CN109272216B

CN109272216B - Statistical process control method for zero excess particle number in ultra-clean room

Info

Publication number: CN109272216B
Application number: CN201811013365.2A
Authority: CN
Inventors: 游海龙; 张金力; 田文星; 贾新章; 顾铠
Original assignee: Xidian University
Current assignee: Xidian University
Priority date: 2018-08-31
Filing date: 2018-08-31
Publication date: 2021-09-10
Anticipated expiration: 2038-08-31
Also published as: CN109272216A

Abstract

The invention discloses a statistical process control method for the number of particles with excessive zero in an ultra-clean room, which mainly solves the problem that the existing statistical process control technology has excessive false alarm or can not give an alarm in time when the zero proportion of the number of particles in the ultra-clean room is more than 50 percent. The implementation steps are as follows: 1. collecting the particle number in the air by using a particle number counter, wherein the collected data at least comprises 90 non-zero data; 2. obtaining the probability of different particle numbers; 3. calculating the estimated values of the parameters c and T in the threshold Poisson distribution by using an iterative method to obtain an upper control limit; 4. according to the drawing method of the Huhart control chart, drawing the particle number and the control line into the corresponding control chart; 6. whether the particle number in the ultra-clean room is controlled or not is judged by observing whether the control chart slightly exceeds the upper control line or not, whether production is continued or not is really carried out, so that timely early warning when the particle number is out of control is realized, the product quality is improved, the economic loss caused by out of control is reduced, and the method can be used for quality monitoring of semiconductor production.

Description

Statistical process control method for zero excess particle number in ultra-clean room

Technical Field

The invention belongs to the technical field of semiconductors, and particularly relates to a statistical process control method which can be used for monitoring whether the particle number is in a controlled state when the particle number in an ultra-clean room is excessive and zero.

Background

"statistical process control" is today one of the most popular and effective quality improvement methods. The statistical process control technology mainly refers to monitoring the quality characteristics of products in each stage, namely working procedures, in the production process by using a Huhart process control theory, namely a control chart, analyzing the trend of the quality characteristics according to the point distribution condition on the control chart, and taking preventive measures to ensure that the production process is in a statistical control state, thereby achieving the purposes of improving and ensuring the quality.

As chip density increases and the feature size of semiconductor devices decreases in semiconductor manufacturing, particle contamination has an increasing impact on device yield, performance, and reliability, and therefore, there is a need for close monitoring of airborne particles to ensure that airborne particles are in a controlled state. The number of particles in air has different diameters, the larger the diameter the greater the impact on device yield, performance and reliability, while certain particles with particularly small diameters have little impact on the device, and therefore, generally only particles with diameters larger than a certain value are of interest. The method comprises the following steps: the particle counter is used for measuring the number of particles with the particle diameter larger than a set value in the air, and then the control chart is used for monitoring. The c control map, which is the control map that is most easily conceivable when selecting the control map, is used on the premise that the particle count can better comply with the poisson distribution. In the case where the particle diameter of interest is relatively large or the environmental cleanliness is extremely high, the proportion of particles with a zero number is high, typically more than 50%, and this phenomenon is called the zero excess phenomenon. In this case the particle count does not follow the poisson distribution well, which would lead to excessive false alarms if the c-control map is still used. In addition, in practical application, the phenomenon that the Neyman model and the Gamma-Poisson model which are widely applied to describing the defect number cannot well describe the phenomenon that the particle number is excessive and zero is found, and if corresponding control charts are still used, early warning cannot be timely realized.

The prior art explains the zero-over phenomenon as a process that is so advanced that it reaches the "near-zero defect" level where defects only occasionally appear, and Rider (1962) defines this distribution as a zero-inflected distribution. Under the explanation, some zero-affected models such as ZIP, ZINB and GZIP are proposed, but the effect is not ideal when the control chart corresponding to the ZIP, ZINB and GZIP models is used for carrying out statistical process control in the monitoring of zero excess particle number.

Disclosure of Invention

The invention aims to provide a statistical process control method of the number of the excessive zero particles in the ultra-clean room aiming at the defects of the prior art, so as to accurately monitor the number of the excessive zero particles in the ultra-clean room and improve the yield, the performance and the reliability of the semiconductor device.

The technical scheme of the invention is realized as follows:

technical principle

Actual particle number X considered by the invention_ASubject to a Poisson distribution, and the measured particle diameter set value corresponds to a counting threshold T, the measurement data X obtained by the particle counter_B＝max(X_A-T,0)，

The true particle number is assumed to be the poisson distribution obeying a mean value c:

from X_B＝max(X_A-T,0) knowing:

p(X_B＝m)＝p(X_A＝m+T),m＝1,2,…

namely, it is

In the present invention, called X_BThe obeyed distribution is threshold Poisson distribution, model parameters c and T are estimated according to sample data, and then a control chart is established according to the establishment principle of the control chart, so that the statistical process control of the number of the excessive zero particles in the ultra-clean room is completed.

Second, implementation scheme

According to the principle, the implementation steps of the invention comprise the following steps:

(1) collecting a sample: measuring the number of particles with the diameter larger than a set value in the air once every other time by using a particle number counter, wherein the obtained sample data at least comprises 90 non-zero data;

(2) obtaining sample data according to the step (1), and obtaining the sample probability p with the particle number of m_mAnd sample probability mean

(3) Obtaining parameter estimation values of a real particle number mean value c and a threshold value T in a threshold Poisson distribution model by using an iterative method according to sample probabilities and sample probability mean values of different particle numbers:

3a) setting an initial value T of a count threshold T₀＝0；

3b) According to T₀Obtaining the initial value c of the mean value c of the number of real particles₀：

Wherein the content of the first and second substances,

represents a degree of freedom of 2 (T)₀+1) χ²P of distribution₀A lower quantile;

3c) obtaining T by the following formula₀And c₀Corresponding theoretical probability value p of m particle number_0m：

3d) According to p_mAnd p obtained in step 4c)_0mObtaining an iteration point T₀And c₀Corresponding decision coefficient R²(T₀,c₀)：

3e) Let T next iteration point T₁＝T₀+1, according to T₁Value, obtaining the corresponding next iteration point c of c₁：

Wherein the content of the first and second substances,

represents a degree of freedom of 2 (T)₁+1) χ²P of distribution₀A lower quantile;

3f) obtaining T by the following formula₁And c₁Corresponding theoretical probability value p of m particle number_1m：

3g) According to p_mAnd p obtained in step 4f)_1mCalculating an iteration point T₁And c₁Corresponding decision coefficient R²(T₁,c₁)；

3h) R obtained in the above is²(T₀,c₀) And R²(T₁,c₁) And (3) comparison:

if R is²(T₀,c₀)≤R²(T₁,c₁) Then let T₀＝T₁，R²(T₀,c₀)＝R²(T₁,c₁) Returning to the step 3 e);

if R is²(T₀,c₀)＞R²(T₁,c₁) Then, stopping iteration to obtain estimated values of c and T: t ═ T₀,，c＝c₀；

(4) From the estimated values of c and T, the upper control line UCL of the control map is obtained:

(5) drawing the upper control line obtained in the step (4) and the sample data acquired in the step (1) into a control chart according to a drawing method of a Huhart control chart;

(6) judging whether the particle number in the ultra-clean room is controlled according to the control chart obtained in the step (5):

if the data point falls below the upper control line, it indicates that the particle count in the ultraclean is controlled and production continues;

if the data point is above the upper control line, the particle number in the ultra-clean room is out of control, the production needs to be stopped, the out-of-control reason is searched, and corresponding measures are taken.

The invention has the following advantages:

1. distribution of over zero particles

Compared with the existing Poisson distribution, ZIP, GZIP, Neyman and Gamma-Poisson, the threshold Poisson distribution provided by the invention can better describe the distribution with excessive zero particle number.

2. Accurate parameter estimation

The parameter estimation method provided by the invention iterates according to the zero-occupation ratio to obtain the estimated value of the parameter, so that the parameter estimation is influenced little when abnormal points exist, namely certain measured values are extremely large, and the parameter estimation is more accurate.

Drawings

FIG. 1 is a flow chart of an implementation of the present invention;

fig. 2 is a control chart drawn according to the collected sample in the present invention.

Detailed Description

The invention will be further described with reference to the drawings, which take the number of particles collected in an ultra-clean room of an integrated circuit manufacturing company as an example.

Referring to fig. 1, the implementation steps of this example are as follows:

step 1: a sample is collected.

A Lasair II-100 particle counter is used for collecting the number of particles with the particle diameter larger than 0.5um in an ultra-clean room with the cleanliness of 100 grades, and data are collected once per minute. After 250 samples are continuously collected, the obtained zero data is 90, and the sample data is shown in table 1.

TABLE 1 sample data with zero excess particle count

Step 2: obtaining sample data according to the step (1), and obtaining the sample probability p with the particle number of m_mAnd sample probability mean

2a) Counting the sample data in table 1 to obtain the sample data with the total number k of 250, the maximum value M of the particle number of 5 and the number k of the sample with the particle number of M_m；

2b) Calculating the sample probability p of the number of particles m_m

The calculation results of step 2b) above are shown in table 2:

TABLE 2 particle count statistics

Number of particles (m)	0	1	2	3	4	5
							Number of samples k_m	160	49	27	11	2	1
Sample probability p_m	0.640	0.196	0.108	0.044	0.008	0.004

2c) Obtaining mean values of sample probabilities of different particle counts from the data of Table 2

And step 3: sample probability and sample probability mean value according to different particle numbers in table 2

Obtaining the parameter estimation values of the mean value c of the number of real particles and the threshold T in the threshold Poisson distribution model by using an iterative method:

the specific implementation of this step is as follows:

3a) setting an initial value T of a count threshold T₀＝0；

Wherein the content of the first and second substances,

3d) According to p_mAnd p obtained in step 3c)_0mObtaining an iteration point T₀And c₀Corresponding decision coefficient R²(T₀,c₀)：

3e) Let T next iteration point T₁＝T₀+1, according to T₁Value of c, obtaining the next of the corresponding cIteration point c₁：

Wherein the content of the first and second substances,

3g) According to p_mAnd p obtained in step 3f)_1mCalculating an iteration point T₁And c₁Corresponding decision coefficient R²(T₁,c₁)；

T2 and c 2.1357 are calculated.

And 4, step 4: and calculating an upper control line of the control chart according to the result obtained in the step 3:

and 5: and drawing a control chart.

5a) Newly building a plane rectangular coordinate system A, and drawing the upper control line UCL obtained in the step (4) in the plane rectangular coordinate system A;

5b) and (3) marking the sample data obtained in the step (1) on a coordinate system A according to a measurement sequence, and then connecting the data points by using a broken line according to the measurement sequence to obtain a graph which is a control graph, as shown in the attached figure 2.

Step 6: and determining the production condition according to the particle number in the ultra-clean room.

By observing fig. 2, it is judged whether the number of particles in the ultra-clean room is controlled:

if the data point falls below the upper control line, indicating that the particle count in the ultraclean is controlled, production continues;

if the data point is above the upper control line, which indicates that the particle number in the ultra-clean room is out of control, the production is stopped, the reason for the out of control is searched, and corresponding measures are taken.

This example can be seen in fig. 2, all data points are below the upper control line, so the control chart is normal, indicating that the production process is controlled and production can continue.

The effect of the present invention can be illustrated by the sample data collected by 15 Lasair II-100 particle counters in a clean room of an IC manufacturing company:

sample data collected by 15 Lasair II-100 particle counters are shown in Table 3:

TABLE 3 sample data

Particle counter numbering	Number k of samples having particle number m_m
		A-1	k₀＝1067，k₁＝73，k₂＝26，k₃＝11，k₄＝1
A-2	k₀＝880，k₁＝166，k₂＝87，k₃＝38，k₄＝1，k₅＝1，k₁₀＝1，k₁₁＝1
		A-3	k₀＝793，k₁＝129，k₂＝84，k₃＝63，k₄＝3，k₅＝2，k₆＝2，k₁₂＝1
A-4	k₀＝959，k₁＝86，k₂＝24，k₃＝5
		A-5	k₀＝620，k₁＝40，k₂＝17，k₃＝6，k₄＝1
A-6	k₀＝506，k₁＝84，k₂＝49，k₃＝40，k₄＝3，k₅＝1，k₁₀＝1
		A-7	k₀＝588，k₁＝53，k₂＝19，k₃＝10
A-8	k₀＝500，k₁＝79，k₂＝54，k₃＝32，k₄＝3
		A-9	k₀＝186,k₁＝65,k₂＝27,k₃＝17,k₄＝4,k₅＝0,k₆＝1,k₇＝1,k₈＝1,k₉＝1
A-10	k₀＝167，k₁＝28，k₂＝23，k₃＝9，k₄＝5，k₅＝1，k₆＝2，k₇＝2，k₈＝1，k₁₀＝2，k₁₃＝1，k₂₁＝1
		A-11	k₀＝164，k₁＝36，k₂＝17，k₃＝7，k₄＝8，k₅＝2，k₆＝2，k₇＝1，k₁₁＝1，k₃₀＝1
A-12	k₀＝139，k₁＝45，k₂＝26，k₃＝7，k₄＝6，k₅＝4，k₆＝2，k₉＝1，k₁₂＝1，k₁₇＝1，k₁₉＝1，k₂₀＝1，k₄₀＝1
		A-13	k₀＝136，k₁＝37，k₂＝25，k₃＝6，k₄＝5，k₅＝4，k₆＝1，k₈＝1，k₉＝1，k₁₀＝1，k₁₃＝1，k₁₉＝1
A-14	k₀＝137，k₁＝34，k₂＝19，k₃＝9，k₄＝6，k₅＝1，k₁₀＝1，k₁₁＝1，k₁₂＝1，k₁₃＝1，k₂₆＝1，k₂₇＝1，k₅₁＝1
		A-15	k₀＝140，k₁＝37，k₂＝14，k₃＝5，k₄＝6，k₅＝1，k₆＝2，k₈＝3，k₉＝2，k₁₀＝1，k₁₄＝1

The data in table 3 should be compared and analyzed by using the threshold Poisson distribution provided by the invention and Poisson distribution in the prior art, namely ZIP, GZIP, Neyman and Gamma-Poisson, so as to prove that the threshold Poisson distribution provided by the invention is superior to other models.

The quality of the model can be evaluated through a judgment coefficient, the Chichi information content AIC and the Bayesian information content BIC: the closer the decision coefficient is to 1, the better the model is, the smaller the AIC, the better the model is, and the smaller the BIC, the better the model is.

The data in table 3 were analyzed to obtain the corresponding decision coefficients, AIC and BIC of each model, and the results are shown in tables 4, 5 and 6.

TABLE 4 decision coefficients for each model

Particle counter numbering	Threshold Poisson distribution of the invention	Poisson distribution	ZIP	GZIP(n＝2)	GZIP(n＝3)	Neyman	Gamma-poisson
								A-1	1.0000	0.9923	1.0000	0.9981	0.9981	1.0000	0.9996
A-2	0.9997	0.9451	0.8159	0.9872	0.9872	0.9995	0.9974
								A-3	0.9986	0.8842	0.9983	0.9915	0.9915	0.9983	0.9895
A-4	1.0000	0.9957	1.0000	0.9962	0.9962	1.0000	0.9998
								A-5	1.0000	0.9908	1.0000	0.9984	0.9984	1.0000	0.9995
A-6	0.9986	0.8891	0.9982	0.9908	0.9908	0.9981	0.9901
								A-7	0.9999	0.9860	0.9999	0.9972	0.9972	0.9998	0.9992
A-8	0.9989	0.8945	0.9975	0.9929	0.9929	0.9930	0.9826
								A-9	0.9974	0.8899	0.9510	0.9557	0.9565	0.9789	0.9986
A-10	0.9986	0.6552	0.8997	0.9916	0.9916	0.9145	0.9947
								A-11	0.9981	0.7304	0.8354	0.9839	0.9839	0.8445	0.9713
A-12	0.9970	0.5814	0.5813	0.9681	0.9681	0.5948	0.8909
								A-13	0.9971	0.7075	0.8094	0.9759	0.9759	0.8434	0.9833
A-14	0.9987	0.4642	0.6962	0.9832	0.9832	0.7010	0.9087
								A-15	0.9954	0.7187	0.8747	0.9738	0.9738	0.8987	0.9939
Mean value of decision coefficient	0.9985	0.8217	0.8972	0.9856	0.9857	0.9176	0.9799
								Standard deviation of decision coefficients	0.0014	0.1677	0.1294	0.0124	0.0123	0.1254	0.0337

As can be seen from table 4, the minimum value of the decision coefficient is 0.9954, which indicates that the threshold poisson distribution provided by the present invention can well describe real data, and meanwhile, the comparison with other models shows that the average value of the corresponding decision coefficient is the largest, the standard deviation is the smallest, which indicates that the threshold poisson distribution provided by the present invention is superior to other models.

TABLE 5 AIC for each model

Particle counter numbering	Threshold Poisson distribution of the invention	Poisson distribution	ZIP	GZIP(n＝2)	GZIP(n＝3)	Neyman	Gamma-poisson
								A-1	940.52	1052.63	938.83	970.96	974.96	942.42	945.82
A-2	1980.93	2168.48	1995.28	2048.08	2052.08	1985.57	1990.55
								A-3	1976.83	2265.15	1972.02	2008.55	2012.55	1966.83	1999.53
A-4	894.16	941.80	893.12	1035.22	1039.22	893.89	897.10
								A-5	547.96	624.44	549.10	642.25	646.25	549.58	554.70
A-6	1239.67	1413.71	1235.53	1292.95	1296.95	1233.83	1252.97
								A-7	652.22	722.35	653.51	759.09	763.09	652.49	657.63
A-8	1157.24	1284.82	1142.76	1227.66	1231.66	1154.29	1175.62
								A-9	717.01	783.01	742.95	744.89	734.57	717.65	725.88
A-10	624.09	868.03	800.50	587.26	591.26	723.99	577.80
								A-11	628.90	818.86	984.22	599.38	594.57	883.96	563.83
A-12	953.53	1142.07	1788.36	683.51	687.51	1579.48	698.63
								A-13	634.90	781.83	794.85	610.48	614.48	708.31	574.23
A-14	938.87	1224.97	2178.30	602.96	606.96	1986.25	609.39
								A-15	608.18	731.71	673.38	587.86	591.86	616.77	518.49

As can be seen from table 5, the AIC corresponding to the threshold poisson distribution proposed by the present invention is smaller than that of other models in most cases, which can show that the threshold poisson distribution proposed by the present invention is superior to that of other models.

TABLE 6 BIC for each model

Particle counter numbering	Threshold Poisson distribution of the invention	Poisson distribution	ZIP	GZIP(n＝2)	GZIP(n＝3)	Neyman	Gamma-poisson
								A-1	950.67	1062.77	948.98	991.25	1005.39	952.57	955.96
A-2	1991.07	2178.62	2005.41	2068.35	2082.49	1995.71	2000.69
								A-3	1986.79	2275.11	1981.98	2028.48	2042.44	1976.79	2009.49
A-4	904.12	951.76	903.07	1055.14	1069.10	903.85	907.06
								A-5	557.02	633.49	558.16	660.37	673.42	558.64	563.75
A-6	1248.73	1422.77	1244.59	1311.06	1324.12	1242.89	1262.03
								A-7	661.24	731.37	662.53	777.12	790.13	661.50	666.64
A-8	1166.25	1293.82	1151.77	1245.68	1258.69	1163.30	1184.63
								A-9	724.44	790.44	750.38	759.75	756.85	725.08	733.30
A-10	631.07	875.01	807.48	601.22	612.19	730.96	584.78
								A-11	635.85	825.81	991.17	613.29	615.43	890.91	570.78
A-12	960.45	1148.99	1795.28	697.35	708.27	1586.40	705.55
								A-13	641.68	788.61	801.62	624.04	634.82	715.09	581.00
A-14	945.59	1231.70	2185.02	616.41	627.13	1992.98	616.11
								A-15	614.89	738.42	680.10	601.28	612.00	623.48	525.20

As can be seen from table 6, the BIC corresponding to the threshold poisson distribution proposed by the present invention is smaller than that of other models in most cases, which indicates that the threshold poisson distribution proposed by the present invention is superior to that of other models.

Claims

1. A statistical process control method for zero excess particle number in ultra-clean room is characterized by comprising the following steps:

3a) setting an initial value T of a count threshold T₀＝0；

Wherein the content of the first and second substances,

Wherein the content of the first and second substances,

if R is²(T₀,c₀)＞R²(T₁,c₁) Then, stopping iteration to obtain estimated values of c and T: t ═ T₀,c＝c₀；

2. The method of claim 1, wherein step (2) is performed by:

2a) counting the sample data obtained in the step (1), recording the total number of the sample data as k, recording the maximum value of the sample data as M, and recording the number of the samples with the particle number of M as k_m，m＝0,1,…,M；

2b) Calculating the sample probability p of the number of particles m_m

2c) Obtaining mean values of sample probabilities of different particle counts

3. The method of claim 1, wherein step (5) is performed by:

5b) and (3) marking the sample data obtained in the step (1) on a coordinate system A according to the measurement sequence, and then connecting the data points by using a broken line according to the measurement sequence.