JPH0752153B2 - Ellipsometry - Google Patents

Description

The present invention relates to an ellipsometry method suitable for measuring the physical properties of a multilayer film sample.

[Prior Art] The ellipsometry is a method for determining the physical properties of a sample by measuring the change in the polarization state when polarized light passes through or is reflected by the sample. The basic properties are the film thickness and optical constant of the thin film sample, and the optical constant of the substrate. These are collectively called structural parameters of the sample. This method has the advantage that it has the highest accuracy as a method for measuring the film thickness and that it can be measured nondestructively.

In this method, the polarization state is quantified by two numerical values of a phase difference Δ and an amplitude reflectance ratio tan ψ. As measurement conditions, the incident angle of the light beam, the wavelength of the light beam, etc. are set to appropriate values.

Although the method has a long history and has been studied since the end of the last century, many analysis methods have been proposed corresponding to the sample to be measured.

The method generally used in the past is a method mainly intended for analysis of a single-layer film sample. Two unknowns such as the film thickness and the refractive index of the one-layer film can be obtained by solving the simultaneous equations from the two measurement values Δ and ψ measured under the single measurement condition. Earl, M. A. Asam (RMAAzzam) and N. M. Bashala (NMBasha
r). Ellipsometry and Polarized Light, North Holland Publishing Kampany Amsterdam 1977
('Ellipsometry and polarized light', North−Holla
nd publishing Co., Amsterdam, 1977)) as an example.

However, the above method cannot obtain three or more structural parameters and is not suitable for the analysis of a multilayer film sample. In a general equation, the solution can be found only if the number of measurements is greater than or equal to the number of unknowns. Since a large number of parameters are required to quantify the structure of a multilayer film sample (at least (3m + 2) parameters are needed to quantitatively represent the structure of an m-layer multilayer film), It is necessary to obtain a measured value, and the following method was invented.

Journal of Applied Physics Volume 60 3293
Pages 3302 (P.G.Snyder Etoar;
(PGSnyder et al .; J. Appl. Phys., 60,3293-3302,198
In the SE method (Spectroscopic Ellipsometry) described in 6), the angle of incidence is fixed, measurement is performed using a number of different wavelengths, and the minimum two Multilayer structure parameters are determined by multiplication.

Shin Solid Films, Vol. 28, pp. 167-177 (J. Sarakinos and J. Spirideris (J.
Sarakinos and J. Spyridelis; Thin Solid Films, 28,167
-177,1975)), the MAIE method (Multiple Angle of Incident Ellipsometry)
Murutiple Angle of Incidense Ellipsometry) uses a single-wavelength light source, measures at multiple incident angles, and determines the multi-layer structure parameter by the most squares method from the obtained multiple measured values.

[Problems to be solved by the invention]

In the SE method, which is the first example, the incident angle is fixed and measurement is performed using a number of different wavelengths to determine the multilayer structure parameter.

However, among the multi-layer structure parameters, the refractive index takes different values for each wavelength, so that there is a problem that only a sample whose wavelength dependence of the refractive index is known in advance can be analyzed.

Further, in order to analyze the structure of the multilayer film sample, there is an optimum incident angle specific to each wavelength and each structure of the multilayer film to be measured. The incident angle will be called a characteristic incident angle. Measurement near the characteristic incident angle is important for the analysis of the multilayer structure. However, the SE method has a problem that the measurement accuracy is lowered because the measurement is performed for a plurality of wavelengths at a single incident angle.

In the second example, the MAIE method, the multi-layer structure parameters are determined by measuring at a single wavelength and multiple incident angles. Since only the measured values near the characteristic incident angle are important for determining the parameter, sufficient information cannot be obtained even if a large number of measured values are obtained by the MAIE method. Further, since a single wavelength is used, there is a problem that only a plurality of solutions due to the phase difference of light rays can be obtained. Further, among the multi-layer structure parameters, the film thickness and the refractive index have a large correlation, so that there is a problem that they cannot be accurately determined simultaneously.

An object of the present invention is to analyze the physical properties of a multilayer film sample with high accuracy.

[Means for solving problems]

In the ellipsometry, the above-mentioned purpose is to perform measurement under a plurality of measurement conditions in which the wavelength and the incident angle of the light beam incident on the sample surface are variable, and input all the obtained measured values to the least squares method at the same time. It is achieved by optimizing the structural parameters that represent the physical properties of the surface film layer.

[Action]

In order to obtain sufficient information to determine the structural parameters of the multilayer film sample, independent ellipsometry measurements are performed under multiple measurement conditions. By measuring at multiple incident angles, it is possible to obtain information near the characteristic incident angle necessary for determining the parameter, and by measuring at multiple wavelengths, the problem of multiple solutions caused by the phase difference of light rays can be obtained. Can be solved.

〔Example〕

An embodiment of the present invention will be described in detail below with reference to the drawings.

(1) Measurement FIG. 2 is a schematic diagram of the polarization analyzer used in this example, FIG. 3 is a schematic diagram of the optical system, and FIG. 4 is a block diagram of the polarization analyzer. In this example, an example of the reflection polarization analyzer using the rotation analyzer method is shown, but the same measurement can be performed using another polarization analyzer. Reference numeral 1 is a light source capable of generating by switching a plurality of wavelengths. Light source 1
The generated light ray passes through the polarizer 2 and becomes an incident light ray 7.
The incident light beam 7 is incident on the surface of the sample 4. At this time, the angle formed by the incident light beam 7 and the vertical line of the sample surface is the incident angle 10. The incident ray 7 is reflected by the surface of the sample 4 and becomes a reflected ray 8. When the light rays are reflected, the polarization state changes according to the physical properties of the sample 4. The reflected light beam 8 enters the detector 6 through the rotation analyzer 5 controlled by the computer 20 and is detected. The detection intensity is analyzed by the computer 20 to obtain measured values Δ and ψ.

In the present embodiment, for example, using light rays having a wavelength of 633 nm and 700 nm,
2 at incident angles from 70 to 84 degrees at each wavelength
Measure in steps. Generally, the measurement is performed under l different measurement conditions, and the i-th measurement condition is λ _i and φ.
_{Let i} denote it. However, let λ _{i be} the wavelength of the i-th measurement and φ _{i be} the incident angle of the i-th measurement. At this time, it is assumed that a set of measured values {Δ ₁ , Δ ₂ , ..., Δ _l , ψ ₁ , ψ ₂ , ..., ψ _l } (1) is obtained.

(2) Structure of the multilayer film In order to measure the physical properties of the multilayer film sample, it is necessary to clearly define the structure of the multilayer film. FIG. 3 shows a model of the m-layer multilayer film used in this example. The interface of the film is a parallel plane with an infinite area, and the refractive index changes discontinuously at the interface. The refractive index _S of the substrate and the film thickness d _j and the refractive index _j of each layer are taken as parameters representing the physical properties of this model for light of a certain wavelength. . Note that _j is a complex index of refraction, which is expressed by two equations by the refractive index n _j and the extinction coefficient k _j . _{j = n j -ik j ...... (} 2) the structural parameters of the sample for one wavelength with (3m + 2)
It is assumed that the original vertical vector c is expressed as in Equation 3. Hereafter, vertical vectors are shown in lowercase and boldface, but they are written horizontally for the sake of space.

c = (n ₁ , ..., n _m , n _s , k ₁ , ..., km _m , k _s , d ₁ , ..., d _m ) = (c ₁ , ..., c _{3m + 2} ) ... (3) Since the refractive index and extinction coefficient have different values depending on the wavelength, the structural parameter of the sample for two wavelengths is (5m + 4)
The original vertical vector c'is expressed as in Equation 4.

c ′ = (n ₁ , ..., n _m , n _s , k ₁ , ..., k _m , k _s , n ', ..., n _m ′, n _s ′, k ₁ ′, ..., k _m ′, k _s' d _1,
,, d _m ) = (c ₁ , ..., C _{5m + 4} ) (4) In this embodiment, an example for two wavelengths is shown, but the same can be considered when three or more types of wavelengths are used.

(3) Analysis In order to analyze a multilayer film sample, it is necessary to find the relationship between the structure of the multilayer film and the measured values. The measured values Δ _i and ψ _i are
Structural parameters c ′ and measurement conditions (wavelength λ _i and incident angle φ
_i ) as a function of

Δ _i = f ¹ (λ _i , φ _i ; c ′) ≡f ¹ _i (c ′) …… (5) ψ _i = f ² (λ _i , φ _i ; c ′) ≡f ² _i (c ′ ) (6) (i = ^{1 to} l) The functions f ¹ and f ² are, for example, the above-mentioned “Ellipsometry and polarization”.
It is described in.

A nonlinear least squares method is applied as a method for calculating structural parameters from a large number of measurement values under a large number of measurement conditions.

The estimated value of the structural parameter is c, and the calculated value for c is And the residual r _i is defined as in Equation 9.

However, σ ² (Δ _i ) and σ ² (ψ _i ) are measurement errors of the i-th measurement values Δ _i and ψ _i , respectively.

The least squares condition is a condition that minimizes the residual sum of squares Q.

When initial values of appropriate parameters are given, a set c of parameters satisfying the least squares condition can be obtained by the nonlinear least squares method. In this embodiment, the iterative refinement method (Gaus
s-Newton method). The least squares method is described in "Experimental data analysis by the least squares method" by Nakagawa and Koyanagi (The University of Tokyo Press, 1982) as an example.

If the kth-order estimated value of the parameter is c ^(k) , the improved parameter is c ^{(k + 1)} , and the correction vector is σc ^(k) , then Equation 11 holds.

c ^{(k + 1)} = c ^(k) + δc ^(k) (11) The correction vector can be obtained by the following equation.

δc ^(k) = ( ^(k) W ¹ A ^(K) + ^(k) W ² B ^(k) ) ^-1 · ( ^(k) W ¹ δy ^{1 (k)} + ^(k) W ² δy ^2
(k) ) (12) A ^(k) and B ^(k) are l-by-m matrices called the Jacobian matrix, and their i, j components Aij ^(k) and Bij ^(k) are As shown, it is a partial differential coefficient with respect to the parameter cj of the function f.

^(k) and ^(k) are transposed matrices of A ^(k) and B ^(k) . δy
^{1 (k)} and δy ^{2 (k)} are f-element vertical vectors, and their i components δy ^{1 (k)} i and δy ^{2 (k)} i are given by Equations 15 and 16.

^{δy 1 (k) i = △} i -f 1 i (c (k)) ...... (15) δy 2 (k) i = ψ i -f 2 i (c (k)) ...... (16) (i = ¹ to ¹ ) W ¹ and W ² are called weighting matrices and are diagonal matrices with l rows and 1 columns.
The i-th diagonal components of W ¹ and W ² are 1 / σ ² (△ i), respectively
And 1 / σ ² (ψi).

The Gauss-Newton method has poor stability when the initial value is far from the true value and the nonlinearity of the objective function is large. Therefore, as a stabilizing method, a reduction factor α (0 <α ≦ 1) is introduced into the correction vector. When Q is increased by the improved value of the parameter, if α is reduced, Q
A correction vector is obtained that reduces

c ^{(k + 1)} = c ^(k) + αδc ^(k) (17) (4) Analysis procedure An example of analysis by the above-described Gauss-Newton method algorithm will be described with reference to the flowchart of FIG. .

In step 100, 2l measured under 1 kind of measurement conditions
Enter individual measurements. In step 101, the initial value (estimated value) C ^(o) of the structural parameter is input. Step 102
Now, input the cycle number ICMAX of the least squares method and select the parameter to be optimized. In the above-mentioned Gauss-Newton method algorithm, all the elements of the structure parameter vector C are simplified and described, but in reality, known structure parameters are fixed and unknown. Only the parameters need to be optimized. Here, in order to prevent the formula from becoming complicated, hereinafter, a vector consisting of only parameters to be optimized will be represented by C.

In step 103, the cycle number IC and the reduction factor α are set to an initial value 1. In step 104, the initial value of the parameter Is substituted into C in equations (7) and (8), To calculate. In step 105, the calculated value And the measured values Δi, ψi (i = 1 to l) are substituted into the equation (9),
Calculate the residual ri (i = 1 to 1) and use the equation (10). To calculate. In step 101, partial differential coefficients with respect to the parameters of the functions f ¹ and f ² are calculated, and according to the equations (13) and (14), To calculate. Also, according to equations (15) and (16) To calculate. Substituting the results obtained above into equation (12),
Correction vector Is calculated (step 112). this Is substituted into the equation (17) to obtain the improved value C ⁽¹⁾ of the structural parameter (step 113).

After adding 1 to the value of IC in step 114, the process proceeds to the next cycle, and the process returns to step 104. After that, the same processing is continued. If the number of cycles exceeds the limit value ICMAX,
The process ends (step 106). If the number of cycles IC is neither 1 nor ICMAX, it is determined whether or not Q has increased from the previous cycle (step 108), and if it has increased, α is decreased (step 110) and decreased. In this case, α is increased (step 109).

Although the most basic example of carrying out the present invention using the Gauss-Newton method has been described above, there are many other non-linear least squares algorithms and similar results can be obtained.

〔The invention's effect〕

As described above, in the method of the present invention, in order to obtain sufficient information for determining the structural parameters of the multilayer film sample,
Independent ellipsometry measurements are performed under multiple measurement conditions. By measuring at multiple incident angles, it is possible to obtain information near the characteristic incident angle necessary for determining the parameter, and by measuring at multiple wavelengths, the problem of multiple solutions caused by the phase difference of light rays can be obtained. Can be solved. This allows
Physical properties such as film thickness and refractive index of a multilayer film sample can be determined with high accuracy.

[Brief description of drawings]

FIG. 1 is a flow chart showing an embodiment of the processing procedure of the present invention, FIG. 2 is a schematic diagram of a polarization analyzer according to the present invention, and FIG. 3 is a schematic diagram of an optical system of the polarization analyzer according to the present invention. FIG. 4 is a block diagram of an ellipsometer according to the present invention, and FIG. 5 is a structural schematic diagram of a multilayer film sample measured by the present invention. 1 ... Multiple light sources, 2 ... Polarizer, 3 ... 1/4 wavelength plate, 4 ... Sample, 5 ... Rotation analyzer, 6 ... Detector, 7
…… Incident light, 8 …… Reflected light, 9 …… Sample position drive mechanism, 10 …… Incident angle.

Claims (1)

[Claims] 1. A method comprising: analyzing a polarization state of a light ray that is reflected by a sample or transmitted through the sample and reflected by the sample or transmitted through the sample, wherein in the step, a wavelength and an incident angle of the incident light ray are measured. A plurality of sets of phase differences and amplitude reflectance ratios under a plurality of variable measurement conditions are measured, and all measured values under the plurality of measurement conditions obtained in the step are simultaneously input to the nonlinear least squares method, An ellipsometry method characterized by analyzing physical properties.