CN114238845B - Temperature compensation correction method for film thickness measurement airflow sensor in wafer measurement - Google Patents
Temperature compensation correction method for film thickness measurement airflow sensor in wafer measurement Download PDFInfo
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- CN114238845B CN114238845B CN202111568748.8A CN202111568748A CN114238845B CN 114238845 B CN114238845 B CN 114238845B CN 202111568748 A CN202111568748 A CN 202111568748A CN 114238845 B CN114238845 B CN 114238845B
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- 238000005259 measurement Methods 0.000 title claims abstract description 24
- 238000000034 method Methods 0.000 title claims abstract description 20
- 238000012937 correction Methods 0.000 title claims abstract description 10
- 238000012360 testing method Methods 0.000 claims abstract description 6
- 238000004364 calculation method Methods 0.000 claims abstract description 4
- 239000011159 matrix material Substances 0.000 claims description 12
- 238000002474 experimental method Methods 0.000 claims description 3
- 238000012545 processing Methods 0.000 claims description 3
- 238000013499 data model Methods 0.000 abstract description 3
- 238000012067 mathematical method Methods 0.000 abstract description 3
- 239000004065 semiconductor Substances 0.000 abstract description 3
- 230000000694 effects Effects 0.000 abstract 1
- 230000007613 environmental effect Effects 0.000 abstract 1
- 238000001179 sorption measurement Methods 0.000 abstract 1
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- 238000005516 engineering process Methods 0.000 description 3
- XUIMIQQOPSSXEZ-UHFFFAOYSA-N Silicon Chemical group [Si] XUIMIQQOPSSXEZ-UHFFFAOYSA-N 0.000 description 1
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
- G06F17/12—Simultaneous equations, e.g. systems of linear equations
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
- G01B13/00—Measuring arrangements characterised by the use of fluids
- G01B13/02—Measuring arrangements characterised by the use of fluids for measuring length, width or thickness
- G01B13/06—Measuring arrangements characterised by the use of fluids for measuring length, width or thickness for measuring thickness
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization
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- Indication And Recording Devices For Special Purposes And Tariff Metering Devices (AREA)
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Abstract
The invention discloses a measuring method for testing the thickness of a wafer through an airflow sensor, which utilizes the adsorption effect of gas to change the conductivity of a semiconductor, performs data calculation correction by assuming an initial value, and utilizes a fitting curve to further measure the thickness of the wafer. The environmental temperature, dielectric constant and the like in the factory can be determined, the corresponding multi-element linear model can be obtained only through multiple experimental tests, and the linear data model is calibrated with certain precision by a mathematical method, so that the measurement precision is improved.
Description
Technical Field
The invention relates to the technical field of wafer films, in particular to a temperature compensation correction method for a film thickness measurement airflow sensor in wafer measurement.
Background
An air flow sensor is a device that can convert air flow information, including gas concentration and type, into acoustic, electrical, optical, or digital information that can be utilized by an operator, instrument, computer, etc. The film thickness was measured by the change in air density and flow direction of the film medium.
A wafer is a silicon wafer used in the fabrication of semiconductor circuits, on which various electronic device circuits can be etched. Compared with developed western countries, the semiconductor industry in China is slow to develop, the technology is still immature in the aspect of measuring the thickness of the wafer film, the measured error value is large, and the industrial detection requirement cannot be met.
Disclosure of Invention
The invention aims to: the invention provides a temperature compensation correction method for a film thickness measurement air flow sensor in wafer measurement, which aims to solve the defects in the prior art.
The technical scheme is as follows: temperature compensation correction method for film thickness measurement airflow sensor in wafer measurement, and measuring wafer film thickness Y=β 0+β1x1+α2T2 +epsilon (1) according to temperature T 2 measured by sensor and dielectric constant beta 1
Wherein beta 0,β1,α2 is a constant, x 1 is a dielectric constant, T 2 is a temperature constant, and epsilon is an error compensation value;
As can be derived from the desired formula,
EY＝β0+β1x1+α2T2 (2)
Recording device
N observations were made for the variable x 1,T2, Y to obtain n sets of observations:
(xi1,Ti2,yi)i＝1,2,……n
If the observation of Y corresponding to (x i1,Ti2) i=1, 2, … … n is considered as a random variable, then it is possible to obtain
(xi1,Ti2,Yi)i＝1,2,……n
And then can obtain
Formula (3) can be written as Y= [ X T ] beta+epsilon (4)
Recording device
The error exists between the value of beta 0,β1,α2 obtained by the equation set and the true value, and the value is an estimated value;
therefore, the above formula processing needs to be rewritten as
The equation is expressed in a matrix form, A and B are expressed as the following matrix, and the estimated value of beta is expressed as
A＝xτx B＝xτy
The method can obtain the following steps:
Obtaining Then, the data curve of the thickness of the obtained wafer film is corrected in the following calculation mode to improve the curve precision;
K1＝h f(xi,yi)
K2＝hf (xi+h,yi+K1)
a further improvement of the present invention is that the temperature compensation test results are obtained by performing an experiment to obtain experimental data, and the values are brought into formula (1).
A further improvement of the present invention is that the value of x 1 is a known constant.
A further improvement of the invention is that the value of T 2 is a known constant.
A further improvement of the invention is that the values of matrix a can be calculated.
A further improvement of the invention is that the values of matrix B can be calculated.
Compared with the prior art, the temperature compensation correction method for the film thickness measurement air flow sensor in wafer measurement at least has the following beneficial effects:
Through experimental tests, a corresponding multiple linear model can be obtained, a certain precision calibration is carried out on the linear data model by using a mathematical method, the measurement precision is improved, the wafer film thickness measurement technology is improved, the measured error value is small, and the industrial detection requirement is met.
Drawings
FIG. 1 is a flow chart of the wafer thickness measurement calibration of the air flow sensor of the present invention;
FIG. 2 is a graph showing the comparison of the calibration algorithm of the air flow sensor according to the present invention with the sample parameters;
Fig. 3 is a temperature compensation comparison chart.
Detailed Description
Various exemplary embodiments of the invention will now be described in detail. It should be noted that: the relative arrangement of the components and steps, numerical expressions and numerical values set forth in these embodiments are not limiting to the scope of the present invention unless it is specifically stated otherwise, and the following description of at least one exemplary embodiment is merely illustrative in nature and is in no way intended to limit the invention, its application, or uses. Techniques, methods, and apparatus known to one of ordinary skill in the relevant art may not be discussed in detail, but are intended to be part of the specification where appropriate. In all examples shown and discussed herein, any specific values should be construed as merely illustrative, and not a limitation.
Referring to fig. 1-3, a temperature compensation correction method for a film thickness measuring air flow sensor in wafer measurement measures the thickness y=β 0+β1x1+α2T2 +ε (1) of a wafer film according to the measured temperature T 2 and the dielectric constant β 1 of the sensor
Wherein beta 0,β1,α2 is a constant, x 1 is a dielectric constant, T 2 is a temperature constant, and epsilon is an error compensation value;
As can be derived from the desired formula,
EY＝β0+β1x1+α2T2 (2)
Recording device
N observations were made for the variable x 1,T2, Y to obtain n sets of observations:
(xi1,Ti2,yi)i＝1,2,……n
If the observation of Y corresponding to (x i1,Ti2) i=1, 2, … … n is considered as a random variable, then it is possible to obtain
(xi1,Ti2,Yi)i＝1,2,……n
And then can obtain
Formula (3) can be written as Y= [ X T ] beta+epsilon (4)
Recording device
The error exists between the value of beta 0,β1,α2 obtained by the equation set and the true value, and the value is an estimated value;
therefore, the above formula processing needs to be rewritten as
The equation is expressed in a matrix form, A and B are expressed as the following matrix, and the estimated value of beta is expressed as
A＝xτx B＝xτy
The method can obtain the following steps:
The value of x 1T2 is a known constant, and the values in the matrix A and the matrix B can be calculated and can be obtained correspondingly Then, the data curve of the thickness of the obtained wafer film is corrected in the following calculation mode to improve the curve precision;
K1＝h f(xi,yi)
K2＝h f(xi+h,yi+K1)。
the temperature compensation test results experimental data were obtained by performing a plurality of experiments, and the values were brought into formula (1).
As shown in fig. 3, the distance fitting curve without temperature compensation has a large deviation value, so that the setting of the temperature compensation parameter is important.
In summary, the temperature compensation correction method for the film thickness measurement air flow sensor in wafer measurement provided by the invention uses a mathematical method to calibrate a linear data model to a certain degree of accuracy, improves measurement accuracy, improves the wafer film thickness measurement technology, has small measured error value, and meets the industrial detection requirement.
While certain specific embodiments of the invention have been described in detail by way of example, it will be appreciated by those skilled in the art that the above examples are for illustration only and are not intended to limit the scope of the invention. It will be appreciated by those skilled in the art that modifications may be made to the above embodiments without departing from the scope and spirit of the invention. The scope of the invention is defined by the appended claims.
Claims (6)
1. A temperature compensation correction method for a film thickness measurement airflow sensor in wafer measurement is characterized in that the film thickness Y=β 0+β1x1+α2T2 +epsilon (1) of a wafer is measured according to the measured temperature T 2 and the dielectric constant beta 1 of the sensor
Wherein beta 0,β1,α2 is a constant, x 1 is a dielectric constant, T 2 is a temperature constant, and epsilon is an error compensation value;
As can be derived from the desired formula,
EY＝β0+β1x1+α2T2 (2)
Recording device
N observations were made for the variable x 1,T2, Y to obtain n sets of observations:
(xi1,Ti2,yi) i＝1,2,……n
If the observation of Y corresponding to (x i1,Ti2) i=1, 2, … … n is considered as a random variable, then it is possible to obtain
(xi1,Ti2,Yi) i＝1,2,……n
And then can obtain
Formula (3) can be written as Y= [ X T ] beta+epsilon (4)
Recording device
The error exists between the value of beta 0,β1,α2 obtained by the equation set and the true value, and the value is an estimated value;
therefore, the above formula processing needs to be rewritten as
The equation is expressed in a matrix form, A and B are expressed as the following matrix, and the estimated value of beta is expressed as
A＝xτx B＝xτy
The method can obtain the following steps:
Obtaining Then, the data curve of the thickness of the obtained wafer film is corrected in the following calculation mode to improve the curve precision;
K1＝hf(xi,yi)
K2＝hf(xi+h,yi+K1)。
2. The method for correcting temperature compensation of a film thickness measuring air flow sensor in wafer measurement according to claim 1, wherein the temperature compensation test result is obtained by performing an experiment to obtain experimental data, and the value is brought into (1).
3. The method of claim 1, wherein the value of x 1 is a known constant.
4. The method of claim 1, wherein the value of T 2 is a known constant.
5. The method of claim 1, wherein the values of the matrix a are calculated.
6. The method of claim 1, wherein the values of matrix B are calculated.
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