Plasma fluid modeling with transient to stochastic transformation

1. A computer method for simulating plasma density distribution within a plasma reactor that is responsive to a first variable having a first time basis and a second variable having a second time basis, wherein the first time basis is different from the second time basis the computer method comprising executing computer program codes according to a process comprising:

approximating a steady state of a transient sheath behavior of a plasma occurring in the plasma reactor in response to the first variable, the approximating including:

defining a model domain that is characteristic of the plasma reactor;

determining a response function of a time dependent variable across the model domain with respect to the second time basis;

constructing a probability function for the response function;

randomly sampling the probability function; and

transferring the random sampling into a spatial matrix representing the steady state; and

performing, on the spatial matrix, a time-domain simulation of the transient plasma sheath behavior occurring in the plasma reactor in response to the first variable to arrive at a steady state plasma density distribution.

2. The computer method of claim 1 , wherein the first variable is a first power coupled to the plasma and operating at a first frequency, and the second variable is a second power coupled to the plasma and operating at a second frequency.

3. The computer method of claim 1 , wherein defining the model domain includes establishing at least one initial value and at least one boundary condition for the model domain.

4. The computer method of claim 2 , wherein the first frequency is greater than the second frequency.

5. The computer method of claim 4 , wherein the first frequency ranges from 10 MHz to 200 MHz and the second frequency ranges from 1 MHz to 5 MHz.

6. The computer method of claim 1 , wherein defining the model domain includes determining at least one characteristic of an electromagnetic effect of the plasma within the plasma reactor.

7. The computer method of claim 1 , wherein defining the model domain includes at least two dimensions.

8. The computer method of claim 1 , wherein randomly sampling includes calculating a character value of each random sampling.

9. The computer method of claim 8 , wherein transferring the random sampling includes a randomly generated union of the character values.

10. The computer method of claim 1 , wherein performing the time-domain simulation further comprises:

simulating non-elastic collisions and the diffusion of at least one particle within the plasma.

11. The computer method of claim 2 , wherein the first frequency is a radiofrequency.

12. The computer method of claim 2 , wherein:

the first power is a radiofrequency power provided by a plasma-generating antenna;

the second power is a biasing voltage provided by a biasing electrode; and

the time dependent variable is at least one of a particle velocity, a particle distribution, or a particle angular momentum.

13. A method of operating a plasma reactor, the method comprising:

(a) igniting a plasma within the plasma reactor with a dual frequency power supply having a first variable with a first time basis and a second variable at a second time basis, the dual frequency power supply operably controlled by a controller;

(b) processing at least one wafer with the plasma in accordance with a processing recipe;

(c) extinguishing the plasma;

(d) simulating the plasma reactor with the controller over the first time basis by:

(i) defining a model domain representative of the plasma reactor and including a profile of the processed at least one wafer;

(ii) determining a response function of a time-dependent variable across the model domain with respect to the second time basis;

(iii) constructing a probability function for the response function;

(iv) randomly sampling the probability function to define a plurality of character values;

(v) generating a spatial matrix from a randomly-generated union of the plurality of character values; and

(vi) performing, on the spatial matrix, a time-domain simulation of the transient plasma sheath behavior occurring in the plasma reactor in response to the first time basis to arrive at a steady state plasma density distribution; and

(e) adjusting the processing recipe in accordance with the arrived steady state plasma density before further processing.

14. The method of claim 13 , wherein the first variable is a first power coupled to the plasma and operating at a first frequency, and the second variable is a second power coupled to the plasma and operating at a second frequency.

15. The method of claim 13 , wherein defining the model domain includes establishing at least one initial value and at least one boundary condition for the model domain.

16. The method of claim 14 , wherein the first frequency of the dual frequency power supply is greater than the second frequency of the dual frequency power supply.

17. The method of claim 16 , wherein the first frequency of the dual frequency power supply ranges from 10 MHz to 200 MHz and the second frequency of the dual frequency power supply ranges from 1 MHz to 5 MHz.

18. The method of claim 13 , wherein defining the model domain includes determining at least one characteristic of an electromagnetic effect of the plasma within the plasma reactor.

19. The method of claim 13 , wherein defining the model domain includes at least two dimensions.

20. The method of claim 13 , wherein randomly sampling includes calculating a character value of each random sampling.

21. The method of claim 20 , wherein transferring the random sampling includes a randomly generated union of the character values.

22. The method of claim 13 , wherein performing the time-domain simulation further comprises:

simulating non-elastic collisions and the diffusion of at least one particle within the plasma.

23. The method of claim 14 , wherein the first frequency is a radiofrequency.

24. The method of claim 14 , wherein:

the first power is a radiofrequency power provided by a plasma-generating antenna;

the second power is a biasing voltage provided by a biasing electrode; and

the time dependent variable is at least one of a particle velocity, a particle distribution, or a particle angular momentum.