In-plane distribution data compression method, in-plane distribution measurement method, in-plane distribution optimization method, process apparatus control method, and process control method

1. A process control method used in a production process of a semiconductor device, comprising the steps of:

performing a process over a substrate;

measuring an in-plane distribution over the substrate;

calculating a model formula of the in-plane distribution from measurement data obtained by measurement;

comparing the measurement data with the model formula to calculate a set of parameters of the model formula;

storing the parameters as data of the in-plane distribution over the substrate;

comparing the measurement data with the model formula to detect measurement data which is inconsistent with the model formula;

performing remeasurement for the inconsistent measurement data to update the inconsistent measurement data; and

determining an abnormality in a semiconductor device production apparatus from a variation in the stored parameters,

wherein the measurement data includes measurement coordinates over the substrate, and

the model formula is obtained by modeling the tendency that the in-plane distribution concentrically varies and the tendency that the in-plane distribution varies along a diameter direction and along a tilt of the substrate.

2. The process control method of claim 1 , wherein the model formula is expressed as follows:

F ( x, y )= A 0 +A 1 R+A 2 R 2 +A x x+A y y

where F(x, y) is a measured value at a point (x, y) over the substrate, A is a coefficient of the zeroth order term, A 1 is a coefficient of the first order term, A 2 is a coefficient of the second order term, R is a roundness factor, A x is a X-direction tilt coefficient, and A y is a Y-direction tilt factor.

3. The process control method of claim 1 , wherein in the model formula, centers of concentric circles in the in-plane distribution do not correspond to each other.

4. The process control method of claim 3 , wherein a point (x, y) over the substrate is rotated by θ degree with respect to X-Y coordinate of the substrate.

5. The process control method of claim 4 , wherein the point rotated by θ degree is expressed by the following formula:

X=x cons θ− y cos θ, Y=x sin θ+ y cos θ.

6. The process control method of claim 1 , wherein the model formula is expressed by the following formulas:

F ( x, y )= A 0 +A 1 R+A 2 R 2 +A x x+A y y

R =(X−X 0 ) 2 /B+(Y−Y 0 ) 2 /(1 −B )

where F(x, y) is a measured value at a point (x, y) over the substrate, A is a coefficient of the zeroth order term, A 1 is a coefficient of the first order term, A 2 is a coefficient of the second order term, R is a roundness factor, A x is a X-direction tilt coefficient, A y is a Y-direction tilt factor, X 0 is a shift amount in X direction, Y 0 is a shift amount in Y direction, and θ is a rotation factor.

7. A process control method used in a production process of a semiconductor device, comprising the steps of:

performing a process over a substrate;

measuring an in-plane distribution over the substrate;

calculating a model formula of the in-plane distribution from measurement data obtained by measurement;

comparing the measurement data with the model formula to calculate a set of parameters of the model formula;

storing the parameters as data of the in-plane distribution over the substrate; and

determining an abnormality in a semiconductor device production apparatus from a variation in the stored parameters,

wherein the measurement data includes measurement coordinates over the substrate, and

the model formula is obtained by modeling the tendency that the in-plane distribution concentrically varies and the tendency that the in-plane distribution varies along a diameter direction,

wherein the step of performing over the substrate is a step of performing a first process over the substrate;

the step of measuring an in-plane distribution over the substrate is a step of measuring an in-plane distribution A over the substrate;

the step of calculating the model formula of the in-plane distribution from measurement data obtained by measurement is a step of calculating a model formula A of the in-plane distribution A from measurement data A obtained by measurement;

the step of comparing the measurement data with the model formula to calculate the set of parameters of the model formula is a step of comparing the measurement data A with the model formula A to calculate a set of parameters A of the model formula A; and

the step of storing the parameters as data of the in-plane distribution over the substrate is a step of storing the parameters A as data A of the in-plane distribution A over the substrate,

wherein the process control method further comprises the steps of:

performing a second process over the substrate;

measuring an in-plane distribution B over the substrate;

calculating a model formula B of the in-plane distribution B from measurement data B obtained by measurement;

comparing the measurement data B with the model formula B to calculate a set of parameters B of the model formula B;

storing the parameters B as data B of the in-plane distribution B over the substrate;

comparing the in-plane distribution A and the in-plane distribution B based on the data A and the data B to calculate a correction value such that the uniformity of the in-plane distribution B in the second process improves; and

adjusting wafer orientation based on the correction value,

wherein the measurement data A and the measurement data B include measurement coordinates over the substrate, and

the model formula A and the model formula B are obtained by modeling the tendency that the in-plane distribution A and the in-plane distribution B concentrically vary and the tendency that the in-plane distribution A and the in-plane distribution B vary along a diameter direction.

8. A process control method used in a production process of a semiconductor device, comprising the steps of:

performing a process over a substrate;

measuring an in-plane distribution over the substrate;

calculating a model formula of the in-plane distribution from measurement data obtained by measurement;

comparing the measurement data with the model formula to calculate a set of parameters of the model formula;

storing the parameters as data of the in-plane distribution over the substrate; and

determining an abnormality in a semiconductor device production apparatus from a variation in the stored parameters,

wherein the measurement data includes measurement coordinates over the substrate,

the model formula is obtained by modeling the tendency that the in-plane distribution concentrically varies and the tendency that the in-plane distribution varies along a diameter direction and along a tilt of the substrate,

in the model formula, centers of concentric circles in the in-plane distribution do not correspond to each other, and

a point (x, y) over the substrate is rotated by θ degree with respect to X-Y coordinate of the substrate.

9. The process control method of claim 8 , wherein the point rotated by θ degree is expressed by the following formula:

X=x cons θ− y cos θ, Y=x sin θ+ y cos θ.

10. A process control method used in a production process of a semiconductor device, comprising the steps of:

performing a first process over a substrate;

measuring an in-plane distribution A over the substrate;

calculating a model formula A of the in-plane distribution A from measurement data A obtained by measurement;

comparing the measurement data A with the model formula A to calculate a set of parameters A of the model formula A;

storing the parameters A as data A of the in-plane distribution A over the substrate;

comparing the measurement data A with the model formula A to detect measurement data A which is inconsistent with the model formula A;

performing remeasurement for the inconsistent measurement data A to update the inconsistent measurement data A; and

determining an abnormality in a semiconductor device production apparatus from a variation in the stored parameters,

wherein the measurement data includes measurement coordinates over the substrate, and

the model formula is obtained by modeling the tendency that the in-plane distribution concentrically varies and the tendency that the in-plane distribution varies along a diameter direction,

wherein the process control method further comprises the steps of:

performing a second process over the substrate;

measuring an in-plane distribution B over the substrate;

calculating a model formula B of the in-plane distribution B from measurement data B obtained by measurement;

comparing the measurement data B with the model formula B to calculate a set of parameters B of the model formula B;

storing the parameters B as data B of the in-plane distribution B over the substrate;

comparing the measurement data B with the model formula B to detect measurement data B which is inconsistent with the model formula B;

performing remeasurement for the inconsistent measurement data B to update the inconsistent measurement data B; and

comparing the in-plane distribution A and the in-plane distribution B based on the data A and the data B to calculate a correction value such that the uniformity of the in-plane distribution B in the second process improves,

wherein the measurement data A and the measurement data B include measurement coordinates over the substrate, and

the model formula A and the model formula B are obtained by modeling the tendency that the in-plane distribution A and the in-plane distribution B concentrically vary and the tendency that the in-plane distribution A and the in-plane distribution B vary along a diameter direction.

11. A process control method used in a production process of a semiconductor device, comprising the steps of:

performing a process over a substrate;

measuring an in-plane distribution over the substrate;

calculating a model formula of the in-plane distribution from measurement data obtained by measurement;

comparing the measurement data with the model formula to calculate a set of parameters of the model formula;

storing the parameters as data of the in-plane distribution over the substrate; and

determining an abnormality in a semiconductor device production apparatus from a variation in the stored parameters; and

performing remeasurement for inconsistent measurement data detected by the comparison of the measurement data with the model formula to update the inconsistent measurement data,

wherein the measurement data includes measurement coordinates over the substrate,

the model formula is obtained by modeling the tendency that the in-plane distribution concentrically varies and the tendency that the in-plane distribution varies along a diameter direction and along a tilt of the substrate,

in the model formula, centers of concentric circles in the in-plane distribution do not correspond to each other, and

the model formula is expressed by the following formulas:

F ( x, y )= A 0 +A 1 R+A 2 R 2 +A x x+A y y

R =(X−X 0 ) 2 /B+(Y−Y 0 ) 2 /(1 −B )

where F(x, y) is a measured value at a point (x, y) over the substrate, A is a coefficient of the zeroth order term, A 1 is a coefficient of the first order term, A 2 is a coefficient of the second order term, R is a roundness factor, A x is a X-direction tilt coefficient, A y is a Y-direction tilt factor, X 0 is a shift amount in X direction, Y 0 is a shift amount in Y direction, and θ is a rotation factor.

12. A process control method used in a production process of a semiconductor device, comprising the steps of:

performing a process over a substrate;

measuring an in-plane distribution over the substrate;

calculating a model formula of the in-plane distribution from measurement data obtained by measurement;

comparing the measurement data with the model formula to calculate a set of parameters of the model formula;

storing the parameters as data of the in-plane distribution over the substrate;

determining an abnormality in a semiconductor device production apparatus from a variation in the stored parameters; and

outputting the determination result of the abnormality in the semiconductor device production apparatus,

wherein the measurement data includes measurement coordinates over the substrate, and

the model formula is obtained by modeling the tendency that the in-plane distribution concentrically varies and the tendency that the in-plane distribution varies along a diameter direction and along a tilt of the substrate.

13. The process control method of claim 12 , wherein the model formula is expressed as follows:

F ( x, y )= A 0 +A 1 R+A 2 R 2 +A x x+A y y

where F(x, y) is a measured value at a point (x, y) over the substrate, A is a coefficient of the zeroth order term, A 1 is a coefficient of the first order term, A 2 is a coefficient of the second order term, R is a roundness factor, A x is a X-direction tilt coefficient, and A y is a Y-direction tilt factor.

14. The process control method of claim 12 , wherein in the model formula, centers of concentric circles in the in-plane distribution do not correspond to each other.

15. The process control method of claim 12 , wherein the model formula is expressed by the following formulas:

F ( x, y )= A 0 +A 1 R+A 2 R 2 +A x x+A y y

R =(X−X 0 ) 2 /B+(Y−Y 0 ) 2 /(1 −B )

where F(x, y) is a measured value at a point (x, y) over the substrate, A is a coefficient of the zeroth order term, A 1 is a coefficient of the first order term, A 2 is a coefficient of the second order term, R is a roundness factor, A x is a X-direction tilt coefficient, A y is a Y-direction tilt factor, X 0 is a shift amount in X direction, Y 0 is a shift amount in Y direction, and θ is a rotation factor.