modeling spectrum by using a nonlinear regression algorithm, and extracting the optical parameters and the geometric parameters of the sample to be detected.
2. The method of claim 1, wherein in Step1, the measuring a sample to be measured with a muller spectroscopic ellipsometer to obtain a reflection muller matrix of the sample to be measured, specifically comprises:
and measuring the sample to be measured by the dual-rotation Mueller matrix ellipsometer to obtain a reflection Mueller matrix of the sample to be measured, thereby obtaining depolarization information of the sample to be measured.
3. The method of claim 1, wherein in Step2, the deriving an average mueller matrix of reflections of samples to be tested with different thicknesses according to the reflection mueller matrices of the samples to be tested specifically comprises:
assuming that the incident light of the muller spectrum ellipsometer is completely polarized light, the longitudinal height fluctuation characteristic dimension of the film under the elliptical light spot is obviously different, light beams in different polarization states are reflected by the sample, and finally the average muller matrix of the sample is collected by the polarization analysis end of the muller spectrum ellipsometer
Figure FDA0003789311710000021
Comprises the following steps:
Figure FDA0003789311710000022
wherein < R > is the average light intensity reflectance; < psi > is the average amplitude ratio angle; < Δ > is the average phase difference angle.
4. The method of claim 3, wherein the Step3 of creating the multilayer thin film stack thickness non-uniformity optical model based on the average Mueller matrices for the reflections of samples with different thicknesses comprises:
assuming that the sample to be tested is a stack structure of m layers of thin films, each layer has an average thickness
Figure FDA0003789311710000023
And non-uniform standard deviation σ i I is 1,2 … m; assuming that the distribution density function is w (t), the Mueller matrix corresponding to the optical model of uneven thickness
Figure FDA0003789311710000024
Comprises the following steps:
Figure FDA0003789311710000025
Figure FDA0003789311710000026
wherein S is an elliptical spot area, and M (t) is a Mueller matrix of an ideal thickness stack t;
Figure FDA0003789311710000027
is the thickness t of the ith layer i Film average thickness of i-th layer
Figure FDA00037893117100000210
Standard deviation of distribution is σ i A distribution density function of (a); sigma i％ The standard deviation of the thickness distribution of the ith layer is the percentage of the average thickness of the ith layer.
5. The method as claimed in claim 4, wherein in Step4, the Mueller matrix corresponding to the optical model with uneven thickness is subjected to Gaussian numerical integration, and the Gaussian integral formula is:
Figure FDA0003789311710000028
Figure FDA0003789311710000029
Figure FDA0003789311710000031
in the formula, S is an elliptical light spot area; n is a radical of hydrogen 1 Of the thickness of the first filmThe number of integral nodes; n is a radical of 2 The number of integral nodes of the thickness of the second layer of film; n is a radical of m The number of integral nodes of the thickness of the mth layer of film; when the thickness of the m layers selects the same node number and distribution form, N is the simplified integral node number; m (T) i ) Corresponding thickness vector T for ith node i The lower ideal sample mueller matrix; w is a i Is the weight of the ith node;
Figure FDA0003789311710000032
is the average thickness vector of m layers of films; sigma ％ Is a vector of the percentage of standard deviation of the thickness distribution of the m layers of film.
6. The method as claimed in claim 4, wherein Step5, using a 4 × 4 transmission matrix method for forward modeling, calculates a jones matrix for each node of the optical model with uneven thickness, and converts the jones matrix into a sample forward modeling average mueller matrix, specifically includes:
Figure FDA0003789311710000033
Figure FDA0003789311710000034
wherein T is a thin film transmission matrix; m is the number of layers of the film stack; jones is the Jones matrix; p is the vibration direction of the polarized light parallel to the incident plane, and s is the vibration direction of the polarized light vertical to the incident plane; r is pp The Fresnel reflection coefficient in the pp direction; r is ps Fresnel reflection coefficient in ps direction; r is sp The fresnel reflection coefficient in the sp direction; r is ss The Fresnel reflection coefficient in the ss direction; l is a radical of an alcohol i -1 An incident matrix of an environmental layer; t is jp (-d j ) Transmitting a matrix for the jth film portion; l is t Setting a matrix for the substrate;
Figure FDA0003789311710000035
is kronecker product; j is the complex conjugate matrix of Jones matrix J, A -1 Is the inverse of matrix A; m is an average Mueller matrix of the forward modeling of the sample;
the matrix A is:
Figure FDA0003789311710000036
7. a spectrum ellipsometry thickness unevenness source depolarization effect correction modeling device is characterized by comprising:
the measurement module is used for measuring a sample to be measured by using the muller spectrum ellipsometer to obtain a reflection muller matrix of the sample to be measured;
the average Mueller matrix derivation module is used for deriving the average Mueller matrices reflected by the samples to be detected with different thicknesses according to the reflected Mueller matrices of the samples to be detected;
the model establishing module is used for establishing a multilayer thin film stack thickness-nonuniform optical model based on the average Mueller matrixes reflected by samples to be measured with different thicknesses, and obtaining a Mueller matrix corresponding to the thickness-nonuniform optical model;
the Gaussian numerical integration module is used for performing Gaussian numerical integration on the Mueller matrix corresponding to the uneven thickness optical model to obtain a Mueller matrix numerical solution of the uneven thickness optical model;
the ellipsometry parameter calculation module is used for forward modeling by utilizing a 4 x 4 transmission matrix method based on the Mueller matrix numerical solution of the uneven thickness optical model, calculating each node Jones matrix of the uneven thickness optical model, and converting the Jones matrix into a sample forward modeling average Mueller matrix;
and the parameter extraction module is used for matching the measurement spectrum after the separation of the scattering depolarization effect with the modeling spectrum by using a nonlinear regression algorithm, and extracting the optical parameters and the geometric parameters of the sample to be detected.
8. An electronic device comprising a memory, a processor and a computer program stored on the memory and executable on the processor, wherein the steps of the method according to any of claims 1 to 6 are performed when the program is executed by the processor.
9. A non-transitory computer readable storage medium, on which a computer program is stored, the computer program, when being executed by a processor, implementing the steps of the method according to any one of claims 1 to 6.
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