Application of statistical technology in semiconductor process

Transcription

1 Application of statistical technology in semiconductor process Zhang, Shangui Beijing North Microelectronics 2008 Beijing ASTS 28~29, October 1

2 Outline Application of SPC method in semiconductor process Application of FDC method in semiconductor process Application of sensitivity analysis in the process Introduction of the process sensitivity analysis method Result of process AEI sensitivity analysis Result of process CDB sensitivity analysis Conclusion 2

3 Application of SPC method in semiconductor process SPC(Statistical Process Control)is a Fault Detection method based on data statistical analysis. 3

4 Application of SPC method in semiconductor process The main function of SPC method in the Advanced Process Control system is to monitor various parameters in the process in real time, and to send alarms in the case of abnormity. 4

5 Application of SPC method in semiconductor process Under the same process monitor condition, the alarming probability of X-chart and D-chart is smaller than that of S-chart and R-chart. 5

6 Application of SPC method in semiconductor process X-chart is suitable to monitor whether each parameter in the process has shifted tremendously for long time. 6

7 Application of SPC method in semiconductor process S-chart and R-chart are suitable to monitor whether each parameter in the process has fluctuated tremendously. 7

8 Application of SPC method in semiconductor process S-chart and R-chart are suitable to monitor whether each parameter in the process has fluctuated tremendously. 8

9 Application of SPC method in semiconductor process D-chart is suitable to monitor whether each parameter in the two continuous processes has shifted tremendously. D-chart is a method derived from X-chart, the function of this two are similar. 9

10 Application of FDC method in semiconductor process 10

11 Application of FDC method in semiconductor process FDC method is a multivariate fault detection and classification method based on principal components analysis. Hotelling T2 and Q(SPE)are the most common two methods to realize FDC. 11

12 Application of FDC method in semiconductor process 12

13 Application of FDC method in semiconductor process 13

14 Introduction of the process sensitivity analysis method DOE Process input variables Param. Param. Param. p1 p2 pm Process output variables Param. Param. Param. p1 p2 pn EXP1 x11 x12 X1m y11 y12 y1n EXP2 x21 x21 X2m y21 y22 y2n EXP3 x31 x32 X3m y31 y32 y3n EXPk xk1 xk2 xkm yk1 yk2 ykn 14

15 Introduction of the process sensitivity analysis method The relationship of process input variables and output variables can be written as the following matrix transformation equation: Process input variable Etching Process output variable According to the cause and effect principle of statistics, there exists potential and definite transfer function between input variables and out put variables. 15

16 Introduction of the process sensitivity analysis method Suppose that the transfer function equation between input and out put variables is as following: y prediction = f(x1, x2,, xk)+ D the above function is a multiple input and single output function, in which: y is a process output variable, x1, x2,, xk are the process output variables, constant D is the estimate of disturbance. In the equation, function f normally is a linear function, its specific expressions can be determined by Multiple linear regression, Ordinary least squares and Neural network regression. 16

17 Introduction of the process sensitivity analysis method On the other hand, as for a multiple input and single output prediction model, the true output variable(the measurement value)is equal to the model prediction value plus model error: y metrology = y prediction +ε= a 1 x 1 + a 2 x a k x k + D +ε Now suppose we have done many groups of process experiment, the above equation can be written as the following matrix format: Y = XA + E In the equation, A is the regression coefficient, E is the error between measurement value and prediction value. 17

18 Introduction of the process sensitivity analysis method The condition of parameter A getting the optimal value is the Sum of Squared Error between the measured value and the predicted value is minimal, which is also to make parameter E satisfying: min J(A)= E T *E =(Y-XA) T *(Y-XA) Simplify the above equation and conduct Partial Derivative on A on both side, we can get: α(j)/α(a)= 0-2X T Y+2X T XA According to the Lagrange Multiplication Principle, we know that the target function J(A)can reach the minimal when α(j)/α(a) Is equal to zero(α(j)/α(a)= 0 ), so we can get: A = inv(x T X)*X T *Y 18

19 Result of process AEI sensitivity analysis SRFPower BRFPower ADI AEI CL2 HBr HeO Case 1:Considering the effect of ADI in the process, the result of AEI sensitivity analysis is as following: AEI = 0.009*Pre *SRFP *BRFP 1.388*CL *HBr *HeO *ADI

20 Result of process AEI sensitivity analysis Model Performance Analysis:the correlation coefficient between the predicted output and the real output is 0.966, the predicted absolute error of the model is 1.51, the predicted relative error is

21 Result of process AEI sensitivity analysis SRFPower BRFPower ADI AEI CL2 HBr HeO Case 2:Without considering the effect of ADI in the process, the result of AEI sensitivity analysis is as following: AEI = 0.053*Pre 0.042*SRFP *BRFP 1.218*CL *HBr *HeO

22 Result of process AEI sensitivity analysis Model Performance Analysis:the correlation coefficient between the predicted output and the real output is 0.941, the predicted absolute error of the model is 1.96, the predicted relative error is

23 Result of process AEI sensitivity analysis The comparison of the results of AEI sensitivity analysis considering ADI and not considering ADI: AEISAR PRE SRFP BRFP CL2 HBr HeO ADI Case Case NaN Conclusion: 1)BRFPower, HeO, ADI are directly proportional to AEI, SRFPower and CL2, HBr are inversely proportional to AEI, the relationship between Pressure and AEI is undecided. 23

24 Result of process AEI sensitivity analysis 2)The order of the effect weight of the input variables to the output variable(aei)is as the above graph. 3)If the spectrum of AEI is known, the adjustment window of the in put variables(process recipe)can be calculated correspondingly. 24

25 Result of process CDB sensitivity analysis SRFPower BRFPower ADI AEI CL2 HBr HeO CDB Case:the number of process experiment samples is 20, sensitivity analysis method is multiple linear regression. CDB = *Pre *SRFP *BRFP 1.34*CL *HBr *HeO

26 Result of process CDB sensitivity analysis Model Performance Analysis:the correlation coefficient between the predicted output and the real output is 0.971, the predicted absolute error of the model is 1.53, the predicted relative error is

27 Result of process CDB sensitivity analysis Conclusion: 1)BRFPower and HeO is directly proportional to CDB, SRFPower and Pressure, CL2, HBr are inversely proportional to CDB. 2)The order of the effect weight of the input variables to the output variable(cdb)is as the above graph. 3)If the spectrum of CDB is known, the adjustment window of the input variable(process recipe)can be calculated correspondingly. 27

28 Conclusion 28